Solutions%20Of%20Engineering%20Electromagnetics%207th%20ed-Hayt%20(2002)

Solutions%20Of%20Engineering%20Electromagnetics%207th%20ed-Hayt%20(2002)

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Unformatted text preview: CHAPTER 1 1.1. Given the vectors M = -10ax + 4ay - 8az and N = 8ax + 7ay - 2az , find: a) a unit vector in the direction of -M + 2N. -M + 2N = 10ax - 4ay + 8az + 16ax + 14ay - 4az = (26, 10, 4) Thus a= b) the magnitude of 5ax + N - 3M: (5, 0, 0) + (8, 7, -2) - (-30, 12, -24) = (43, -5, 22), and |(43, -5, 22)| = 48.6. c) |M||2N|(M + N): |(-10, 4, -8)||(16, 14, -4)|(-2, 11, -10) = (13.4)(21.6)(-2, 11, -10) = (-580.5, 3193, -2902) 1.2. Given three points, A(4, 3, 2), B(-2, 0, 5), and C(7, -2, 1): a) Specify the vector A extending from the origin to the point A. A = (4, 3, 2) = 4ax + 3ay + 2az b) Give a unit vector extending from the origin to the midpoint of line AB. The vector from the origin to the midpoint is given by M = (1/2)(A + B) = (1/2)(4 - 2, 3 + 0, 2 + 5) = (1, 1.5, 3.5) The unit vector will be m= (1, 1.5, 3.5) = (0.25, 0.38, 0.89) |(1, 1.5, 3.5)| (26, 10, 4) = (0.92, 0.36, 0.14) |(26, 10, 4)| c) Calculate the length of the perimeter of triangle ABC: Begin with AB = (-6, -3, 3), BC = (9, -2, -4), CA = (3, -5, -1). Then |AB| + |BC| + |CA| = 7.35 + 10.05 + 5.91 = 23.32 1.3. The vector from the origin to the point A is given as (6, -2, -4), and the unit vector directed from the origin toward point B is (2, -2, 1)/3. If points A and B are ten units apart, find the coordinates of point B. With A = (6, -2, -4) and B = 1 B(2, -2, 1), we use the fact that |B - A| = 10, or 3 |(6 - 2 B)ax - (2 - 2 B)ay - (4 + 1 B)az | = 10 3 3 3 Expanding, obtain 36 - 8B + 4 B 2 + 4 - 8 B + 4 B 2 + 16 + 8 B + 1 B 2 = 100 9 3 9 3 9 or B 2 - 8B - 44 = 0. Thus B = B= 8 64-176 2 = 11.75 (taking positive option) and so 2 1 2 (11.75)ax - (11.75)ay + (11.75)az = 7.83ax - 7.83ay + 3.92az 3 3 3 1 1.4. given points A(8, -5, 4) and B(-2, 3, 2), find: a) the distance from A to B. |B - A| = |(-10, 8, -2)| = 12.96 b) a unit vector directed from A towards B. This is found through aAB = B-A = (-0.77, 0.62, -0.15) |B - A| c) a unit vector directed from the origin to the midpoint of the line AB. a0M = (3, -1, 3) (A + B)/2 = = (0.69, -0.23, 0.69) |(A + B)/2| 19 d) the coordinates of the point on the line connecting A to B at which the line intersects the plane z = 3. Note that the midpoint, (3, -1, 3), as determined from part c happens to have z coordinate of 3. This is the point we are looking for. 1.5. A vector field is specified as G = 24xyax + 12(x 2 + 2)ay + 18z2 az . Given two points, P (1, 2, -1) and Q(-2, 1, 3), find: a) G at P : G(1, 2, -1) = (48, 36, 18) b) a unit vector in the direction of G at Q: G(-2, 1, 3) = (-48, 72, 162), so aG = (-48, 72, 162) = (-0.26, 0.39, 0.88) |(-48, 72, 162)| c) a unit vector directed from Q toward P : aQP = (3, -1, 4) P-Q = = (0.59, 0.20, -0.78) |P - Q| 26 d) the equation of the surface on which |G| = 60: We write 60 = |(24xy, 12(x 2 + 2), 18z2 )|, or 10 = |(4xy, 2x 2 + 4, 3z2 )|, so the equation is 100 = 16x 2 y 2 + 4x 4 + 16x 2 + 16 + 9z4 2 1.6. For the G field in Problem 1.5, make sketches of Gx , Gy , Gz and |G| along the line y = 1, z = 1, for 0 x 2. We find G(x, 1, 1) = (24x, 12x 2 + 24, 18), from which Gx = 24x, Gy = 12x 2 + 24, Gz = 18, and |G| = 6 4x 4 + 32x 2 + 25. Plots are shown below. 1.7. Given the vector field E = 4zy 2 cos 2xax + 2zy sin 2xay + y 2 sin 2xaz for the region |x|, |y|, and |z| less than 2, find: a) the surfaces on which Ey = 0. With Ey = 2zy sin 2x = 0, the surfaces are 1) the plane z = 0, with |x| < 2, |y| < 2; 2) the plane y = 0, with |x| < 2, |z| < 2; 3) the plane x = 0, with |y| < 2, |z| < 2; 4) the plane x = /2, with |y| < 2, |z| < 2. b) the region in which Ey = Ez : This occurs when 2zy sin 2x = y 2 sin 2x, or on the plane 2z = y, with |x| < 2, |y| < 2, |z| < 1. c) the region in which E = 0: We would have Ex = Ey = Ez = 0, or zy 2 cos 2x = zy sin 2x = y 2 sin 2x = 0. This condition is met on the plane y = 0, with |x| < 2, |z| < 2. 1.8. Two vector fields are F = -10ax + 20x(y - 1)ay and G = 2x 2 yax - 4ay + zaz . For the point P (2, 3, -4), find: a) |F|: F at (2, 3, -4) = (-10, 80, 0), so |F| = 80.6. b) |G|: G at (2, 3, -4) = (24, -4, -4), so |G| = 24.7. c) a unit vector in the direction of F - G: F - G = (-10, 80, 0) - (24, -4, -4) = (-34, 84, 4). So a= F-G (-34, 84, 4) = = (-0.37, 0.92, 0.04) |F - G| 90.7 d) a unit vector in the direction of F + G: F + G = (-10, 80, 0) + (24, -4, -4) = (14, 76, -4). So a= F+G (14, 76, -4) = = (0.18, 0.98, -0.05) |F + G| 77.4 3 1.9. A field is given as G= (x 2 25 (xax + yay ) + y2) Find: a) a unit vector in the direction of G at P (3, 4, -2): Have Gp = 25/(9 + 16) (3, 4, 0) = 3ax + 4ay , and |Gp | = 5. Thus aG = (0.6, 0.8, 0). b) the angle between G and ax at P : The angle is found through aG ax = cos . So cos = (0.6, 0.8, 0) (1, 0, 0) = 0.6. Thus = 53 . c) the value of the following double integral on the plane y = 7: 4 0 4 0 0 2 0 2 G ay dzdx 4 0 0 2 25 (xax + yay ) ay dzdx = x2 + y2 = 350 25 7 dzdx = x 2 + 49 - 0 = 26 4 0 350 dx x 2 + 49 1 tan-1 7 4 7 1.10. Use the definition of the dot product to find the interior angles at A and B of the triangle defined by the three points A(1, 3, -2), B(-2, 4, 5), and C(0, -2, 1): a) Use RAB = (-3, 1, 7) and RAC = (-1, -5, 3) to form RAB RAC = |RAB ||RAC | cos A . Obtain 3 + 5 + 21 = 59 35 cos A . Solve to find A = 65.3 . b) Use RBA = (3, -1, -7) and RBC = (2, -6, -4) to form RBA RBC = |RBA ||RBC | cos B . Obtain 6 + 6 + 28 = 59 56 cos B . Solve to find B = 45.9 . 1.11. Given the points M(0.1, -0.2, -0.1), N(-0.2, 0.1, 0.3), and P (0.4, 0, 0.1), find: a) the vector RMN : RMN = (-0.2, 0.1, 0.3) - (0.1, -0.2, -0.1) = (-0.3, 0.3, 0.4). b) the dot product RMN RMP : RMP = (0.4, 0, 0.1) - (0.1, -0.2, -0.1) = (0.3, 0.2, 0.2). RMN RMP = (-0.3, 0.3, 0.4) (0.3, 0.2, 0.2) = -0.09 + 0.06 + 0.08 = 0.05. c) the scalar projection of RMN on RMP : RMN aRMP = (-0.3, 0.3, 0.4) d) the angle between RMN and RMP : M = cos-1 RMN RMP |RMN ||RMP | = cos-1 0.05 0.34 0.17 = 78 0.05 = = 0.12 0.09 + 0.04 + 0.04 0.17 (0.3, 0.2, 0.2) 4 1.12. Given points A(10, 12, -6), B(16, 8, -2), C(8, 1, -4), and D(-2, -5, 8), determine: a) the vector projection of RAB + RBC on RAD : RAB + RBC = RAC = (8, 1, 4) - (10, 12, -6) = (-2, -11, 10) Then RAD = (-2, -5, 8) - (10, 12, -6) = (-12, -17, 14). So the projection will be: (RAC aRAD )aRAD = (-2, -11, 10) (-12, -17, 14) (-12, -17, 14) = (-6.7, -9.5, 7.8) 629 629 b) the vector projection of RAB + RBC on RDC : RDC = (8, -1, 4) - (-2, -5, 8) = (10, 6, -4). The projection is: (RAC aRDC )aRDC = (-2, -11, 10) (10, 6, -4) (10, 6, -4) = (-8.3, -5.0, 3.3) 152 152 c) the angle between RDA and RDC : Use RDA = -RAD = (12, 17, -14) and RDC = (10, 6, -4). The angle is found through the dot product of the associated unit vectors, or: D = cos-1 (aRDA aRDC ) = cos-1 (12, 17, -14) (10, 6, -4) 629 152 = 26 1.13. a) Find the vector component of F = (10, -6, 5) that is parallel to G = (0.1, 0.2, 0.3): F||G = FG (10, -6, 5) (0.1, 0.2, 0.3) (0.1, 0.2, 0.3) = (0.93, 1.86, 2.79) G= 2 |G| 0.01 + 0.04 + 0.09 b) Find the vector component of F that is perpendicular to G: FpG = F - F||G = (10, -6, 5) - (0.93, 1.86, 2.79) = (9.07, -7.86, 2.21) c) Find the vector component of G that is perpendicular to F: GpF = G - G||F = G - GF 1.3 F = (0.1, 0.2, 0.3) - (10, -6, 5) = (0.02, 0.25, 0.26) 2 |F| 100 + 36 + 25 1.14. The four vertices of a regular tetrahedron are located at O(0, 0, 0), A(0, 1, 0), B(0.5 3, 0.5, 0), and C( 3/6, 0.5, 2/3). a) Find a unit vector perpendicular (outward) to the face ABC: First find RBA RBC = [(0, 1, 0) - (0.5 3, 0.5, 0)] [( 3/6, 0.5, 2/3) - (0.5 3, 0.5, 0)] = (-0.5 3, 0.5, 0) (- 3/3, 0, 2/3) = (0.41, 0.71, 0.29) The required unit vector will then be: RBA RBC = (0.47, 0.82, 0.33) |RBA RBC | b) Find the area of the face ABC: Area = 1 |RBA RBC | = 0.43 2 5 1.15. Three vectors extending from the origin are given as r1 = (7, 3, -2), r2 = (-2, 7, -3), and r3 = (0, 2, 3). Find: a) a unit vector perpendicular to both r1 and r2 : ap12 = (5, 25, 55) r1 r2 = = (0.08, 0.41, 0.91) |r1 r2 | 60.6 b) a unit vector perpendicular to the vectors r1 - r2 and r2 - r3 : r1 - r2 = (9, -4, 1) and r2 - r3 = (-2, 5, -6). So r1 - r2 r2 - r3 = (19, 52, 32). Then ap = (19, 52, 32) (19, 52, 32) = = (0.30, 0.81, 0.50) |(19, 52, 32)| 63.95 c) the area of the triangle defined by r1 and r2 : Area = 1 |r1 r2 | = 30.3 2 d) the area of the triangle defined by the heads of r1 , r2 , and r3 : Area = 1 1 |(r2 - r1 ) (r2 - r3 )| = |(-9, 4, -1) (-2, 5, -6)| = 32.0 2 2 1.16. Describe the surfaces defined by the equations: a) r ax = 2, where r = (x, y, z): This will be the plane x = 2. b) |r ax | = 2: r ax = (0, z, -y), and |r ax | = centered on the x axis, and of radius 2. z2 + y 2 = 2. This is the equation of a cylinder, 1.17. Point A(-4, 2, 5) and the two vectors, RAM = (20, 18, -10) and RAN = (-10, 8, 15), define a triangle. a) Find a unit vector perpendicular to the triangle: Use ap = (350, -200, 340) RAM RAN = = (0.664, -0.379, 0.645) |RAM RAN | 527.35 The vector in the opposite direction to this one is also a valid answer. b) Find a unit vector in the plane of the triangle and perpendicular to RAN : aAN = Then apAN = ap aAN = (0.664, -0.379, 0.645) (-0.507, 0.406, 0.761) = (-0.550, -0.832, 0.077) The vector in the opposite direction to this one is also a valid answer. c) Find a unit vector in the plane of the triangle that bisects the interior angle at A: A non-unit vector in the required direction is (1/2)(aAM + aAN ), where aAM = (20, 18, -10) = (0.697, 0.627, -0.348) |(20, 18, -10)| 6 (-10, 8, 15) = (-0.507, 0.406, 0.761) 389 1.17c. (continued) Now 1 1 (aAM + aAN ) = [(0.697, 0.627, -0.348) + (-0.507, 0.406, 0.761)] = (0.095, 0.516, 0.207) 2 2 Finally, abis = (0.095, 0.516, 0.207) = (0.168, 0.915, 0.367) |(0.095, 0.516, 0.207)| 1.18. Given points A( = 5, = 70 , z = -3) and B( = 2, = -30 , z = 1), find: a) unit vector in cartesian coordinates at A toward B: A(5 cos 70 , 5 sin 70 , -3) = A(1.71, 4.70, -3), In the same manner, B(1.73, -1, 1). So RAB = (1.73, -1, 1) - (1.71, 4.70, -3) = (0.02, -5.70, 4) and therefore (0.02, -5.70, 4) aAB = = (0.003, -0.82, 0.57) |(0.02, -5.70, 4)| b) a vector in cylindrical coordinates at A directed toward B: aAB a = 0.003 cos 70 - 0.82 sin 70 = -0.77. aAB a = -0.003 sin 70 - 0.82 cos 70 = -0.28. Thus aAB = -0.77a - 0.28a + 0.57az . c) a unit vector in cylindrical coordinates at B directed toward A: Use aBA = (-0, 003, 0.82, -0.57). Then aBA a = -0.003 cos(-30 ) + 0.82 sin(-30 ) = -0.43, and aBA a = 0.003 sin(-30 ) + 0.82 cos(-30 ) = 0.71. Finally, aBA = -0.43a + 0.71a - 0.57az 1.19 a) Express the field D = (x 2 + y 2 )-1 (xax + yay ) in cylindrical components and cylindrical variables: Have x = cos , y = sin , and x 2 + y 2 = 2 . Therefore D= Then D = D a = and D = D a = Therefore D= 1 1 cos (ax a ) + sin (ay a ) = [cos (- sin ) + sin cos ] = 0 1 a 1 (cos ax + sin ay ) 1 1 1 cos (ax a ) + sin (ay a ) = cos2 + sin2 = 7 1.19b. Evaluate D at the point where = 2, = 0.2, and z = 5, expressing the result in cylindrical and cartesian coordinates: At the given point, and in cylindrical coordinates, D = 0.5a . To express this in cartesian, we use D = 0.5(a ax )ax + 0.5(a ay )ay = 0.5 cos 36 ax + 0.5 sin 36 ay = 0.41ax + 0.29ay 1.20. Express in cartesian components: a) the vector at A( = 4, = 40 , z = -2) that extends to B( = 5, = -110 , z = 2): We have A(4 cos 40 , 4 sin 40 , -2) = A(3.06, 2.57, -2), and B(5 cos(-110 ), 5 sin(-110 ), 2) = B(-1.71, -4.70, 2) in cartesian. Thus RAB = (-4.77, -7.30, 4). b) a unit vector at B directed toward A: Have RBA = (4.77, 7.30, -4), and so aBA = (4.77, 7.30, -4) = (0.50, 0.76, -0.42) |(4.77, 7.30, -4)| Have rB = (-1.71, -4.70, 2), and so -rB = c) a unit vector at B directed toward the origin: (1.71, 4.70, -2). Thus a= (1.71, 4.70, -2) = (0.32, 0.87, -0.37) |(1.71, 4.70, -2)| 1.21. Express in cylindrical components: a) the vector from C(3, 2, -7) to D(-1, -4, 2): C(3, 2, -7) C( = 3.61, = 33.7 , z = -7) and D(-1, -4, 2) D( = 4.12, = -104.0 , z = 2). Now RCD = (-4, -6, 9) and R = RCD a = -4 cos(33.7) - 6 sin(33.7) = -6.66. Then R = RCD a = 4 sin(33.7) - 6 cos(33.7) = -2.77. So RCD = -6.66a - 2.77a + 9az b) a unit vector at D directed toward C: RCD = (4, 6, -9) and R = RDC a = 4 cos(-104.0) + 6 sin(-104.0) = -6.79. Then R = RDC a = 4[- sin(-104.0)] + 6 cos(-104.0) = 2.43. So RDC = -6.79a + 2.43a - 9az Thus aDC = -0.59a + 0.21a - 0.78az c) a unit vector at D directed toward the origin: Start with rD = (-1, -4, 2), and so the vector toward the origin will be -rD = (1, 4, -2). Thus in cartesian the unit vector is a = (0.22, 0.87, -0.44). Convert to cylindrical: a = (0.22, 0.87, -0.44) a = 0.22 cos(-104.0) + 0.87 sin(-104.0) = -0.90, and a = (0.22, 0.87, -0.44) a = 0.22[- sin(-104.0)] + 0.87 cos(-104.0) = 0, so that finally, a = -0.90a - 0.44az . 1.22. A field is given in cylindrical coordinates as F= 2 40 + 3(cos + sin ) a + 3(cos - sin )a - 2az +1 where the magnitude of F is found to be: |F| = FF= 1600 240 (cos + sin ) + 22 + 2 2 + 1)2 ( +1 8 1/2 Sketch |F|: a) vs. with = 3: in this case the above simplifies to |F( = 3)| = |F a| = [38 + 24(cos + sin )]1/2 b) vs. with = 0, in which: 1600 240 |F( = 0)| = |F b| = + 22 + 2 2 + 1)2 ( +1 c) vs. with = 45 , in which |F( = 45 )| = |F c| = 1/2 1600 240 2 + 22 + 2 ( 2 + 1)2 +1 1/2 9 1.23. The surfaces = 3, = 5, = 100 , = 130 , z = 3, and z = 4.5 define a closed surface. a) Find the enclosed volume: Vol = 3 4.5 130 100 3 5 d d dz = 6.28 NOTE: The limits on the integration must be converted to radians (as was done here, but not shown). b) Find the total area of the enclosing surface: Area = 2 + 3 4.5 130 100 130 100 3 5 130 100 5 3 d d + 3 4.5 3 4.5 3 d dz 5 d dz + 2 d dz = 20.7 c) Find the total length of the twelve edges of the surfaces: Length = 4 1.5 + 4 2 + 2 30 30 2 3 + 2 5 = 22.4 360 360 d) Find the length of the longest straight line that lies entirely within the volume: This will be between the points A( = 3, = 100 , z = 3) and B( = 5, = 130 , z = 4.5). Performing point transformations to cartesian coordinates, these become A(x = -0.52, y = 2.95, z = 3) and B(x = -3.21, y = 3.83, z = 4.5). Taking A and B as vectors directed from the origin, the requested length is Length = |B - A| = |(-2.69, 0.88, 1.5)| = 3.21 1.24. At point P (-3, 4, 5), express the vector that extends from P to Q(2, 0, -1) in: a) rectangular coordinates. Then |RP Q | = RP Q = Q - P = 5ax - 4ay - 6az 25 + 16 + 36 = 8.8 b) cylindrical coordinates. At P , = 5, = tan-1 (4/ - 3) = -53.1 , and z = 5. Now, RP Q a = (5ax - 4ay - 6az ) a = 5 cos - 4 sin = 6.20 RP Q a = (5ax - 4ay - 6az ) a = -5 sin - 4 cos = 1.60 Thus 6.202 + 1.602 + 62 = 8.8 c) spherical coordinates. At P , r = 9 + 16 + 25 = 50 = 7.07, = cos-1 (5/7.07) = 45 , and = tan-1 (4/ - 3) = -53.1 . RP Q ar = (5ax - 4ay - 6az ) ar = 5 sin cos - 4 sin sin - 6 cos = 0.14 RP Q a = (5ax - 4ay - 6az ) a = 5 cos cos - 4 cos sin - (-6) sin = 8.62 RP Q a = (5ax - 4ay - 6az ) a = -5 sin - 4 cos = 1.60 10 and |RP Q | = RP Q = 6.20a + 1.60a - 6az 1.24. (continued) Thus and |RP Q | = RP Q = 0.14ar + 8.62a + 1.60a 0.142 + 8.622 + 1.602 = 8.8 d) Show that each of these vectors has the same magnitude. Each does, as shown above. 1.25. Given point P (r = 0.8, = 30 , = 45 ), and E= 1 r2 cos ar + sin a sin a) Find E at P : E = 1.10a + 2.21a . b) Find |E| at P : |E| = 1.102 + 2.212 = 2.47. c) Find a unit vector in the direction of E at P : aE = E = 0.45ar + 0.89a |E| 1.26. a) Determine an expression for ay in spherical coordinates at P (r = 4, = 0.2, = 0.8 ): Use ay ar = sin sin = 0.35, ay a = cos sin = 0.48, and ay a = cos = -0.81 to obtain ay = 0.35ar + 0.48a - 0.81a b) Express ar in cartesian components at P : Find x = r sin cos = -1.90, y = r sin sin = 1.38, and z = r cos = -3.24. Then use ar ax = sin cos = -0.48, ar ay = sin sin = 0.35, and ar az = cos = 0.81 to obtain ar = -0.48ax + 0.35ay + 0.81az 1.27. The surfaces r = 2 and 4, = 30 and 50 , and = 20 and 60 identify a closed surface. a) Find the enclosed volume: This will be Vol = 60 20 50 30 2 4 r 2 sin drdd = 2.91 where degrees have been converted to radians. b) Find the total area of the enclosing surface: Area = 60 20 50 30 (4 + 2 ) sin dd + 2 2 2 4 60 20 50 r(sin 30 + sin 50 )drd 4 +2 c) Find the total length of the twelve edges of the surface: Length = 4 2 4 30 rdrd = 12.61 2 dr + 2 50 30 (4 + 2)d + 60 20 (4 sin 50 + 4 sin 30 + 2 sin 50 + 2 sin 30 )d = 17.49 11 1.27. (continued) d) Find the length of the longest straight line that lies entirely within the surface: This will be from A(r = 2, = 50 , = 20 ) to B(r = 4, = 30 , = 60 ) or A(x = 2 sin 50 cos 20 , y = 2 sin 50 sin 20 , z = 2 cos 50 ) to B(x = 4 sin 30 cos 60 , y = 4 sin 30 sin 60 , z = 4 cos 30 ) or finally A(1.44, 0.52, 1.29) to B(1.00, 1.73, 3.46). Thus B - A = (-0.44, 1.21, 2.18) and Length = |B - A| = 2.53 1.28. a) Determine the cartesian components of the vector from A(r = 5, = 110 , = 200 ) to B(r = 7, = 30 , = 70 ): First transform the points to cartesian: xA = 5 sin 110 cos 200 = -4.42, yA = 5 sin 110 sin 200 = -1.61, and zA = 5 cos 110 = -1.71; xB = 7 sin 30 cos 70 = 1.20, yB = 7 sin 30 sin 70 = 3.29, and zB = 7 cos 30 = 6.06. Now RAB = B - A = 5.62ax + 4.90ay + 7.77az b) Find the spherical components of the vector at P (2, -3, 4) extending to Q(-3, 2, 5): First, RP Q = Q - P = (-5, 5, 1). Then at P , r = 4 + 9 + 16 = 5.39, = cos-1 (4/ 29) = 42.0 , and = tan-1 (-3/2) = -56.3 . Now RP Q ar = -5 sin(42 ) cos(-56.3 ) + 5 sin(42 ) sin(-56.3 ) + 1 cos(42 ) = -3.90 RP Q a = -5 cos(42 ) cos(-56.3 ) + 5 cos(42 ) sin(-56.3 ) - 1 sin(42 ) = -5.82 RP Q a = -(-5) sin(-56.3 ) + 5 cos(-56.3 ) = -1.39 So finally, RP Q = -3.90ar - 5.82a - 1.39a c) If D = 5ar - 3a + 4a , find D a at M(1, 2, 3): First convert a to cartesian coordinates at the specified point. Use a = (a ax )ax + (a ay )ay . At A(1, 2, 3), = 5, = tan-1 (2) = 63.4 , r = 14, and = cos-1 (3/ 14) = 36.7 . So a = cos(63.4 )ax + sin(63.4 )ay = 0.45ax + 0.89ay . Then (5ar - 3a + 4a ) (0.45ax + 0.89ay ) = 5(0.45) sin cos + 5(0.89) sin sin - 3(0.45) cos cos - 3(0.89) cos sin + 4(0.45)(- sin ) + 4(0.89) cos = 0.59 1.29. Express the unit vector ax in spherical components at the point: a) r = 2, = 1 rad, = 0.8 rad: Use ax = (ax ar )ar + (ax a )a + (ax a )a = sin(1) cos(0.8)ar + cos(1) cos(0.8)a + (- sin(0.8))a = 0.59ar + 0.38a - 0.72a 12 1.29 (continued) Express the unit vector ax in spherical components at the point: b) x = 3, y = 2, z = -1: First, transform the point to spherical coordinates. Have r = = cos-1 (-1/ 14) = 105.5 , and = tan-1 (2/3) = 33.7 . Then ax = sin(105.5 ) cos(33.7 )ar + cos(105.5 ) cos(33.7 )a + (- sin(33.7 ))a = 0.80ar - 0.22a - 0.55a c) = 2.5, = 0.7 rad, z = 1.5: Again, convert the point to spherical coordinates. r = 8.5, = cos-1 (z/r) = cos-1 (1.5/ 8.5) = 59.0 , and = 0.7 rad = 40.1 . Now ax = sin(59 ) cos(40.1 )ar + cos(59 ) cos(40.1 )a + (- sin(40.1 ))a = 0.66ar + 0.39a - 0.64a 1.30. Given A(r = 20, = 30 , = 45 ) and B(r = 30, = 115 , = 160 ), find: 14, 2 + z2 = a) |RAB |: First convert A and B to cartesian: Have xA = 20 sin(30 ) cos(45 ) = 7.07, yA = 20 sin(30 ) sin(45 ) = 7.07, and zA = 20 cos(30 ) = 17.3. xB = 30 sin(115 ) cos(160 ) = -25.6, yB = 30 sin(115 ) sin(160 ) = 9.3, and zB = 30 cos(115 ) = -12.7. Now RAB = RB - RA = (-32.6, 2.2, -30.0), and so |RAB | = 44.4. b) |RAC |, given C(r = 20, = 90 , = 45 ). Again, converting C to cartesian, obtain xC = 20 sin(90 ) cos(45 ) = 14.14, yC = 20 sin(90 ) sin(45 ) = 14.14, and zC = 20 cos(90 ) = 0. So RAC = RC - RA = (7.07, 7.07, -17.3), and |RAC | = 20.0. c) the distance from A to C on a great circle path: Note that A and C share the same r and coordinates; thus moving from A to C involves only a change in of 60 . The requested arc length is then distance = 20 60 2 360 = 20.9 13 CHAPTER 2 2.1. Four 10nC positive charges are located in the z = 0 plane at the corners of a square 8cm on a side. A fifth 10nC positive charge is located at a point 8cm distant from the other charges. Calculate the magnitude of the total force on this fifth charge for = 0 : Arrange the charges in the xy plane at locations (4,4), (4,-4), (-4,4), and (-4,-4). Then the fifth charge will be on the z axis at location z = 4 2, which puts it at 8cm distance from the other four. By symmetry, the force on the fifth charge will be z-directed, and will be four times the z component of force produced by each of the four other charges. 4 4 q2 (10-8 )2 F = = = 4.0 10-4 N 2 4 0 d 2 2 4(8.85 10-12 )(0.08)2 2.2. A charge Q1 = 0.1 C is located at the origin, while Q2 = 0.2 C is at A(0.8, -0.6, 0). Find the locus of points in the z = 0 plane at which the x component of the force on a third positive charge is zero. To solve this problem, the z coordinate of the third charge is immaterial, so we can place it in the xy plane at coordinates (x, y, 0). We take its magnitude to be Q3 . The vector directed from the first charge to the third is R13 = xax + yay ; the vector directed from the second charge to the third is R23 = (x - 0.8)ax + (y + 0.6)ay . The force on the third charge is now F3 = Q1 R13 Q2 R23 + 3 |R13 | |R23 |3 Q3 10-6 0.1(xax + yay ) 0.2[(x - 0.8)ax + (y + 0.6)ay ] = + 4 0 (x 2 + y 2 )1.5 [(x - 0.8)2 + (y + 0.6)2 ]1.5 Q3 4 0 We desire the x component to be zero. Thus, 0= or x[(x - 0.8)2 + (y + 0.6)2 ]1.5 = 2(0.8 - x)(x 2 + y 2 )1.5 2.3. Point charges of 50nC each are located at A(1, 0, 0), B(-1, 0, 0), C(0, 1, 0), and D(0, -1, 0) in free space. Find the total force on the charge at A. The force will be: F= (50 10-9 )2 4 0 RCA RDA RBA + + 3 3 |RCA | |RDA | |RBA |3 (x 2 0.1xax 0.2(x - 0.8)ax + 2 )1.5 +y [(x - 0.8)2 + (y + 0.6)2 ]1.5 where RCA = ax - ay , RDA = ax + ay , and RBA = 2ax . The magnitudes are |RCA | = |RDA | = and |RBA | = 2. Substituting these leads to F= (50 10-9 )2 4 0 1 1 2 ax = 21.5ax N + + 2 2 2 2 8 2, where distances are in meters. 14 2.4. Let Q1 = 8 C be located at P1 (2, 5, 8) while Q2 = -5 C is at P2 (6, 15, 8). Let a) Find F2 , the force on Q2 : This force will be F2 = = 0. Q1 Q2 R12 (8 10-6 )(-5 10-6 ) (4ax + 10ay ) = = (-1.15ax - 2.88ay ) mN 4 0 |R12 |3 4 0 (116)1.5 b) Find the coordinates of P3 if a charge Q3 experiences a total force F3 = 0 at P3 : This force in general will be: Q3 Q1 R13 Q2 R23 + F3 = 3 4 0 |R13 | |R23 |3 where R13 = (x - 2)ax + (y - 5)ay and R23 = (x - 6)ax + (y - 15)ay . Note, however, that all three charges must lie in a straight line, and the location of Q3 will be along the vector R12 extended past Q2 . The slope of this vector is (15 - 5)/(6 - 2) = 2.5. Therefore, we look for P3 at coordinates (x, 2.5x, 8). With this restriction, the force becomes: 8[(x - 2)ax + 2.5(x - 2)ay ] 5[(x - 6)ax + 2.5(x - 6)ay ] Q3 - F3 = 4 0 [(x - 2)2 + (2.5)2 (x - 2)2 ]1.5 [(x - 6)2 + (2.5)2 (x - 6)2 ]1.5 where we require the term in large brackets to be zero. This leads to 8(x - 2)[((2.5)2 + 1)(x - 6)2 ]1.5 - 5(x - 6)[((2.5)2 + 1)(x - 2)2 ]1.5 = 0 which reduces to 8(x - 6)2 - 5(x - 2)2 = 0 6 8-2 5 x= = 21.1 8- 5 The coordinates of P3 are thus P3 (21.1, 52.8, 8) or 2.5. Let a point charge Q1 25 nC be located at P1 (4, -2, 7) and a charge Q2 = 60 nC be at P2 (-3, 4, -2). a) If = 0, find E at P3 (1, 2, 3): This field will be E= 10-9 4 0 25R13 60R23 + 3 |R13 | |R23 |3 where R13 = -3ax +4ay -4az and R23 = 4ax -2ay +5az . Also, |R13 | = 41 and |R23 | = 45. So 10-9 25 (-3ax + 4ay - 4az ) 60 (4ax - 2ay + 5az ) E= + 4 0 (41)1.5 (45)1.5 = 4.58ax - 0.15ay + 5.51az b) At what point on the y axis is Ex = 0? P3 is now at (0, y, 0), so R13 = -4ax + (y + 2)ay - 7az and R23 = 3ax + (y - 4)ay + 2az . Also, |R13 | = 65 + (y + 2)2 and |R23 | = 13 + (y - 4)2 . Now the x component of E at the new P3 will be: 25 (-4) 10-9 60 3 + 2 ]1.5 4 0 [65 + (y + 2) [13 + (y - 4)2 ]1.5 To obtain Ex = 0, we require the expression in the large brackets to be zero. This expression simplifies to the following quadratic: Ex = 0.48y 2 + 13.92y + 73.10 = 0 which yields the two values: y = -6.89, -22.11 15 2.6. Point charges of 120 nC are located at A(0, 0, 1) and B(0, 0, -1) in free space. a) Find E at P (0.5, 0, 0): This will be EP = 120 10-9 4 0 RAP RBP + |RAP |3 |RBP |3 where RAP = 0.5ax - az and RBP = 0.5ax + az . Also, |RAP | = |RBP | = EP = 120 10-9 ax = 772 V/m 4(1.25)1.5 0 1.25. Thus: b) What single charge at the origin would provide the identical field strength? We require Q0 = 772 4 0 (0.5)2 from which we find Q0 = 21.5 nC. 2.7. A 2 C point charge is located at A(4, 3, 5) in free space. Find E , E , and Ez at P (8, 12, 2). Have 2 10-6 RAP 2 10-6 4ax + 9ay - 3az = 65.9ax + 148.3ay - 49.4az = 4 0 |RAP |3 4 0 (106)1.5 Then, at point P , = 82 + 122 = 14.4, = tan-1 (12/8) = 56.3 , and z = z. Now, EP = E = Ep a = 65.9(ax a ) + 148.3(ay a ) = 65.9 cos(56.3 ) + 148.3 sin(56.3 ) = 159.7 and E = Ep a = 65.9(ax a ) + 148.3(ay a ) = -65.9 sin(56.3 ) + 148.3 cos(56.3 ) = 27.4 Finally, Ez = -49.4 2.8. Given point charges of -1 C at P1 (0, 0, 0.5) and P2 (0, 0, -0.5), and a charge of 2 C at the origin, find E at P (0, 2, 1) in spherical components, assuming = 0 . The field will take the general form: EP = R1 10-6 2R2 R3 - + - 3 3 4 0 |R1 | |R2 | |R3 |3 where R1 , R2 , R3 are the vectors to P from each of the charges in their original listed order. Specifically, R1 = (0, 2, 0.5), R2 = (0, 2, 1), and R3 = (0, 2, 1.5). The magnitudes are |R1 | = 2.06, |R2 | = 2.24, and |R3 | = 2.50. Thus -(0, 2, 0.5) 2(0, 2, 1) (0, 2, 1.5) = 89.9ay + 179.8az + - (2.06)3 (2.24)3 (2.50)3 Now, at P , r = 5, = cos-1 (1/ 5) = 63.4 , and = 90 . So EP = Er = EP ar = 89.9(ay ar ) + 179.8(az ar ) = 89.9 sin sin + 179.8 cos = 160.9 E = EP a = 89.9(ay a ) + 179.8(az a ) = 89.9 cos sin + 179.8(- sin ) = -120.5 E = EP a = 89.9(ay a ) + 179.8(az a ) = 89.9 cos = 0 16 10-6 4 0 2.9. A 100 nC point charge is located at A(-1, 1, 3) in free space. a) Find the locus of all points P (x, y, z) at which Ex = 500 V/m: The total field at P will be: EP = 100 10-9 RAP 4 0 |RAP |3 where RAP = (x + 1)ax + (y - 1)ay + (z - 3)az , and where |RAP | = [(x + 1)2 + (y - 1)2 + (z - 3)2 ]1/2 . The x component of the field will be Ex = 100 10-9 4 0 + 1)2 (x + 1) = 500 V/m + (y - 1)2 + (z - 3)2 ]1.5 [(x And so our condition becomes: (x + 1) = 0.56 [(x + 1)2 + (y - 1)2 + (z - 3)2 ]1.5 b) Find y1 if P (-2, y1 , 3) lies on that locus: At point P , the condition of part a becomes 3.19 = 1 + (y1 - 1)2 from which (y1 - 1)2 = 0.47, or y1 = 1.69 or 0.31 2.10. Charges of 20 and -20 nC are located at (3, 0, 0) and (-3, 0, 0), respectively. Let Determine |E| at P (0, y, 0): The field will be EP = 20 10-9 4 0 R1 R2 - 3 |R1 | |R2 |3 = 0. 3 where R1 , the vector from the positive charge to point P is (-3, y, 0), and R2 , the vector from the negative charge to point P , is (3, y, 0). The magnitudes of these vectors are |R1 | = |R2 | = 9 + y 2 . Substituting these into the expression for EP produces EP = from which |EP | = 20 10-9 4 0 -6ax (9 + y 2 )1.5 1079 V/m (9 + y 2 )1.5 2.11. A charge Q0 located at the origin in free space produces a field for which Ez = 1 kV/m at point P (-2, 1, -1). a) Find Q0 : The field at P will be EP = Q0 4 0 -2ax + ay - az 61.5 Since the z component is of value 1 kV/m, we find Q0 = -4 0 61.5 103 = -1.63 C. 17 2.11. (continued) b) Find E at M(1, 6, 5) in cartesian coordinates: This field will be: EM = -1.63 10-6 4 0 ax + 6ay + 5az [1 + 36 + 25]1.5 1 + 36 = 6.08, = tan-1 (6/1) = or EM = -30.11ax - 180.63ay - 150.53az . c) Find E at M(1, 6, 5) in cylindrical coordinates: At M, = 80.54 , and z = 5. Now E = EM a = -30.11 cos - 180.63 sin = -183.12 E = EM a = -30.11(- sin ) - 180.63 cos = 0 (as expected) so that EM = -183.12a - 150.53az . d) Find E at M(1, 6, 5) in spherical coordinates: At M, r = 1 + 36 + 25 = 7.87, = 80.54 (as before), and = cos-1 (5/7.87) = 50.58 . Now, since the charge is at the origin, we expect to obtain only a radial component of EM . This will be: Er = EM ar = -30.11 sin cos - 180.63 sin sin - 150.53 cos = -237.1 2.12. The volume charge density v = 0 e-|x|-|y|-|z| exists over all free space. Calculate the total charge present: This will be 8 times the integral of v over the first octant, or Q=8 0 0 0 0 e-x-y-z dx dy dz = 80 2.13. A uniform volume charge density of 0.2 C/m3 (note typo in book) is present throughout the spherical shell extending from r = 3 cm to r = 5 cm. If v = 0 elsewhere: a) find the total charge present throughout the shell: This will be Q= 0 2 0 .05 .03 r3 0.2 r sin dr d d = 4(0.2) 3 2 .05 .03 = 8.21 10-5 C = 82.1 pC b) find r1 if half the total charge is located in the region 3 cm < r < r1 : If the integral over r in part a is taken to r1 , we would obtain 4(0.2) Thus r1 = 3 4.105 10-5 + (.03)3 0.2 4 r3 3 r1 .03 = 4.105 10-5 1/3 = 4.24 cm 18 2.14. Let 1 C/m3 + 10 in the region 0 10, - < < , all z, and v = 0 elsewhere. v = 5e-0.1 ( - ||) z2 a) Determine the total charge present: This will be the integral of v over the region where it exists; specifically, 10 1 5e-0.1 ( - ||) 2 d d dz Q= z + 10 - - 0 which becomes e-0.1 (-0.1 - 1) Q=5 (0.1)2 or Q = 5 26.4 Finally, 1 Q = 5 26.4 2 tan-1 10 z 10 - 10 0 - - 2 0 ( - ) 1 dz + 10 z2 1 d dz + 10 2 z2 = 5(26.4) 3 = 1.29 103 C = 1.29 mC 10 b) Calculate the charge within the region 0 4, -/2 < < /2, -10 < z < 10: With the limits thus changed, the integral for the charge becomes: Q = 10 -10 /2 4 0 2 0 5e-0.1 ( - ) 1 d d dz z2 + 10 Following the same evaulation procedure as in part a, we obtain Q = 0.182 mC. 2.15. A spherical volume having a 2 m radius contains a uniform volume charge density of 1015 C/m3 . a) What total charge is enclosed in the spherical volume? This will be Q = (4/3)(2 10-6 )3 1015 = 3.35 10-2 C. b) Now assume that a large region contains one of these little spheres at every corner of a cubical grid 3mm on a side, and that there is no charge between spheres. What is the average volume charge density throughout this large region? Each cube will contain the equivalent of one little sphere. Neglecting the little sphere volume, the average density becomes v,avg = 3.35 10-2 = 1.24 106 C/m3 (0.003)3 2.16. The region in which 4 < r < 5, 0 < < 25 , and 0.9 < < 1.1 contains the volume charge density of v = 10(r - 4)(r - 5) sin sin(/2). Outside the region, v = 0. Find the charge within the region: The integral that gives the charge will be Q = 10 1.1 .9 0 25 4 5 (r - 4)(r - 5) sin sin(/2) r 2 sin dr d d 19 2.16. (continued) Carrying out the integral, we obtain r5 r4 r3 Q = 10 - 9 + 20 5 4 3 5 4 1 1 - sin(2) 2 4 25 -2 cos 0 2 1.1 .9 = 10(-3.39)(.0266)(.626) = 0.57 C 2.17. A uniform line charge of 16 nC/m is located along the line defined by y = -2, z = 5. If a) Find E at P (1, 2, 3): This will be EP = RP l 2 0 |RP |2 = 0: where RP = (1, 2, 3) - (1, -2, 5) = (0, 4, -2), and |RP |2 = 20. So EP = 16 10-9 2 0 4ay - 2az = 57.5ay - 28.8az V/m 20 b) Find E at that point in the z = 0 plane where the direction of E is given by (1/3)ay - (2/3)az : With z = 0, the general field will be Ez=0 = l 2 0 (y + 2)ay - 5az (y + 2)2 + 25 We require |Ez | = -|2Ey |, so 2(y + 2) = 5. Thus y = 1/2, and the field becomes: Ez=0 = l 2 0 2.5ay - 5az = 23ay - 46az (2.5)2 + 25 2.18. Uniform line charges of 0.4 C/m and -0.4 C/m are located in the x = 0 plane at y = -0.6 and y = 0.6 m respectively. Let = 0 . a) Find E at P (x, 0, z): In general, we have EP = l 2 0 R-P R+P - |R+P | |R-P | where R+P and R-P are, respectively, the vectors directed from the positive and negative line charges to the point P , and these are normal to the z axis. We thus have R+P = (x, 0, z) - (0, -.6, z) = (x, .6, 0), and R-P = (x, 0, z) - (0, .6, z) = (x, -.6, 0). So EP = l 2 0 xax + 0.6ay xax - 0.6ay - 2 2 + (0.6)2 x x + (0.6)2 = 0.4 10-6 2 0 x2 8.63ay 1.2ay = 2 kV/m + 0.36 x + 0.36 20 2.18. (continued) b) Find E at Q(2, 3, 4): This field will in general be: EQ = l 2 0 R-Q R+Q - |R+Q | |R-Q | where R+Q = (2, 3, 4) - (0, -.6, 4) = (2, 3.6, 0), and R-Q = (2, 3, 4) - (0, .6, 4) = (2, 2.4, 0). Thus 2ax + 3.6ay 2ax + 2.4ay l EQ = = -625.8ax - 241.6ay V/m - 2 2 + (3.6)2 2 0 2 2 + (2.4)2 2.19. A uniform line charge of 2 C/m is located on the z axis. Find E in cartesian coordinates at P (1, 2, 3) if the charge extends from a) - < z < : With the infinite line, we know that the field will have only a radial component in cylindrical coordinates (or x and y components in cartesian). The field from an infinite line on the z axis is generally E = [l /(2 0 )]a . Therefore, at point P : EP = l RzP (2 10-6 ) ax + 2ay = 7.2ax + 14.4ay kV/m = 2 0 |RzP |2 2 0 5 where RzP is the vector that extends from the line charge to point P , and is perpendicular to the z axis; i.e., RzP = (1, 2, 3) - (0, 0, 3) = (1, 2, 0). b) -4 z 4: Here we use the general relation EP = l dz r - r 4 0 |r - r |3 4 -4 where r = ax + 2ay + 3az and r = zaz . So the integral becomes EP = (2 10-6 ) 4 0 ax + 2ay + (3 - z)az dz [5 + (3 - z)2 ]1.5 4 -4 Using integral tables, we obtain: EP = 3597 (ax + 2ay )(z - 3) + 5az (z2 - 6z + 14) V/m = 4.9ax + 9.8ay + 4.9az kV/m The student is invited to verify that when evaluating the above expression over the limits - < z < , the z component vanishes and the x and y components become those found in part a. 2.20. Uniform line charges of 120 nC/m lie along the entire extent of the three coordinate axes. Assuming free space conditions, find E at P (-3, 2, -1): Since all line charges are infinitely-long, we can write: EP = l 2 0 RyP RxP RzP + + 2 2 |RyP | |RxP | |RzP |2 where RxP , RyP , and RzP are the normal vectors from each of the three axes that terminate on point P . Specifically, RxP = (-3, 2, -1) - (-3, 0, 0) = (0, 2, -1), RyP = (-3, 2, -1) - (0, 2, 0) = (-3, 0, -1), and RzP = (-3, 2, -1) - (0, 0, -1) = (-3, 2, 0). Substituting these into the expression for EP gives EP = l 2 0 -3ax + 2ay 2ay - az -3ax - az + + 5 10 13 21 = -1.15ax + 1.20ay - 0.65az kV/m 2.21. Two identical uniform line charges with l = 75 nC/m are located in free space at x = 0, y = 0.4 m. What force per unit length does each line charge exert on the other? The charges are parallel to the z axis and are separated by 0.8 m. Thus the field from the charge at y = -0.4 evaluated at the location of the charge at y = +0.4 will be E = [l /(2 0 (0.8))]ay . The force on a differential length of the line at the positive y location is dF = dqE = l dzE. Thus the force per unit length acting on the line at postive y arising from the charge at negative y is F= 0 1 l2 dz ay = 1.26 10-4 ay N/m = 126 ay N/m 2 0 (0.8) The force on the line at negative y is of course the same, but with -ay . 2.22. A uniform surface charge density of 5 nC/m2 is present in the region x = 0, -2 < y < 2, and all z. If = 0 , find E at: a) PA (3, 0, 0): We use the superposition integral: E= s da r - r 4 0 |r - r |3 where r = 3ax and r = yay + zaz . The integral becomes: EP A = s 4 0 2 - -2 3ax - yay - zaz dy dz [9 + y 2 + z2 ]1.5 Since the integration limits are symmetric about the origin, and since the y and z components of the integrand exhibit odd parity (change sign when crossing the origin, but otherwise symmetric), these will integrate to zero, leaving only the x component. This is evident just from the symmetry of the problem. Performing the z integration first on the x component, we obtain (using tables): Ex,P A 3s = 4 0 = 3s 2 0 2 -2 dy (9 + y 2 ) z 9 + y2 2 -2 + z2 = - 3s 2 0 2 -2 dy (9 + y 2 ) 1 y tan-1 3 3 = 106 V/m The student is encouraged to verify that if the y limits were - to , the result would be that of the infinite charged plane, or Ex = s /(2 0 ). b) PB (0, 3, 0): In this case, r = 3ay , and symmetry indicates that only a y component will exist. The integral becomes Ey,P B = 2 (3 - y) dy dz s s = 4 0 - -2 [(z2 + 9) - 6y + y 2 ]1.5 2 0 s =- ln(3 - y) 2 = 145 V/m -2 2 0 2 -2 (3 - y) dy (3 - y)2 22 2.23. Given the surface charge density, s = 2 C/m2 , in the region < 0.2 m, z = 0, and is zero elsewhere, find E at: a) PA ( = 0, z = 0.5): First, we recognize from symmetry that only a z component of E will be present. Considering a general point z on the z axis, we have r = zaz . Then, with r = a , we obtain r - r = zaz - a . The superposition integral for the z component of E will be: Ez,PA s = 4 0 = 2 0 0 0.2 z d d 2s =- z 2 + z2 )1.5 4 0 ( 1 z2 + 2 0.2 0 s 1 1 z - 2 + 0.4 2 2 0 z z With z = 0.5 m, the above evaluates as Ez,PA = 8.1 kV/m. b) With z at -0.5 m, we evaluate the expression for Ez to obtain Ez,PB = -8.1 kV/m. 2.24. Surface charge density is positioned in free space as follows: 20 nC/m2 at x = -3, -30 nC/m2 at y = 4, and 40 nC/m2 at z = 2. Find the magnitude of E at the three points, (4, 3, -2), (-2, 5, -1), and (0, 0, 0). Since all three sheets are infinite, the field magnitude associated with each one will be s /(2 0 ), which is position-independent. For this reason, the net field magnitude will be the same everywhere, whereas the field direction will depend on which side of a given sheet one is positioned. We take the first point, for example, and find EA = 20 10-9 30 10-9 40 10-9 ax + ay - az = 1130ax + 1695ay - 2260az V/m 2 0 2 0 2 0 The magnitude of EA is thus 3.04 kV/m. This will be the magnitude at the other two points as well. 2.25. Find E at the origin if the following charge distributions are present in free space: point charge, 12 nC at P (2, 0, 6); uniform line charge density, 3nC/m at x = -2, y = 3; uniform surface charge density, 0.2 nC/m2 at x = 2. The sum of the fields at the origin from each charge in order is: E= (12 10-9 ) (-2ax - 6az ) (0.2 10-9 )ax (3 10-9 ) (2ax - 3ay ) - + 4 0 2 0 (4 + 9) 2 0 (4 + 36)1.5 = -3.9ax - 12.4ay - 2.5az V/m 2.26. A uniform line charge density of 5 nC/m is at y = 0, z = 2 m in free space, while -5 nC/m is located at y = 0, z = -2 m. A uniform surface charge density of 0.3 nC/m2 is at y = 0.2 m, and -0.3 nC/m2 is at y = -0.2 m. Find |E| at the origin: Since each pair consists of equal and opposite charges, the effect at the origin is to double the field produce by one of each type. Taking the sum of the fields at the origin from the surface and line charges, respectively, we find: E(0, 0, 0) = -2 so that |E| = 96.1 V/m. 0.3 10-9 5 10-9 az = -33.9ay - 89.9az ay - 2 2 0 2 0 (2) 23 2.27. Given the electric field E = (4x - 2y)ax - (2x + 4y)ay , find: a) the equation of the streamline that passes through the point P (2, 3, -4): We write Ey dy -(2x + 4y) = = dx Ex (4x - 2y) Thus 2(x dy + y dx) = y dy - x dx or 2 d(xy) = So C1 + 2xy = or y 2 - x 2 = 4xy + C2 Evaluating at P (2, 3, -4), obtain: 9 - 4 = 24 + C2 , or C2 = -19 Finally, at P , the requested equation is y 2 - x 2 = 4xy - 19 b) a unit vector specifying the direction E at Q(3, -2, 5): Have EQ = [4(3) + 2(2)]ax - [2(3) - of 4(2)]ay = 16ax + 2ay . Then |E| = 162 + 4 = 16.12 So aQ = 16ax + 2ay = 0.99ax + 0.12ay 16.12 1 1 d(y 2 ) - d(x 2 ) 2 2 1 2 1 2 y - x 2 2 2.28. Let E = 5x 3 ax - 15x 2 y ay , and find: a) the equation of the streamline that passes through P (4, 2, 1): Write Ey -15x 2 y -3y dy = = = 3 dx Ex 5x x So dy dx = -3 ln y = -3 ln x + ln C y x y = e-3 ln x eln C = At P , have 2 = C/(4)3 C = 128. Finally, at P , y= 128 x3 C x3 Thus 24 2.28. (continued) b) a unit vector aE specifying the direction of E at Q(3, -2, 5): At Q, EQ = 135ax + 270ay , and |EQ | = 301.9. Thus aE = 0.45ax + 0.89ay . c) a unit vector aN = (l, m, 0) that is perpendicular to aE at Q: Since this vector is to have no z component, we can find it through aN = (aE az ). Performing this, we find aN = (0.89ax - 0.45ay ). 2.29. If E = 20e-5y cos 5xax - sin 5xay , find: a) |E| at P (/6, 0.1, 2): Substituting this point, we obtain EP = -10.6ax - 6.1ay , and so |EP | = 12.2. b) a unit vector in the direction of EP : The unit vector associated with E is just cos 5xax - sin 5xay , which evaluated at P becomes aE = -0.87ax - 0.50ay . c) the equation of the direction line passing through P : Use dy - sin 5x = = - tan 5x dy = - tan 5x dx dx cos 5x Thus y = 1 5 ln cos 5x + C. Evaluating at P , we find C = 0.13, and so y= 1 ln cos 5x + 0.13 5 2.30. Given the electric field intensity E = 400yax + 400xay V/m, find: a) the equation of the streamline passing through the point A(2, 1, -2): Write: Ey dy x = = x dx = y dy dx Ex y Thus x 2 = y 2 + C. Evaluating at A yields C = 3, so the equation becomes x2 y2 - =1 3 3 b) the equation of the surface on which |E| = 800 V/m: Have |E| = 400 x 2 + y 2 = 800. Thus x 2 + y 2 = 4, or we have a circular-cylindrical surface, centered on the z axis, and of radius 2. c) A sketch of the part a equation would yield a parabola, centered at the origin, whose axis is the positive x axis, and for which the slopes of the asymptotes are 1. d) A sketch of the trace produced by the intersection of the surface of part b with the z = 0 plane would yield a circle centered at the origin, of radius 2. 25 2.31. In cylindrical coordinates with E(, ) = E (, )a + E (, )a , the differential equation describing the direction lines is E /E = d/(d) in any constant-z plane. Derive the equation of the line passing through the point P ( = 4, = 10 , z = 2) in the field E = 2 2 cos 3a + 2 2 sin 3a : Using the given information, we write E d = cot 3 = E d Thus 1 d = cot 3 d ln = ln sin 3 + ln C 3 or = C(sin 3)1/3 . Evaluate this at P to obtain C = 7.14. Finally, 3 = 364 sin 3 26 CHAPTER 3 3.1. An empty metal paint can is placed on a marble table, the lid is removed, and both parts are discharged (honorably) by touching them to ground. An insulating nylon thread is glued to the center of the lid, and a penny, a nickel, and a dime are glued to the thread so that they are not touching each other. The penny is given a charge of +5 nC, and the nickel and dime are discharged. The assembly is lowered into the can so that the coins hang clear of all walls, and the lid is secured. The outside of the can is again touched momentarily to ground. The device is carefully disassembled with insulating gloves and tools. a) What charges are found on each of the five metallic pieces? All coins were insulated during the entire procedure, so they will retain their original charges: Penny: +5 nC; nickel: 0; dime: 0. The penny's charge will have induced an equal and opposite negative charge (-5 nC) on the inside wall of the can and lid. This left a charge layer of +5 nC on the outside surface which was neutralized by the ground connection. Therefore, the can retained a net charge of -5 nC after disassembly. b) If the penny had been given a charge of +5 nC, the dime a charge of -2 nC, and the nickel a charge of -1 nC, what would the final charge arrangement have been? Again, since the coins are insulated, they retain their original charges. The charge induced on the inside wall of the can and lid is equal to negative the sum of the coin charges, or -2 nC. This is the charge that the can/lid contraption retains after grounding and disassembly. 3.2. A point charge of 12 nC is located at the origin. four uniform line charges are located in the x = 0 plane as follows: 80 nC/m at y = -1 and -5 m, -50 nC/m at y = -2 and -4 m. a) Find D at P (0, -3, 2): Note that this point lies in the center of a symmetric arrangement of line charges, whose fields will all cancel at that point. Thus D arise from the point charge alone, and will be D= 12 10-9 (-3ay + 2az ) = -6.11 10-11 ay + 4.07 10-11 az C/m2 4(32 + 22 )1.5 = -61.1ay + 40.7az pC/m2 b) How much electric flux crosses the plane y = -3 and in what direction? The plane intercepts all flux that enters the -y half-space, or exactly half the total flux of 12 nC. The answer is thus 6 nC and in the -ay direction. c) How much electric flux leaves the surface of a sphere, 4m in radius, centered at C(0, -3, 0)? This sphere encloses the point charge, so its flux of 12 nC is included. The line charge contributions are most easily found by translating the whole assembly (sphere and line charges) such that the sphere is centered at the origin, with line charges now at y = 1 and 2. The flux from the line charges will equal the total line charge that lies within the sphere. The length of each of the inner two line charges (at y = 1) will be h1 = 2r cos 1 = 2(4) cos sin-1 1 4 = 1.94 m That of each of the outer two line charges (at y = 2) will be h2 = 2r cos 2 = 2(4) cos sin-1 2 4 = 1.73 m 27 3.2c. (continued) The total charge enclosed in the sphere (and the outward flux from it) is now Ql + Qp = 2(1.94)(-50 10-9 ) + 2(1.73)(80 10-9 ) + 12 10-9 = 348 nC 3.3. The cylindrical surface = 8 cm contains the surface charge density, s = 5e-20|z| nC/m2 . a) What is the total amount of charge present? We integrate over the surface to find: Q=2 0 0 2 5e -20z -1 -20z e (.08)d dz nC = 20(.08) 20 = 0.25 nC 0 b) How much flux leaves the surface = 8 cm, 1 cm < z < 5cm, 30 < < 90 ? We just integrate the charge density on that surface to find the flux that leaves it. =Q = .05 .01 90 30 -20z 5e (.08) d dz nC = -1 -20z 90 - 30 2(5)(.08) e 360 20 .05 = 9.45 10-3 nC = 9.45 pC .01 3.4. The cylindrical surfaces = 1, 2, and 3 cm carry uniform surface charge densities of 20, -8, and 5 nC/m2 , respectively. a) How much electric flux passes through the closed surface = 5 cm, 0 < z < 1 m? Since the densities are uniform, the flux will be = 2(as1 + bs2 + cs3 )(1 m) = 2 [(.01)(20) - (.02)(8) + (.03)(5)] 10-9 = 1.2 nC b) Find D at P (1 cm, 2 cm, 3 cm): This point lies at radius 5 cm, and is thus inside the outermost charge layer. This layer, being of uniform density, will not contribute to D at P . We know that in cylindrical coordinates, the layers at 1 and 2 cm will produce the flux density: D = D a = or D = as1 + bs2 a (.01)(20) + (.02)(-8) = 1.8 nC/m2 .05 At P , = tan-1 (2/1) = 63.4 . Thus Dx = 1.8 cos = 0.8 and Dy = 1.8 sin = 1.6. Finally, DP = (0.8ax + 1.6ay ) nC/m2 28 3.5. Let D = 4xyax + 2(x 2 + z2 )ay + 4yzaz C/m2 and evaluate surface integrals to find the total charge enclosed in the rectangular parallelepiped 0 < x < 2, 0 < y < 3, 0 < z < 5 m: Of the 6 surfaces to consider, only 2 will contribute to the net outward flux. Why? First consider the planes at y = 0 and 3. The y component of D will penetrate those surfaces, but will be inward at y = 0 and outward at y = 3, while having the same magnitude in both cases. These fluxes will thus cancel. At the x = 0 plane, Dx = 0 and at the z = 0 plane, Dz = 0, so there will be no flux contributions from these surfaces. This leaves the 2 remaining surfaces at x = 2 and z = 5. The net outward flux becomes: = 0 5 0 3 0 3 D ax dy dz + x=2 3 0 3 0 0 2 D z=5 az dx dy =5 4(2)y dy + 2 4(5)y dy = 360 C 3.6. Two uniform line charges, each 20 nC/m, are located at y = 1, z = 1 m. Find the total flux leaving a sphere of radius 2 m if it is centered at a) A(3, 1, 0): The result will be the same if we move the sphere to the origin and the line charges to (0, 0, 1). The length of the line charge within the sphere is given by l = 4 sin[cos-1 (1/2)] = 3.46. With two line charges, symmetrically arranged, the total charge enclosed is given by Q = 2(3.46)(20 nC/m) = 139 nC b) B(3, 2, 0): In this case the result will be the same if we move the sphere to the origin and keep the charges where they were. The length of the line joining the origin to the midpoint of the line charge (in the yz plane) is l1 = 2. The length of the line joining the origin to either endpoint of the line charge is then just the sphere radius, or 2. The half-angle subtended at the origin by the line charge is then = cos-1 ( 2/2) = 45 . The length of each line charge in the sphere = is then l2 2 2 sin = 2 2. The total charge enclosed (with two line charges) is now Q = 2(2 2)(20 nC/m) = 113 nC 3.7. Volume charge density is located in free space as v = 2e-1000r nC/m3 for 0 < r < 1 mm, and v = 0 elsewhere. a) Find the total charge enclosed by the spherical surface r = 1 mm: To find the charge we integrate: Q= 0 2 0 0 .001 2e-1000r r 2 sin dr d d Integration over the angles gives a factor of 4. The radial integration we evaluate using tables; we obtain Q = 8 -r 2 e-1000r 1000 .001 0 + 2 e-1000r (-1000r - 1) 1000 (1000)2 .001 0 = 4.0 10-9 nC b) By using Gauss's law, calculate the value of Dr on the surface r = 1 mm: The gaussian surface is a spherical shell of radius 1 mm. The enclosed charge is the result of part a. We thus write 4 r 2 Dr = Q, or Q 4.0 10-9 Dr = = = 3.2 10-4 nC/m2 4r 2 4(.001)2 29 3.8. Uniform line charges of 5 nC/m ar located in free space at x = 1, z = 1, and at y = 1, z = 0. a) Obtain an expression for D in cartesian coordinates at P (0, 0, z). In general, we have D(z) = s 2 r1 - r1 r2 - r 2 + 2 |r1 - r1 | |r2 - r2 |2 where r1 = r2 = zaz , r1 = ay , and r2 = ax + az . Thus D(z) = s 2 s = 2 [zaz - ay ] [(z - 1)az - ax ] + [1 + z2 ] [1 + (z - 1)2 ] ay -ax (z - 1) z - + + az [1 + (z - 1)2 ] [1 + z2 ] [1 + (z - 1)2 ] [1 + z2 ] b) Plot |D| vs. z at P , -3 < z < 10: Using part a, we find the magnitude of D to be s |D| = 2 1 1 + + 2 ]2 [1 + (z - 1) [1 + z2 ]2 z (z - 1) + 2] [1 + (z - 1) [1 + z2 ] 2 1/2 A plot of this over the specified range is shown in Prob3.8.pdf. 3.9. A uniform volume charge density of 80 C/m3 is present throughout the region 8 mm < r < 10 mm. Let v = 0 for 0 < r < 8 mm. a) Find the total charge inside the spherical surface r = 10 mm: This will be Q= 0 2 0 .010 .008 -10 (80 10-6 )r 2 sin dr d d = 4 (80 10-6 ) r3 3 .010 .008 = 1.64 10 C = 164 pC b) Find Dr at r = 10 mm: Using a spherical gaussian surface at r = 10, Gauss' law is written as 4 r 2 Dr = Q = 164 10-12 , or Dr (10 mm) = 164 10-12 = 1.30 10-7 C/m2 = 130 nC/m2 4(.01)2 c) If there is no charge for r > 10 mm, find Dr at r = 20 mm: This will be the same computation as in part b, except the gaussian surface now lies at 20 mm. Thus Dr (20 mm) = 164 10-12 = 3.25 10-8 C/m2 = 32.5 nC/m2 4(.02)2 3.10. Let s = 8 C/m2 in the region where x = 0 and -4 < z < 4 m, and let s = 0 elsewhere. Find D at P (x, 0, z), where x > 0: The sheet charge can be thought of as an assembly of infinitely-long parallel strips that lie parallel to the y axis in the yz plane, and where each is of thickness dz. The field from each strip is that of an infinite line charge, and so we can construct the field at P from a single strip as: dDP = s dz r - r 2 |r - r |2 30 3.10 (continued) where r = xax + zaz and r = z az We distinguish between the fixed coordinate of P , z, and the variable coordinate, z , that determines the location of each charge strip. To find the net field at P , we sum the contributions of each strip by integrating over z : DP = 4 -4 8 10-6 dz (xax + (z - z )az ) 2[x 2 + (z - z )2 ] We can re-arrange this to determine the integral forms: DP = 8 10-6 (xax + zaz ) 2 4 -4 dz - az 2 + z2 ) - 2zz + (z )2 (x 4 -4 z dz (x 2 + z2 ) - 2zz + (z )2 Using integral tables, we find DP = - which evaluates as DP = 4 10-6 tan-1 z+4 x - tan-1 z-4 x ax + x 2 + (z + 4)2 1 ln 2 az 2 x + (z - 4)2 C/m2 4 10-6 1 (xax + zaz ) tan-1 x 2z - 2z 2x 2z - 2z 2x 4 2z 1 1 ln(x 2 + z2 - 2zz + (z )2 ) + tan-1 2 2 x az -4 The student is invited to verify that for very small x or for a very large sheet (allowing z to approach infinity), the above expression reduces to the expected form, DP = s /2. Note also that the expression is valid for all x (positive or negative values). 3.11. In cylindrical coordinates, let v = 0 for < 1 mm, v = 2 sin(2000) nC/m3 for 1 mm < < 1.5 mm, and v = 0 for > 1.5 mm. Find D everywhere: Since the charge varies only with radius, and is in the form of a cylinder, symmetry tells us that the flux density will be radially-directed and will be constant over a cylindrical surface of a fixed radius. Gauss' law applied to such a surface of unit length in z gives: a) for < 1 mm, D = 0, since no charge is enclosed by a cylindrical surface whose radius lies within this range. b) for 1 mm < < 1.5 mm, we have 2D = 2 .001 2 10-9 sin(2000 ) d 1 sin(2000) - cos(2000) 2 (2000) 2000 .001 = 4 10-9 or finally, D = 10-15 sin(2000) + 2 1 - 103 cos(2000) 2 2 C/m2 (1 mm < < 1.5 mm) 31 3.11. (continued) c) for > 1.5 mm, the gaussian cylinder now lies at radius outside the charge distribution, so the integral that evaluates the enclosed charge now includes the entire charge distribution. To accomplish this, we change the upper limit of the integral of part b from to 1.5 mm, finally obtaining: 2.5 10-15 C/m2 ( > 1.5 mm) D = 3.12. A nonuniform volume charge density, v = 120r C/m3 , lies within the spherical surface r = 1 m, and v = 0 everywhere else. a) Find Dr everywhere. For r < 1 m, we apply Gauss' law to a spherical surface of radius r within this range to find 4r 2 Dr = 4 r 120r (r )2 dr = 120r 4 0 Thus Dr = (30r 2 ) for r < 1 m. For r > 1 m, the gaussian surface lies outside the charge distribution. The set up is the same, except the upper limit of the above integral is 1 instead of r. This results in Dr = (30/r 2 ) for r > 1 m. b) What surface charge density, s2 , should be on the surface r = 2 such that Dr,r=2- = 2Dr,r=2+ ? At r = 2- , we have Dr,r=2- = 30/22 = 15/2, from part a. The flux density in the region r > 2 arising from a surface charge at r = 2 is found from Gauss' law through 4r 2 Drs = 4(2)2 s2 Drs = 4s2 r2 The total flux density in the region r > 2 arising from the two distributions is DrT = 30 4s2 + 2 r2 r Our requirement that Dr,r=2- = 2Dr,r=2+ becomes 30 30 =2 + s2 2 2 22 s2 = - 15 C/m2 4 c) Make a sketch of Dr vs. r for 0 < r < 5 m with both distributions present. With both charges, Dr (r < 1) = 30r 2 , Dr (1 < r < 2) = 30/r 2 , and Dr (r > 2) = 15/r 2 . These are plotted on the next page. 32 . 3.13. Spherical surfaces at r = 2, 4, and 6 m carry uniform surface charge densities of 20 nC/m2 , -4 nC/m2 , and s0 , respectively. a) Find D at r = 1, 3 and 5 m: Noting that the charges are spherically-symmetric, we ascertain that D will be radially-directed and will vary only with radius. Thus, we apply Gauss' law to spherical shells in the following regions: r < 2: Here, no charge is enclosed, and so Dr = 0. 2 < r < 4 : 4r 2 Dr = 4(2)2 (20 10-9 ) Dr = So Dr (r = 3) = 8.9 10-9 C/m2 . 4 < r < 6 : 4r 2 Dr = 4(2)2 (20 10-9 ) + 4(4)2 (-4 10-9 ) Dr = So Dr (r = 5) = 6.4 10-10 C/m2 . b) Determine s0 such that D = 0 at r = 7 m. Since fields will decrease as 1/r 2 , the question could be re-phrased to ask for s0 such that D = 0 at all points where r > 6 m. In this region, the total field will be 16 10-9 s0 (6)2 Dr (r > 6) = + r2 r2 Requiring this to be zero, we find s0 = -(4/9) 10-9 C/m2 . 3.14. If v = 5 nC/m3 for 0 < < 1 mm and no other charges are present: a) find D for < 1 mm: Applying Gauss' law to a cylindrical surface of unit length in z, and of radius < 1 mm, we find 2D = 2 (5 10-9 ) D = 2.5 10-9 C/m2 16 10-9 r2 80 10-9 C/m2 r2 33 3.14b. find D for > 1 mm: The Gaussian cylinder now lies outside the charge, so 2D = (.001)2 (5 10-9 ) D = 2.5 10-15 C/m2 c) What line charge L at = 0 would give the same result for part b? The line charge field will be Dr = L 2.5 10-15 = (part b) 2 Thus L = 5 10-15 C/m. In all answers, is expressed in meters. 3.15. Volume charge density is located as follows: v = 0 for < 1 mm and for > 2 mm, v = 4 C/m3 for 1 < < 2 mm. a) Calculate the total charge in the region 0 < < 1 , 0 < z < L, where 1 < 1 < 2 mm: We find Q= 0 L 0 2 1 .001 4 d d dz = 8L 3 [1 - 10-9 ] C 3 where 1 is in meters. b) Use Gauss' law to determine D at = 1 : Gauss' law states that 21 LD = Q, where Q is the result of part a. Thus 3 4(1 - 10-9 ) D (1 ) = C/m2 31 where 1 is in meters. c) Evaluate D at = 0.8 mm, 1.6 mm, and 2.4 mm: At = 0.8 mm, no charge is enclosed by a cylindrical gaussian surface of that radius, so D (0.8mm) = 0. At = 1.6 mm, we evaluate the part b result at 1 = 1.6 to obtain: 4[(.0016)3 - (.0010)3 ] = 3.6 10-6 C/m2 3(.0016) At = 2.4, we evaluate the charge integral of part a from .001 to .002, and Gauss' law is written as 8L 2LD = [(.002)2 - (.001)2 ] C 3 from which D (2.4mm) = 3.9 10-6 C/m2 . D (1.6mm) = 3.16. Given the electric flux density, D = 2xy ax + x 2 ay + 6z3 az C/m2 : a) use Gauss' law to evaluate the total charge enclosed in the volume 0 < x, y, z < a: We call the surfaces at x = a and x = 0 the front and back surfaces respectively, those at y = a and y = 0 the right and left surfaces, and those at z = a and z = 0 the top and bottom surfaces. To evaluate the total charge, we integrate D n over all six surfaces and sum the results: =Q= D n da = 0 a 0 0 left a a 0 front a 0 0 right a a 2ay dy dz + 0 a 0 a -2(0)y dy dz a back a 0 bottom a 0 0 top a + -x 2 dx dz + x 2 dx dz + 0 -6(0)3 dx dy + 6a 3 dx dy 34 3.16a. (continued) Noting that the back and bottom integrals are zero, and that the left and right integrals cancel, we evaluate the remaining two (front and top) to obtain Q = 6a 5 + a 4 . b) use Eq. (8) to find an approximate value for the above charge. Evaluate the derivatives at . P (a/2, a/2, a/2): In this application, Eq. (8) states that Q = ( D P ) v. We find D = . 2x + 18z2 , which when evaluated at P becomes D P = a + 4.5a 2 . Thus Q = (a + 4.5a 2 )a 3 = 4.5a 5 + a 4 c) Show that the results of parts a and b agree in the limit as a 0. In this limit, both expressions reduce to Q = a 4 , and so they agree. 3.17. A cube is defined by 1 < x, y, z < 1.2. If D = 2x 2 yax + 3x 2 y 2 ay C/m2 : a) apply Gauss' law to find the total flux leaving the closed surface of the cube. We call the surfaces at x = 1.2 and x = 1 the front and back surfaces respectively, those at y = 1.2 and y = 1 the right and left surfaces, and those at z = 1.2 and z = 1 the top and bottom surfaces. To evaluate the total charge, we integrate D n over all six surfaces and sum the results. We note that there is no z component of D, so there will be no outward flux contributions from the top and bottom surfaces. The fluxes through the remaining four are =Q= D n da = 1 1.2 1 1 left 1.2 1.2 1 front 1.2 1 1 right 1.2 1.2 2(1.2)2 y dy dz + 1 1.2 1 1.2 -2(1)2 y dy dz back + -3x 2 (1)2 dx dz + 3x 2 (1.2)2 dx dz = 0.1028 C b) evaluate D at the center of the cube: This is D = 4xy + 6x 2 y = 4(1.1)2 + 6(1.1)3 = 12.83 (1.1,1.1) c) Estimate the total charge enclosed within the cube by using Eq. (8): This is . Q= D center v = 12.83 (0.2)3 = 0.1026 Close! 3.18. Let a vector field by given by G = 5x 4 y 4 z4 ay . Evaluate both sides of Eq. (8) for this G field and the volume defined by x = 3 and 3.1, y = 1 and 1.1, and z = 2 and 2.1. Evaluate the partial derivatives at the center of the volume. First find G= Gy = 20x 4 y 3 z4 y The center of the cube is located at (3.05,1.05,2.05), and the volume is v = (0.1)3 = 0.001. Eq. (8) then becomes . = 20(3.05)4 (1.05)3 (2.05)4 (0.001) = 35.4 35 3.19. A spherical surface of radius 3 mm is centered at P (4, 1, 5) in free space. Let D = xax C/m2 . Use the . results of Sec. 3.4 to estimate the net electric flux leaving the spherical surface: We use = D v, where in this case D = (/x)x = 1 C/m3 . Thus . 4 = (.003)3 (1) = 1.13 10-7 C = 113 nC 3 3.20. A cube of volume a 3 has its faces parallel to the cartesian coordinate surfaces. It is centered at P (3, -2, 4). Given the field D = 2x 3 ax C/m2 : a) calculate div D at P : In the present case, this will be D= Dx dDx = = 54 C/m3 x dx b) evaluate the fraction in the rightmost side of Eq. (13) for a = 1 m, 0.1 m, and 1 mm: With the field having only an x component, flux will pentrate only the two surfaces at x = 3 a/2, each of which has surface area a 2 . The cube volume is v = a 3 . The equation reads: D dS 1 a = 3 2 3+ v a 2 3 a2 - 2 3 - a 2 3 a2 = 2 a a (3 + )3 - (3 - )3 a 2 2 evaluating the above formula at a = 1 m, .1 m, and 1 mm, yields respectively 54.50, 54.01, and 54.00 C/m3 , thus demonstrating the approach to the exact value as v gets smaller. 3.21. Calculate the divergence of D at the point specified if a) D = (1/z2 ) 10xyz ax + 5x 2 z ay + (2z3 - 5x 2 y) az at P (-2, 3, 5): We find D= 10y 10x 2 y +0+2+ z z3 = 8.96 (-2,3,5) b) D = 5z2 a + 10z az at P (3, -45 , 5): In cylindrical coordinates, we have D= Dz 1 D 5z2 1 + (D ) + = + 10 z = 71.67 (3,-45 ,5) c) D = 2r sin sin ar + r cos sin a + r cos a at P (3, 45 , -45 ): In spherical coordinates, we have 1 1 1 D D = 2 (r 2 Dr ) + (sin D ) + r r r sin r sin sin cos 2 sin - = -2 = 6 sin sin + sin sin (3,45 ,-45 ) 36 3.22. Let D = 8 sin a + 4 cos a C/m2 . a) Find div D: Using the divergence formula for cylindrical coordinates (see problem 3.21), we find D = 12 sin . b) Find the volume charge density at P (2.6, 38 , -6.1): Since v = D, we evaluate the result of part a at this point to find vP = 12 sin 38 = 7.39 C/m3 . c) How much charge is located inside the region defined by 0 < < 1.8, 20 < < 70 , 2.4 < z < 3.1? We use Q= vol v dv = 3.1 2.4 70 20 0 1.8 12 sin d d dz = -(3.1 - 2.4)12 cos 70 20 2 2 1.8 0 = 8.13 C 3.23. a) A point charge Q lies at the origin. Show that div D is zero everywhere except at the origin. For a point charge at the origin we know that D = Q/(4r 2 ) ar . Using the formula for divergence in spherical coordinates (see problem 3.21 solution), we find in this case that D= 1 d r 2 dr r2 Q 4r 2 =0 The above is true provided r > 0. When r = 0, we have a singularity in D, so its divergence is not defined. b) Replace the point charge with a uniform volume charge density v0 for 0 < r < a. Relate v0 to Q and a so that the total charge is the same. Find div D everywhere: To achieve the same net charge, we require that (4/3)a 3 v0 = Q, so v0 = 3Q/(4a 3 ) C/m3 . Gauss' law tells us that inside the charged sphere 4 Qr 3 4r 2 Dr = r 3 v0 = 3 3 a Thus Qr 3 3Q Qr 1 d Dr = = C/m2 and D = 2 3 3 4a r dr 4a 4a 3 as expected. Outside the charged sphere, D = Q/(4r 2 ) ar as before, and the divergence is zero. 3.24. Inside the cylindrical shell, 3 < < 4 m, the electric flux density is given as D = 5( - 3)3 a C/m2 a) What is the volume charge density at = 4 m? In this case we have v = D = 1 d 1 d 5( - 3)2 (D ) = [5( - 3)3 ] = (4 - 3) C/m3 d d Evaluating this at = 4 m, we find v (4) = 16.25 C/m3 b) What is the electric flux density at = 4 m? We evaluate the given D at this point to find D(4) = 5 a C/m2 37 3.24c. How much electric flux leaves the closed surface 3 < < 4, 0 < < 2 , -2.5 < z < 2.5? We note that D has only a radial component, and so flux would leave only through the cylinder sides. Also, D does not vary with or z, so the flux is found by a simple product of the side area and the flux density. We further note that D = 0 at = 3, so only the outer side (at = 4) will contribute. We use the result of part b, and write the flux as = [2.5 - (-2.5)]2(4)(5) = 200 C d) How much charge is contained within the volume used in part c? By Gauss' law, this will be the same as the net outward flux through that volume, or again, 200 C. 3.25. Within the spherical shell, 3 < r < 4 m, the electric flux density is given as D = 5(r - 3)3 ar C/m2 a) What is the volume charge density at r = 4? In this case we have v = D = 1 d 2 5 (r Dr ) = (r - 3)2 (5r - 6) C/m3 r 2 dr r which we evaluate at r = 4 to find v (r = 4) = 17.50 C/m3 . b) What is the electric flux density at r = 4? Substitute r = 4 into the given expression to find D(4) = 5 ar C/m2 c) How much electric flux leaves the sphere r = 4? Using the result of part b, this will be 4(4)2 (5) = 320 C = d) How much charge is contained within the sphere, r = 4? From Gauss' law, this will be the same as the outward flux, or again, Q = 320 C. 3.26. Given the field D= find: a) the volume charge density: Use v = D = 1 d 2 5 sin cos (r Dr ) = C/m3 2 dr r r2 5 sin cos ar C/m2 , r b) the total charge contained within the region r < 2 m: To find this, we integrate over the volume: Q= 0 2 0 0 2 5 sin cos 2 r sin dr d d r2 Before plunging into this one notice that the integration is of cos from zero to 2. This yields a zero result, and so the total enclosed charge is Q = 0. c) the value of D at the surface r = 2: Substituting r = 2 into the given field produces D(r = 2) = 5 sin cos ar C/m2 2 38 3.26d. the total electric flux leaving the surface r = 2 Since the total enclosed charge is zero (from part b), the net outward flux is also zero, from Gauss' law. 3.27. Let D = 5.00r 2 ar mC/m2 for r 0.08 m and D = 0.205 ar /r 2 C/m2 for r 0.08 m (note error in problem statement). a) Find v for r = 0.06 m: This radius lies within the first region, and so v = D = 1 d 2 1 d (r Dr ) = 2 (5.00r 4 ) = 20r mC/m3 2 dr r r dr which when evaluated at r = 0.06 yields v (r = .06) = 1.20 mC/m3 . b) Find v for r = 0.1 m: This is in the region where the second field expression is valid. The 1/r 2 dependence of this field yields a zero divergence (shown in Problem 3.23), and so the volume charge density is zero at 0.1 m. c) What surface charge density could be located at r = 0.08 m to cause D = 0 for r > 0.08 m? The total surface charge should be equal and opposite to the total volume charge. The latter is Q= 0 2 0 0 .08 20r(mC/m3 ) r 2 sin dr d d = 2.57 10-3 mC = 2.57 C 2.57 = -32 C/m2 4(.08)2 So now s = - 3.28. The electric flux density is given as D = 20 3 a C/m2 for < 100 m, and k a / for > 100 m. a) Find k so that D is continuous at = 100 m: We require 20 10-12 = k k = 2 10-15 C/m 10-4 b) Find and sketch v as a function of : In cylindrical coordinates, with only a radial component of D, we use 1 1 (D ) = (20 4 ) = 80 2 C/m3 ( < 100 m) v = D = For > 100 m, we obtain k 1 v = ( ) = 0 The sketch of v vs. would be a parabola, starting at the origin, reaching a maximum value of 8 10-7 C/m3 at = 100 m. The plot is zero at larger radii. 3.29. In the region of free space that includes the volume 2 < x, y, z < 3, D= 2 (yz ax + xz ay - 2xy az ) C/m2 z2 a) Evaluate the volume integral side of the divergence theorem for the volume defined above: In cartesian, we find D = 8xy/z3 . The volume integral side is now D dv = vol 2 3 2 3 2 3 8xy 1 1 - dxdydz = (9 - 4)(9 - 4) 3 z 4 9 39 = 3.47 C 3.29b. Evaluate the surface integral side for the corresponding closed surface: We call the surfaces at x = 3 and x = 2 the front and back surfaces respectively, those at y = 3 and y = 2 the right and left surfaces, and those at z = 3 and z = 2 the top and bottom surfaces. To evaluate the surface integral side, we integrate D n over all six surfaces and sum the results. Note that since the x component of D does not vary with x, the outward fluxes from the front and back surfaces will cancel each other. The same is true for the left and right surfaces, since Dy does not vary with y. This leaves only the top and bottom surfaces, where the fluxes are: D dS = 2 3 2 3 -4xy dxdy - 32 top 3 2 2 3 -4xy 1 1 dxdy = (9 - 4)(9 - 4) - 2 2 4 9 bottom = 3.47 C 3.30. If D = 15 2 sin 2 a + 10 2 cos 2 a C/m2 , evaluate both sides of the divergence theorem for the region 1 < < 2 m, 1 < < 2 rad, 1 < z < 2 m: Taking the surface integral side first, the six sides over which the flux must be evaluated are only four, since there is no z component of D. We are left with the sides at = 1 and = 2 rad (left and right sides, respectively), and those at = 1 and = 2 (back and front sides). We evaluate D dS = 1 2 1 1 left 2 1 front 2 2 1 1 right 2 2 15(2)2 sin(2) (2)ddz - 1 2 1 2 15(1)2 sin(2) (1)ddz back - 10 2 cos(2) ddz + 10 2 cos(4) ddz = 6.93 C For the volume integral side, we first evaluate the divergence of D, which is D= Next D dv = vol 1 2 1 2 1 2 1 1 (15 3 sin 2) + (10 2 cos 2) = 25 sin 2 25 3 3 2 1 25 sin(2) d d dz = - cos(2) 2 2 = 6.93 C 1 3.31. Given the flux density 16 cos(2) a C/m2 , r use two different methods to find the total charge within the region 1 < r < 2 m, 1 < < 2 rad, 1 < < 2 rad: We use the divergence theorem and first evaluate the surface integral side. We are evaluating the net outward flux through a curvilinear "cube", whose boundaries are defined by the specified ranges. The flux contributions will be only through the surfaces of constant , however, since D has only a component. On a constant-theta surface, the differential area is da = r sin drd, where is fixed at the surface location. Our flux integral becomes D= D dS = - 1 2 1 2 16 cos(2) r sin(1) drd + r =1 2 1 1 2 16 cos(4) r sin(2) drd r =2 = -16 [cos(2) sin(1) - cos(4) sin(2)] = -3.91 C 40 3.31. (continued) We next evaluate the volume integral side of the divergence theorem, where in this case, D= 1 1 d d (sin D ) = r sin d r sin d 16 16 cos 2 cos cos 2 sin = 2 - 2 sin 2 r r sin We now evaluate: D dv = vol 1 2 1 2 1 2 16 cos 2 cos - 2 sin 2 r 2 sin drd d r2 sin The integral simplifies to 2 1 1 2 1 2 16[cos 2 cos - 2 sin 2 sin ] drdd = 8 1 2 [3 cos 3 - cos ] d = -3.91 C 3.32. If D = 2r ar C/m2 , find the total electric flux leaving the surface of the cube, 0 < x, y, z < 0.4: This is where the divergence theorem really saves you time! First find D= Then the net outward flux will be D dv = 6(0.4)3 = 0.38 C vol 1 d 2 (r 2r) = 6 r 2 dr 41 CHAPTER 4 4.1. The value of E at P ( = 2, = 40 , z = 3) is given as E = 100a - 200a + 300az V/m. Determine the incremental work required to move a 20 C charge a distance of 6 m: a) in the direction of a : The incremental work is given by dW = -q E dL, where in this case, dL = d a = 6 10-6 a . Thus dW = -(20 10-6 C)(100 V/m)(6 10-6 m) = -12 10-9 J = -12 nJ b) in the direction of a : In this case dL = 2 d a = 6 10-6 a , and so dW = -(20 10-6 )(-200)(6 10-6 ) = 2.4 10-8 J = 24 nJ c) in the direction of az : Here, dL = dz az = 6 10-6 az , and so dW = -(20 10-6 )(300)(6 10-6 ) = -3.6 10-8 J = -36 nJ d) in the direction of E: Here, dL = 6 10-6 aE , where aE = Thus dW = -(20 10-6 )[100a - 200a + 300az ] [0.267 a - 0.535 a + 0.802 az ](6 10-6 ) = -44.9 nJ e) In the direction of G = 2 ax - 3 ay + 4 az : In this case, dL = 6 10-6 aG , where aG = So now dW = -(20 10-6 )[100a - 200a + 300az ] [0.371 ax - 0.557 ay + 0.743 az ](6 10-6 ) = -(20 10-6 ) 37.1(a ax ) - 55.7(a ay ) - 74.2(a ax ) + 111.4(a ay ) + 222.9] (6 10-6 ) where, at P , (a ax ) = (a ay ) = cos(40 ) = 0.766, (a ay ) = sin(40 ) = 0.643, and (a ax ) = - sin(40 ) = -0.643. Substituting these results in dW = -(20 10-6 )[28.4 - 35.8 + 47.7 + 85.3 + 222.9](6 10-6 ) = -41.8 nJ 2ax - 3ay + 4az = 0.371 ax - 0.557 ay + 0.743 az [22 + 32 + 42 ]1/2 100a - 200a + 300az = 0.267 a - 0.535 a + 0.802 az [1002 + 2002 + 3002 ]1/2 42 4.2. Let E = 400ax - 300ay + 500az in the neighborhood of point P (6, 2, -3). Find the incremental work done in moving a 4-C charge a distance of 1 mm in the direction specified by: a) ax + ay + az : We write dW = -qE dL = -4(400ax - 300ay + 500az ) =- (4 10-3 ) (400 - 300 + 500) = -1.39 J 3 (ax + ay + az ) (10-3 ) 3 b) -2ax + 3ay - az : The computation is similar to that of part a, but we change the direction: dW = -qE dL = -4(400ax - 300ay + 500az ) =- (4 10-3 ) (-800 - 900 - 500) = 2.35 J 14 (-2ax + 3ay - az ) (10-3 ) 14 4.3. If E = 120 a V/m, find the incremental amount of work done in moving a 50 m charge a distance of 2 mm from: a) P (1, 2, 3) toward Q(2, 1, 4): The vector along this direction will be Q - P = (1, -1, 1) from which aP Q = [ax - ay + az ]/ 3. We now write dW = -qE dL = -(50 10-6 ) 120a (ax - ay + az (2 10-3 ) 3 1 = -(50 10-6 )(120) (a ax ) - (a ay ) (2 10-3 ) 3 At P , = tan-1 (2/1) = 63.4 . Thus (a ax ) = cos(63.4) = 0.447 and (a ay ) = sin(63.4) = 0.894. Substituting these, we obtain dW = 3.1 J. b) Q(2, 1, 4) toward P (1, 2, 3): A little thought is in order here: Note that the field has only a radial component and does not depend on or z. Note also that P and Q are at the same radius ( 5) from the z axis, but have different and z coordinates. We could just as well position the two points at the same z location and the problem would not change. If this were so, then moving along a straight line between P and Q would thus involve moving along a chord of a circle whose radius is 5. Halfway along this line is a point of symmetry in the field (make a sketch to see this). This means that when starting from either point, the initial force will be the same. Thus the answer is dW = 3.1 J as in part a. This is also found by going through the same procedure as in part a, but with the direction (roles of P and Q) reversed. 4.4. Find the amount of energy required to move a 6-C charge from the origin to P (3, 1, -1) in the field E = 2xax - 3y 2 ay + 4az V/m along the straight-line path x = -3z, y = x + 2z: We set up the computation as follows, and find the the result does not depend on the path. W = -q = -6 0 E dL = -6 3 (2xax - 3y 2 ay + 4az ) (dxax + dyay + dzaz ) 1 0 2xdx + 6 3y 2 dy - 6 0 -1 4dz = -24 J 43 4.5. Compute the value of A G dL for G = 2yax with A(1, -1, 2) and P (2, 1, 2) using the path: a) straight-line segments A(1, -1, 2) to B(1, 1, 2) to P (2, 1, 2): In general we would have P A P G dL = P A 2y dx The change in x occurs when moving between B and P , during which y = 1. Thus P A G dL = P B 2y dx = 1 2 2(1)dx = 2 b) straight-line segments A(1, -1, 2) to C(2, -1, 2) to P (2, 1, 2): In this case the change in x occurs when moving from A to C, during which y = -1. Thus P A G dL = C A 2y dx = 1 2 2(-1)dx = -2 4.6. Let G = 4xax +2zay +2yaz . Given an initial point P (2, 1, 1) and a final point Q(4, 3, 1), find GdL using the path: a) straight line: y = x - 1, z = 1; b) parabola: 6y = x 2 + 2, z = 1: With G as given, the line integral will be G dL = 2 4 4x dx + 1 3 2z dy + 1 1 2y dz Clearly, we are going nowhere in z, so the last integral is zero. With z = 1, the first two evaluate as G dL = 2x 2 4 2 + 2y 3 1 = 28 The paths specified in parts a and b did not play a role, meaning that the integral between the specified points is path-independent. 4.7. Repeat Problem 4.6 for G = 3xy 3 ax + 2zay . Now things are different in that the path does matter: a) straight line: y = x - 1, z = 1: We obtain: G dL = 2 4 3xy 2 dx + 1 3 2z dy = 2 4 3x(x - 1)2 dx + 1 3 2(1) dy = 90 b) parabola: 6y = x 2 + 2, z = 1: We obtain: G dL = 2 4 3xy 2 dx + 1 3 2z dy = 2 4 1 x(x 2 + 2)2 dx + 12 3 1 2(1) dy = 82 44 4.8. A point charge Q1 is located at the origin in free space. Find the work done in carrying a charge Q2 from: (a) B(rB , B , B ) to C(rA , B , B ) with and held constant; (b) C(rA , B , B ) to D(rA , A , B ) with r and held constant; (c) D(rA , A , B ) to A(rA , A , A ) with r and held constant: The general expression for the work done in this instance is W = -Q2 E dL = -Q2 Q1 Q1 Q2 a (drar + rd a + r sin da ) = - 2 r 4 0 r 4 0 dr r2 We see that only changes in r will produce non-zero results. Thus for part a we have W =- Q1 Q2 4 0 rA rB dr Q1 Q2 = 2 r 4 0 1 1 - rA rB J The answers to parts b and c (involving paths over which r is held constant) are both 0. 4.9. A uniform surface charge density of 20 nC/m2 is present on the spherical surface r = 0.6 cm in free space. a) Find the absolute potential at P (r = 1 cm, = 25 , = 50 ): Since the charge density is uniform and is spherically-symmetric, the angular coordinates do not matter. The potential function for r > 0.6 cm will be that of a point charge of Q = 4a 2 s , or V (r) = 0.081 4(0.6 10-2 )2 (20 10-9 ) = V with r in meters 4 0 r r At r = 1 cm, this becomes V (r = 1 cm) = 8.14 V b) Find VAB given points A(r = 2 cm, = 30 , = 60 ) and B(r = 3 cm, = 45 , = 90 ): Again, the angles do not matter because of the spherical symmetry. We use the part a result to obtain 1 1 VAB = VA - VB = 0.081 - = 1.36 V 0.02 0.03 4.10. Given a surface charge density of 8 nC/m2 on the plane x = 2, a line charge density of 30 nC/m on the line x = 1, y = 2, and a 1-C point charge at P (-1, -1, 2), find VAB for points A(3, 4, 0) and B(4, 0, 1): We need to find a potential function for the combined charges. That for the point charge we know to be Q Vp (r) = 4 0 r Potential functions for the sheet and line charges can be found by taking indefinite integrals of the electric fields for those distributions. For the line charge, we have Vl () = - For the sheet charge, we have Vs (x) = - s s dx + C2 = - x + C2 2 0 2 0 l l ln() + C1 d + C1 = - 2 0 2 0 45 4.10. (continued) The total potential function will be the sum of the three. Combining the integration constants, we obtain: l s Q - ln() - x+C V = 2 0 2 0 4 0 r The terms in this expression are not referenced to a common origin, since the charges are at different positions. The parameters r, , and x are scalar distances from the charges, and will be treated as such here. For point A we have rA = (3 - (-1))2 + (4 - (-1))2 + (-2)2 = 45, A = (3 - 1)2 + (4 - 2)2 = 8, and its distance from the sheet charge is xA = 3 - 2 = 1. The potential at A is then 10-6 8 10-9 30 10-9 VA = ln 8 - (1) + C - 2 0 2 0 4 0 45 At point B, rB = (4 - (-1))2 + (0 - (-1))2 + (1 - 2)2 = 27, B = (4 - 1)2 + (0 - 2)2 = 13, and the distance from the sheet charge is xB = 4 - 2 = 2. The potential at A is then VB = Then VA - VB = 10-6 4 0 1 30 10-9 1 ln - - 2 0 27 45 8 13 - 8 10-9 (1 - 2) = 193 V 2 0 10-6 8 10-9 30 10-9 ln 13 - (2) + C - 2 0 2 0 4 0 27 4.11. Let a uniform surface charge density of 5 nC/m2 be present at the z = 0 plane, a uniform line charge density of 8 nC/m be located at x = 0, z = 4, and a point charge of 2 C be present at P (2, 0, 0). If V = 0 at M(0, 0, 5), find V at N(1, 2, 3): We need to find a potential function for the combined charges which is zero at M. That for the point charge we know to be Vp (r) = Q 4 0 r Potential functions for the sheet and line charges can be found by taking indefinite integrals of the electric fields for those distributions. For the line charge, we have Vl () = - For the sheet charge, we have Vs (z) = - s s dz + C2 = - z + C2 2 0 2 0 l l d + C1 = - ln() + C1 2 0 2 0 The total potential function will be the sum of the three. Combining the integration constants, we obtain: l s Q - ln() - z+C V = 4 0 r 2 0 2 0 46 4.11. (continued) The terms in this expression are not referenced to a common origin, since the charges are at different positions. The parameters r, , and z are scalar distances from the charges, and will be treated as such here. To evaluate the constant, C, we first look at point M, where VT = 0. At M, r = 22 + 52 = 29, = 1, and z = 5. We thus have 0= 5 10-9 2 10-6 8 10-9 ln(1) - 5 + C C = -1.93 103 V - 2 0 2 0 4 0 29 1 + 4 + 9 = 14, = 2, and z = 3. The potential at N is thus At point N, r = VN = 2 10-6 5 10-9 8 10-9 ln( 2) - (3) - 1.93 103 = 1.98 103 V = 1.98 kV - 2 0 2 0 4 0 14 4.12. Three point charges, 0.4 C each, are located at (0, 0, -1), (0, 0, 0), and (0, 0, 1), in free space. a) Find an expression for the absolute potential as a function of z along the line x = 0, y = 1: From a point located at position z along the given line, the distances to the three charges are R1 = (z - 1)2 + 1, R2 = z2 + 1, and R3 = (z + 1)2 + 1. The total potential will be V (z) = Using q = 4 10-7 C, this becomes V (z) = (3.6 103 ) 1 (z - 1)2 + 1 + 1 z2 + 1 + 1 (z + 1)2 + 1 V q 4 1 1 1 + + R1 R2 R3 0 b) Sketch V (z). The sketch will show that V maximizes to a value of 8.68 103 at z = 0, and then monotonically decreases with increasing |z| symmetrically on either side of z = 0. 4.13. Three identical point charges of 4 pC each are located at the corners of an equilateral triangle 0.5 mm on a side in free space. How much work must be done to move one charge to a point equidistant from the other two and on the line joining them? This will be the magnitude of the charge times the potential difference between the finishing and starting positions, or W = (4 10-12 )2 2 0 1 1 - 104 = 5.76 10-10 J = 576 pJ 2.5 5 4.14. two 6-nC point charges are located at (1, 0, 0) and (-1, 0, 0) in free space. a) Find V at P (0, 0, z): Since the charges are positioned symmetrically about the z axis, the potential at z will be double that from one charge. This becomes: V (z) = (2) q q = 2+1 4 0 z 2 0 z2 + 1 b) Find Vmax : It is clear from the part a result that V will maximize at z = 0, or vmax = q/(2 0 ) = 108 V. 47 4.14. (continued) c) Calculate |dV /dz| on the z axis: Differentiating the part a result, we find qz dV = V/m dz 0 (z2 + 1)3/2 d) Find |dV /dz|max : To find this we need to differentiate the part c result and find its zero: d dV q(1 - 2z2 ) 1 = = 0 z = dz dz 0 (z2 + 1)5/2 2 Substituting z = 1/ 2 into the part c result, we find dV dz = q 2 0 (3/2)3/2 = 83.1 V/m max 4.15. Two uniform line charges, 8 nC/m each, are located at x = 1, z = 2, and at x = -1, y = 2 in free space. If the potential at the origin is 100 V, find V at P (4, 1, 3): The net potential function for the two charges would in general be: V =- At the origin, R1 = R2 = l l ln(R1 ) - ln(R2 ) + C 2 0 2 0 5, and V = 100 V. Thus, with l = 8 10-9 , (8 10-9 ) ln( 5) + C C = 331.6 V 2 0 10 and R2 = |(4, 1, 3) - (-1, 2, 3)| = 26. Therefore 100 = -2 At P (4, 1, 3), R1 = |(4, 1, 3) - (1, 1, 2)| = VP = - (8 10-9 ) ln( 10) + ln( 26) + 331.6 = -68.4 V 2 0 48 4.16. Uniform surface charge densities of 6, 4, and 2 nC/m2 are present at r = 2, 4, and 6 cm, respectively, in free space. a) Assume V = 0 at infinity, and find V (r). We keep in mind the definition of absolute potential as the work done in moving a unit positive charge from infinity to location r. At radii outside all three spheres, the potential will be the same as that of a point charge at the origin, whose charge is the sum of the three sphere charges: V (r) (r > 6 cm) = [4(.02)2 (6) + 4(.04)2 (4) + 4(.06)2 (2)] 10-9 q 1 + q2 + q3 = 4 0 r 4 0 r -13 (96 + 256 + 288) 10 1.81 = = V where r is in meters 4(8.85 10-12 )r r As the unit charge is moved inside the outer sphere to positions 4 < r < 6 cm, the outer sphere contribution to the energy is fixed at its value at r = 6. Therefore, V (r) (4 < r < 6 cm) = q1 + q2 q3 0.994 + = + 13.6 V 4 0 r 4 0 (.06) r In moving inside the sphere at r = 4 cm, the contribution from that sphere becomes fixed at its potential function at r = 4: V (r) (2 < r < 4 cm) = q1 q2 q3 0.271 + + = + 31.7 V 4 0 r 4 0 (.04) 4 0 (.06) r 0.271 + 31.7 = 45.3 V .02 Finally, using the same reasoning, the potential inside the inner sphere becomes V (r) (r < 2 cm) = b) Calculate V at r = 1, 3, 5, and 7 cm: Using the results of part a, we substitute these distances (in meters) into the appropriate formulas to obtain: V (1) = 45.3 V, V (3) = 40.7 V, V (5) = 33.5 V, and V (7) = 25.9 V. c) Sketch V versus r for 0 < r < 10 cm. 49 4.17. Uniform surface charge densities of 6 and 2 nC/m2 are present at = 2 and 6 cm respectively, in free space. Assume V = 0 at = 4 cm, and calculate V at: a) = 5 cm: Since V = 0 at 4 cm, the potential at 5 cm will be the potential difference between points 5 and 4: V5 = - 4 5 E dL = - 4 5 (.02)(6 10-9 ) asa 5 d = - ln 4 0 0 = -3.026 V b) = 7 cm: Here we integrate piecewise from = 4 to = 7: V7 = - 4 6 asa d - 0 7 6 (asa + bsb ) d 0 With the given values, this becomes V7 = - (.02)(6 10-9 ) 0 ln 6 4 - (.02)(6 10-9 ) + (.06)(2 10-9 ) 0 ln 7 6 = -9.678 V 4.18. A nonuniform linear charge density, L = 8/(z2 + 1) nC/m lies along the z axis. Find the potential at P ( = 1, 0, 0) in free space if V = 0 at infinity: This last condition enables us to write the potential at P as a superposition of point charge potentials. The result is the integral: VP = - L dz 4 0 R where R = z2 + 1 is the distance from a point z on the z axis to P . Substituting the given charge distribution and R into the integral gives us VP = - 2 10-9 8 10-9 dz z = 2 + 1)3/2 4 0 (z 0 z2 + 1 - = 144 V 4.19. The annular surface, 1 cm < < 3 cm, z = 0, carries the nonuniform surface charge density s = 5 nC/m2 . Find V at P (0, 0, 2 cm) if V = 0 at infinity: We use the superposition integral form: VP = s da 4 0 |r - r | where r = zaz and r = a . We integrate over the surface of the annular region, with da = d d. Substituting the given values, we find VP = 2 0 .03 .01 (5 10-9 ) 2 d d 4 0 2 + z2 Substituting z = .02, and using tables, the integral evaluates as (5 10-9 ) VP = 2 0 2 2 + (.02)2 (.02)2 ln( + - 2 50 .03 2 + (.02)2 ) .01 = .081 V 4.20. Fig. 4.11 shows three separate charge distributions in the z = 0 plane in free space. a) find the total charge for each distribution: Line charge along the y axis: Q1 = 3 5 10-9 dy = 2 10-9 C = 6.28 nC Line charge in an arc at radius = 3: Q2 = Sheet charge: Q3 = 70 10 3.5 1.6 70 10 (10-9 ) 3 d = 4.5 10-9 (70 - 10) 2 = 4.71 10-9 C = 4.71 nC 360 (10-9 ) d d = 5.07 10-9 C = 5.07 nC b) Find the potential at P (0, 0, 6) caused by each of the three charge distributions acting alone: Line charge along y axis: VP 1 = 5 3 L dL = 4 0 R 5 3 10-9 dy 4 0 y 2 + 62 = 103 ln(y + 4 8.854 y 2 + 62 ) 5 3 = 7.83 V Line charge in an arc a radius = 3: VP 2 = Sheet charge: VP 3 = 70 10 3.5 1.6 70 10 (1.5 10-9 ) 3 d Q2 = = 6.31 V 4 0 45 4 0 32 + 62 (10-9 ) d d 4 0 3.5 1.6 2 + 62 = 60 10-9 4(8.854 10-12 2 360 3.5 1.6 d 2 + 36 = 9.42 2 + 36 = 6.93 V c) Find VP : This will be the sum of the three results of part b, or VP = VP 1 + VP 2 + VP 3 = 7.83 + 6.31 + 6.93 = 21.1 V 4.21. Let V = 2xy 2 z3 + 3 ln(x 2 + 2y 2 + 3z2 ) V in free space. Evaluate each of the following quantities at P (3, 2, -1): a) V : Substitute P directly to obtain: V = -15.0 V b) |V |. This will be just 15.0 V. c) E: We have E P = -V + P =- x2 2y 2 z3 + 6x x 2 + 2y 2 + 3z2 az P ax + 4xyz3 + 12y x 2 + 2y 2 + 3z2 ay 6xy 2 z2 + 18z + 2y 2 + 3z2 = 7.1ax + 22.8ay - 71.1az V/m 51 4.21d. |E|P : taking the magnitude of the part c result, we find |E|P = 75.0 V/m. e) aN : By definition, this will be aN P =- E = -0.095 ax - 0.304 ay + 0.948 az |E| f) D: This is D P = 0E P = 62.8 ax + 202 ay - 629 az pC/m2 . 4.22. It is known that the potential is given as V = 80r 0.6 V. Assuming free space conditions, find: a) E: We use dV E = -V = - ar = -(0.6)80r -0.4 ar = -48r -0.4 ar V/m dr b) the volume charge density at r = 0.5 m: Begin by finding D= We next find v = D = Then at r = 0.5 m, v (0.5) = -76.8(8.854 10-12 ) = -1.79 10-9 C/m3 = -1.79 nC/m3 (0.5)1.4 1 d 1 d 76.8 0 r 2 Dr = 2 -48 0 r 1.6 = - 1.4 C/m3 r 2 dr r dr r 0E = -48r -0.4 0 ar C/m2 c) the total charge lying within the surface r = 0.6: The easiest way is to use Gauss'law, and integrate the flux density over the spherical surface r = 0.6. Since the field is constant at constant radius, we obtain the product: Q = 4(0.6)2 (-48 0 (0.6)-0.4 ) = -2.36 10-9 C = -2.36 nC 4.23. It is known that the potential is given as V = 80 .6 V. Assuming free space conditions, find: a) E: We find this through E = -V = - dV a = -48 -.4 V/m d 0 E, b) the volume charge density at = .5 m: Using D = v = [ D].5 = 1 d D d we find the charge density through = -673 pC/m3 .5 .5 = -28.8 0 -1.4 .5 52 4.23c. the total charge lying within the closed surface = .6, 0 < z < 1: The easiest way to do this calculation is to evaluate D at = .6 (noting that it is constant), and then multiply by the cylinder area: Using part a, we have D .6 = -48 0 (.6)-.4 = -521 pC/m2 . Thus Q = -2(.6)(1)52110-12 C = -1.96 nC. 4.24. Given the potential field V = 80r 2 cos and a point P (2.5, = 30 , = 60 ) in free space, find at P : a) V : Substitute the coordinates into the function and find VP = 80(2.5)2 cos(30) = 433 V. b) E: V 1 V ar - a = -160r cos ar + 80r sin a V/m r r Evaluating this at P yields Ep = -346ar + 100a V/m. E = -V = - 0 EP c) D: In free space, DP = d) v : = (-346ar + 100a ) 0 0 = -3.07 ar + 0.885 a nC/m2 . v = D = E= 0 1 1 r 2 Er + 2 (E sin ) 2 r r r sin Substituting the components of E, we find v = - 160 cos 2 1 80r(2 sin cos ) 3r + 2 r r sin 0 = -320 0 cos = -2.45 nC/m3 with = 30 . e) dV /dN: This will be just |E| evaluated at P , which is dV dN f) aN : This will be aN = - EP =- |EP | -346ar + 100a (346)2 + (100)2 = 0.961 ar - 0.278 a P = | - 346ar + 100a | = (346)2 + (100)2 = 360 V/m 4.25. Within the cylinder = 2, 0 < z < 1, the potential is given by V = 100 + 50 + 150 sin V. a) Find V , E, D, and v at P (1, 60 , 0.5) in free space: First, substituting the given point, we find VP = 279.9 V. Then, E = -V = - V 1 V a - a = - [50 + 150 sin ] a - [150 cos ] a Evaluate the above at P to find EP = -179.9a - 75.0a V/m Now D = 0 E, so DP = -1.59a - .664a nC/m2 . Then 1 1 1 D d 1 = - (50 + 150 sin ) + 150 sin D + d 0 v = D = =- 50 0C At P , this is vP = -443 pC/m3 . 53 4.25b. How much charge lies within the cylinder? We will integrate v over the volume to obtain: Q= 0 1 0 2 0 2 - 50 0 d d dz = -2(50) 0 (2) = -5.56 nC 4.26. A dipole having Qd/(4 0 ) = 100 V m2 is located at the origin in free space and aligned so that its moment is in the az direction. a) Sketch |V (r = 1, , = 0)| versus on polar graph paper (homemade if you wish). b) Sketch |E(r = 1, , = 0)| versus on polar graph paper: V = Qd cos 100 cos = |V (r = 1, , = 0)| = |100 cos | 4 0 r 2 r2 Qd 100 (2 cos ar + sin a ) = 3 (2 cos ar + sin a ) 3 4 0 r r 1/2 E= |E(r = 1, , = 0)| = 100 4 cos2 + sin2 These results are plotted below: = 100 1 + 3 cos2 1/2 54 4.27. Two point charges, 1 nC at (0, 0, 0.1) and -1 nC at (0, 0, -0.1), are in free space. a) Calculate V at P (0.3, 0, 0.4): Use VP = q q - +| 4 0 |R 4 0 |R- | where R+ = (.3, 0, .3) and R- = (.3, 0, .5), so that |R+ | = 0.424 and |R- | = 0.583. Thus VP = b) Calculate |E| at P : Use EP = q(.3ax + .3az ) q(.3ax + .5az ) 10-9 - = 2.42ax + 1.41az V/m 4 0 (.424)3 4 0 (.583)3 4 0 10-9 4 0 1 1 - = 5.78 V .424 .583 Taking the magnitude of the above, we find |EP | = 25.2 V/m. c) Now treat the two charges as a dipole at the origin and find V at P : In spherical coordinates, P is located at r = .32 + .42 = .5 and = sin-1 (.3/.5) = 36.9 . Assuming a dipole in far-field, we have qd cos 10-9 (.2) cos(36.9 ) VP = = = 5.76 V 4 0 r 2 4 0 (.5)2 4.28. A dipole located at the origin in free space has a moment p2 10-9 az C m. At what points on the line y = z, x = 0 is: a) |E | = 1 mV/m? We note that the line y = z lies at = 45 . Begin with E= from which E = 10-9 = 10-3 V/m (required) r 3 = 1.27 10-4 or r = 23.3 m 2 0 r 3 2 10-9 10-9 (2 cos ar + sin a ) = (2ar + a ) at = 45 3 4 0 r 3 2 2 0 r The y and z values are thus y = z = 23.3/ 2 = 16.5 m b) |Er | = 1 mV/m? From the above field expression, the radial component magnitude is twice that of the theta component. Using the same development, we then find Er = 2 10-9 = 10-3 V/m (required) r 3 = 2(1.27 10-4 ) or r = 29.4 m 2 0 r 3 The y and z values are thus y = z = 29.4/ 2 = 20.8 m 55 4.29. A dipole having a moment p = 3ax - 5ay + 10az nC m is located at Q(1, 2, -4) in free space. Find V at P (2, 3, 4): We use the general expression for the potential in the far field: V = where r - r = P - Q = (1, 1, 8). So VP = (3ax - 5ay + 10az ) (ax + ay + 8az ) 10-9 = 1.31 V 4 0 [12 + 12 + 82 ]1.5 p (r - r ) 4 0 |r - r |3 4.30. A dipole, having a moment p = 2az nC m is located at the origin in free space. Give the magnitude of E and its direction aE in cartesian components at r = 100 m, = 90 , and =: a) 0 ; b) 30 ; c) 90 . Begin with p E= [2 cos ar + sin a ] 4 0 r 3 from which |E| = Now E x = E ax = then Ey = E ay = and Ez = E az = Since is given as 90 , p 4 0 p 4 p 4 0 0 r3 4 cos2 + sin2 1/2 = p 4 0 r3 1 + 3 cos2 p 4 0 r 3 p 4 0 r 3 p 4 0 r 3 1/2 r3 [2 cos ar ax + sin a ax ] = [3 cos sin cos ] p 4 0 r3 2 cos ar ay + sin a ay = [3 cos sin sin ] r3 2 cos ar az + sin a az = 2 cos2 - sin2 Ex = 0, and the field magnitude becomes 2 2 Ey + Ez = |E( = 90 )| = p 4 0 r 3 9 cos2 sin2 + (2 cos2 - sin2 )2 1/2 Then the unit vector becomes (again at = 90 ): aE = 3 cos sin ay + (2 cos2 - sin2 ) az 9 cos2 sin2 + (2 cos2 - sin2 )2 1/2 Now with r = 100 m and p = 2 10-9 , p 4 0 r 3 = 2 10-9 = 1.80 10-5 4(8.854 10-12 )106 Using the above formulas, we find at = 0 , |E| = (1.80 10-5 )(2) = 36.0 V/m and aE = az . At = 30 , we find |E| = (1.80 10-5 )[1.69 + 1.56]1/2 = 32.5 V/m and aE = (1.30ay + 1.25az )/1.80 = 0.72 ax + 0.69 az . At = 90 , |E| = (1.8010-5 )(1) = 18.0 V/m and aE = -az . 56 4.31. A potential field in free space is expressed as V = 20/(xyz) V. a) Find the total energy stored within the cube 1 < x, y, z < 2. We integrate the energy density over the cube volume, where wE = (1/2) 0 E E, and where E = -V = 20 The energy is now WE = 200 2 0 1 1 1 2 2 ax x 2 yz 1 + 1 1 a + az V/m 2z y xy xyz2 1 x 4 y 2 z2 + 1 x 2 y 4 z2 + 1 x 2 y 2 z4 dx dy dz The integral evaluates as follows: WE = 200 = 200 = 200 = 200 2 0 1 2 0 1 2 0 1 2 0 1 1 1 2 2 - 1 3 7 24 1 1 1 - 4 2- 2 4 3 y 2 z2 x xy z xy z 1 y 2 z2 + 1 6 1 2 1 y 4 z2 1 - 3 z2 y 1 + z2 1 4 + 1 2 1 z4 1 2 2 dy dz 1 1 y 2 z4 1 yz4 2 dy dz - 7 24 7 48 1 - yz2 1 + z2 7 48 dz 1 dz = 200 0 (3) 7 = 387 pJ 96 b) What value would be obtained by assuming a uniform energy density equal to the value at the center of the cube? At C(1.5, 1.5, 1.5) the energy density is wE = 200 0 (3) 1 (1.5)4 (1.5)2 (1.5)2 = 2.07 10-10 J/m3 This, multiplied by a cube volume of 1, produces an energy value of 207 pJ. 4.32. In the region of free space where 2 < r < 3, 0.4 < < 0.6, 0 < < /2, let E = k/r 2 ar . a) Find a positive value for k so that the total energy stored is exactly 1 J: The energy is found through WE = 1 2 0 E dv = 2 .6 .4 /2 0 0.6 0.4 0k 2 2 3 = (- cos ) 2 v 1 2 1 2 - 1 r k2 2 0 2 r sin dr d d r 3 0.616 2 = 0k = 1 J 2 24 Solve for k to find k = 1.18 106 V m. 57 4.32b. Show that the surface = 0.6 is an equipotential surface: This will be the surface of a cone, centered at the origin, along which E, in the ar direction, will exist. Therefore, the given surface cannot be an equipotential (the problem was ill-conceived). Only a surface of constant r could be an equipotential in this field. c) Find VAB , given points A(2, = /2, = /3) and B(3, /2, /4): Use VAB = - A B E dL = - 2 3 k ar ar dr = k r2 1 1 - 2 3 = k 6 Using the result of part a, we find VAB = (1.18 106 )/6 = 197 kV. 4.33. A copper sphere of radius 4 cm carries a uniformly-distributed total charge of 5 C in free space. a) Use Gauss' law to find D external to the sphere: with a spherical Gaussian surface at radius r, D will be the total charge divided by the area of this sphere, and will be ar -directed. Thus D= Q 5 10-6 ar = ar C/m2 4r 2 4r 2 b) Calculate the total energy stored in the electrostatic field: Use WE = 2 1 D E dv = vol 2 0 1 (5 10-6 )2 = (4) 2 16 2 0 1 (5 10-6 )2 2 r sin dr d d 2 4 0 .04 2 16 0 r dr 25 10-12 1 = 2.81 J = 2 8 0 .04 .04 r c) Use WE = Q2 /(2C) to calculate the capacitance of the isolated sphere: We have C= Q2 (5 10-6 )2 = 4.45 10-12 F = 4.45 pF = 2WE 2(2.81) 4.34. Given the potential field in free space, V = 80 V (note that ap hi should not be present), find: a) the energy stored in the region 2 < < 4 cm, 0 < < 0.2, 0 < z < 1 m: First we find E = -V = - Then WE = wE dv = 1 0 0 0.2 .04 .02 1 dV 80 a = - a V/m d v 1 2 0 (80)2 d d dz = 640 2 0 ln .04 .02 = 12.3 nJ b) the potential difference, VAB , for A(3 cm, = 0, z = 0) and B(3cm, 0.2, 1m): Use VAB = - A B E dL = - 0 .2 - 80 a a d = -80(0.2) = -16 V 58 4.34c. the maximum value of the energy density in the specified region: The energy density is wE = 1 1 2 0E = 2 2 0 6400 2 This will maximize at the lowest value of in the specified range, which is = 2 cm. So wE,max = 1 2 0 6400 = 7.1 10-5 J/m3 = 71 J/m3 .022 4.35. Four 0.8 nC point charges are located in free space at the corners of a square 4 cm on a side. a) Find the total potential energy stored: This will be given by WE = 1 2 4 q n Vn n=1 where Vn in this case is the potential at the location of any one of the point charges that arises from the other three. This will be (for charge 1) V1 = V21 + V31 + V41 = q 4 1 1 1 + + .04 .04 .04 2 0 Taking the summation produces a factor of 4, since the situation is the same at all four points. Consequently, WE = 1 1 (.8 10-9 )2 (4)q1 V1 = 2 + = 7.79 10-7 J = 0.779 J 2 2 0 (.04) 2 b) A fifth 0.8 C charge is installed at the center of the square. Again find the total stored energy: This will be the energy found in part a plus the amount of work done in moving the fifth charge into position from infinity. The latter is just the potential at the square center arising from the original four charges, times the new charge value, or WE = The total energy is now WE net = WE (part a) + WE = .779 + .813 = 1.59 J 4(.8 10-9 )2 = .813 J 4 0 (.04 2/2) 59 CHAPTER 5 5.1. Given the current density J = -104 [sin(2x)e-2y ax + cos(2x)e-2y ay ] kA/m2 : a) Find the total current crossing the plane y = 1 in the ay direction in the region 0 < x < 1, 0 < z < 2: This is found through I= J n da = S 0 1 2 0 1 S J ay y=1 dx dz = 0 2 0 1 -104 cos(2x)e-2 dx dz 1 = -104 (2) sin(2x) e-2 = -1.23 MA 0 2 b) Find the total current leaving the region 0 < x, x < 1, 2 < z < 3 by integrating J dS over the surface of the cube: Note first that current through the top and bottom surfaces will not exist, since J has no z component. Also note that there will be no current through the x = 0 plane, since Jx = 0 there. Current will pass through the three remaining surfaces, and will be found through I= 2 3 0 3 2 0 1 J (-ay ) 1 y=0 dx dz + 2 3 0 1 J (ay ) y=1 dx dz + 3 2 0 2 1 3 0 1 J (ax ) x=1 dy dz = 104 = 104 cos(2x)e-0 - cos(2x)e-2 dx dz - 104 sin(2)e-2y dy dz 1 1 sin(2x) (3 - 2) 1 - e-2 + 104 0 2 1 1 sin(2)e-2y (3 - 2) = 0 0 2 c) Repeat part b, but use the divergence theorem: We find the net outward current through the surface of the cube by integrating the divergence of J over the cube volume. We have J= Jy Jx + = -10-4 2 cos(2x)e-2y - 2 cos(2x)e-2y = 0 as expected x y 5.2. Let the current density be J = 2 cos2 a - sin 2a A/m2 within the region 2.1 < < 2.5, 0 < < 0.1 rad, 6 < z < 6.1. Find the total current I crossing the surface: a) = 2.2, 0 < < 0.1, 6 < z < 6.1 in the a direction: This is a surface of constant , so only the radial component of J will contribute: At = 2.2 we write: I= J dS = 6 2 6.1 0 0.1 2(2) cos2 a a 2ddz = 2(2.2)2 (0.1) 0.1 0 0.1 0 1 (1 + cos 2) d 2 = 0.2(2.2) 1 1 (0.1) + sin 2 2 4 = 97 mA b) = 0.05, 2.2 < < 2.5, 6 < z < 6.1 in the a direction: In this case only the component of J will contribute: I= J dS = 6 6.1 2.5 2.2 - sin 2 a a d dz = -(0.1)2 =0.05 2 2 2.5 2.2 = -7 mA 60 5.2c. Evaluate J at P ( = 2.4, = 0.08, z = 6.05): J= 1 1 J 1 1 (J ) + = (2 2 cos2 ) - ( sin 2) = 4 cos2 - 2 cos 2 3 0.08 = 2.0 A/m 5.3. Let J= 400 sin ar A/m2 r2 + 4 a) Find the total current flowing through that portion of the spherical surface r = 0.8, bounded by 0.1 < < 0.3, 0 < < 2: This will be I= = 346.5 Jn da = 0 2 .3 .1 S .3 1 400 sin 400(.8)2 2 (.8)2 sin d d = (.8)2 + 4 4.64 .3 .1 sin2 d .1 2 [1 - cos(2)] d = 77.4 A b) Find the average value of J over the defined area. The area is Area = 0 2 .3 .1 (.8)2 sin d d = 1.46 m2 The average current density is thus Javg = (77.4/1.46) ar = 53.0 ar A/m2 . 5.4. The cathode of a planar vacuum tube is at z = 0. Let E = -4 106 az V/m for z > 0. An electron (e = 1.602 10-19 C, m = 9.11 10-31 kg) is emitted from the cathode with zero initial velocity at t = 0. a) Find v(t): Using Newton's second law, we write: F = ma = qE a = (-1.602 10-19 )(-4 106 )az = 7.0 1017 az m/s2 (9.11 10-31 ) Then v(t) = at = 7.0 1017 t m/s. b) Find z(t), the electron location as a function of time: Use z(t) = 0 t v(t )dt = 1 (7.0 1017 )t 2 = 3.5 1017 t 2 m 2 c) Determine v(z): Solve the result of part b for t, obtaining t= z 3.5 101 7 = 1.7 109 z Substitute into the result of part a to find v(z) = 7.0 1017 (1.7 10-9 ) z = 1.2 109 z m/s. 61 5.4d. Make the assumption that the electrons are emitted continuously as a beam with a 0.25 mm radius and a total current of 60 A. Find J(z) and (z): J(z) = -60 10-6 az = -3.1 102 az A/m2 (0.25)2 (10-6 ) (negative since we have electrons flowing in the positive z direction) Next we use J(z) = v (z)v(z), or v (z) = -3.1 102 2.6 10-7 J -26 = C/m3 = C/m3 =- 9 z v 1.2 10 z z 5.5. Let J= 20 25 a - 2 az A/m2 + 0.01 a) Find the total current crossing the plane z = 0.2 in the az direction for < 0.4: Use I= Jn da = 0 2 0 .4 S z=.2 2 -20 d d + .01 .4 1 20 ln(.01 + 2 ) (2) = -20 ln(17) = -178.0 A =- 0 2 b) Calculate v /t: This is found using the equation of continuity: v 1 1 Jz = - J = (J ) + = (25) + t z z 2 -20 + .01 =0 c) Find the outward current crossing the closed surface defined by = 0.01, = 0.4, z = 0, and z = 0.2: This will be I= + .2 2 .2 2 25 25 a (-a )(.01) d dz + a (a )(.4) d dz .01 .4 0 0 2 .4 -20 -20 az (-az ) d d + a (az ) d d = 0 2 + .01 2 + .01 z 0 0 0 0 2 .4 0 0 since the integrals will cancel each other. d) Show that the divergence theorem is satisfied for J and the surface specified in part b. In part c, the net outward flux was found to be zero, and in part b, the divergence of J was found to be zero (as will be its volume integral). Therefore, the divergence theorem is satisfied. 5.6. Let = 0 and V = 90z4/3 in the region z = 0. a) Obtain expressions for E, D, and v as functions of z: First, E = -V = - dV 4 az = - (90)z1/3 az = -120z1/3 az V/m dz 3 62 5.6a. (continued) Next, D = 0E = 1.06z1/3 az nC/m2 . Then v = D = 1 dDz = - (120) 0 z-2/3 = -354z-2/3 pC/m3 dz 3 b) If the velocity of the charge density is given as vz = 5 106 z2/3 m/s, find Jz at z = 0 and z = 0.1 m (note that vz is written as vx through a missprint): Use Jz = v vz = (-354 10-12 )z-2/3 (5 106 )z2/3 = -1.8 mA/m2 at any z. 5.7. Assuming that there is no transformation of mass to energy or vice-versa, it is possible to write a continuity equation for mass. a) If we use the continuity equation for charge as our model, what quantities correspond to J and v ? These would be, respectively, mass flux density in (kg/m2 - s) and mass density in (kg/m3 ). b) Given a cube 1 cm on a side, experimental data show that the rates at which mass is leaving each of the six faces are 10.25, -9.85, 1.75, -2.00, -4.05, and 4.45 mg/s. If we assume that the cube is an incremental volume element, determine an approximate value for the time rate of change of density at its center. We may write the continuity equation for mass as follows, also invoking the divergence theorem: m dv = - Jm dv = - Jm dS v t v s where s Jm dS = 10.25 - 9.85 + 1.75 - 2.00 - 4.05 + 4.45 = 0.550 mg/s Treating our 1 cm3 volume as differential, we find m . 0.550 10-3 g/s = -550 g/m3 - s =- t 10-6 m3 5.8. The continuity equation for mass equates the divergence of the mass rate of flow (mass per second per square meter) to the negative of the density (mass per cubic meter). After setting up a cartesian coordinate system inside a star, Captain Kirk and his intrepid crew make measurements over the faces of a cube centered at the origin with edges 40 km long and parallel to the coordinate axes. They find the mass rate of flow of material outward across the six faces to be -1112, 1183, 201, -196, 1989, and -1920 kg/m2 s. a) Estimate the divergence of the mass rate of flow at the origin: We make the estimate using the definition of divergence, but without taking the limit as the volume shrinks to zero: . Div Jm = Jm dS (-1112 + 1183 + 201 - 196 + 1989 - 1920)(40)2 = = 3.63 kg/km3 s v (40)3 b) Estimate the rate of change of the density at the origin: The continuity equation for mass reads: Div Jm = - m /t. Therefore, the rate of change of density at the origin will be just the negative . of the part a result, or m /t = - 3.63 kg/km3 s. 63 5.9a. Using data tabulated in Appendix C, calculate the required diameter for a 2-m long nichrome wire that will dissipate an average power of 450 W when 120 V rms at 60 Hz is applied to it: The required resistance will be V2 l R= = P (a 2 ) Thus the diameter will be d = 2a = 2 lP 2(450) =2 = 2.8 10-4 m = 0.28 mm 2 6 )(120)2 V (10 b) Calculate the rms current density in the wire: The rms current will be I = 450/120 = 3.75 A. Thus 3.75 = 6.0 107 A/m2 J = -4 /2 2 2.8 10 5.10. A steel wire has a radius of 2 mm and a conductivity of 2 106 S/m. The steel wire has an aluminum ( = 3.8 107 S/m) coating of 2 mm thickness. Let the total current carried by this hybrid conductor be 80 A dc. Find: a) Jst . We begin with the fact that electric field must be the same in the aluminum and steel regions. This comes from the requirement that E tangent to the boundary between two media must be continuous, and from the fact that when integrating E over the wire length, the applied voltage value must be obtained, regardless of the medium within which this integral is evaluated. We can therefore write JAl Jst Al EAl = Est = = JAl = Jst Al st st The net current is now expressed as the sum of the currents in each region, written as the sum of the products of the current densities in each region times the appropriate cross-sectional area: I = (2 10-3 )2 Jst + [(4 10-3 )2 - (2 10-3 )2 ]JAl = 80 A Using the above relation between Jst and JAl , we find 80 = (2 10-3 )2 1 - 3.8 107 6 106 + (4 10-3 )2 3.8 107 6 106 Jst Solve for Jst to find Jst = 3.2 105 A/m2 . b) JAl = 3.8 107 (3.2 105 ) = 2.0 106 A/m2 6 106 c,d) Est = EAl = Jst /st = JAl /Al = 5.3 10-2 V/m. e) the voltage between the ends of the conductor if it is 1 mi long: Using the fact that 1 mi = 1.61103 m, we have V = El = (5.3 10-2 )(1.61 103 ) = 85.4 V. 64 5.11. Two perfectly-conducting cylindrical surfaces are located at = 3 and = 5 cm. The total current passing radially outward through the medium between the cylinders is 3 A dc. Assume the cylinders are both of length l. a) Find the voltage and resistance between the cylinders, and E in the region between the cylinders, if a conducting material having = 0.05 S/m is present for 3 < < 5 cm: Given the current, and knowing that it is radially-directed, we find the current density by dividing it by the area of a cylinder of radius and length l: 3 a A/m2 J= 2l Then the electric field is found by dividing this result by : E= 3 9.55 a = a V/m 2l l The voltage between cylinders is now: V =- 5 3 E dL = 3 5 9.55 5 9.55 a a d = ln l l 3 4.88 V 1.63 = = I 3l l = 4.88 V l Now, the resistance will be R= b) Show that integrating the power dissipated per unit volume over the volume gives the total dissipated power: We calculate P = E J dv = 0 l 0 2 .05 .03 v 5 32 32 ln d d dz = 2 2 (.05)l 2 (2) 2(.05)l 3 = 14.64 W l We also find the power by taking the product of voltage and current: P = VI = 4.88 14.64 (3) = W l l which is in agreement with the power density integration. 5.12. The spherical surfaces r = 3 and r = 5 cm are perfectly conducting, and the total current passing radially outward through the medium between the surfaces is 3 A dc. a) Find the voltage and resistance between the spheres, and E in the region between them, if a conducting material having = 0.05 S/m is present for 3 < r < 5 cm. We first find J as a function of radius by dividing the current by the area of a sphere of radius r: J= Then E= I 3 a = ar A/m2 2 r 4r 4r 2 J 3 4.77 = a = 2 ar V/m 2 (0.05) r 4r r 65 5.12a. (continued) V =- r1 r2 E dL = - . .03 .05 1 4.77 1 - = 63.7 V dr = 4.77 2 r .03 .05 Finally, R = V /I = 63.7/3 = 21.2 b) Repeat if = 0.0005/r for 3 < r < 5 cm: First, J = 3ar /(4r 2 ) as before. The electric field is now 3rar 477 J ar V/m = E= = 2 4(.0005)r r Now V =- r1 r2 E dL = - . .03 .05 .03 477 dr = -477 ln r .05 = 244 V Finally, R = V /I = 244/3 = 81.3 c) Show that integrating the power dissipated per unit volume in part b over the volume gives the total dissipated power: The dissipated power density is pd = E J = 3 4(.0005)r 3 4r 2 = 114 W/m2 r3 We integrate this over the volume between spheres: Pd = 2 0 0 .05 .03 5 114 2 r sin dr d d = 4(114) ln 3 r 3 = 732 W The dissipated power should be just I 2 R = (3)2 (81.3) = 732 W. So it works. 5.13. A hollow cylindrical tube with a rectangular cross-section has external dimensions of 0.5 in by 1 in and a wall thickness of 0.05 in. Assume that the material is brass, for which = 1.5 107 S/m. A current of 200 A dc is flowing down the tube. a) What voltage drop is present across a 1m length of the tube? Converting all measurements to meters, the tube resistance over a 1 m length will be: R1 = (1.5 107 ) (2.54)(2.54/2) 10-4 1 - 2.54(1 - .1)(2.54/2)(1 - .2) 10-4 = 7.38 10-4 The voltage drop is now V = I R1 = 200(7.38 10-4 = 0.147 V. b) Find the voltage drop if the interior of the tube is filled with a conducting material for which = 1.5 105 S/m: The resistance of the filling will be: R2 = 1 (1.5 105 )(1/2)(2.54)2 10-4 (.9)(.8) = 2.87 10-2 The total resistance is now the parallel combination of R1 and R2 : RT = R1 R2 /(R1 + R2 ) = 7.19 10-4 , and the voltage drop is now V = 200RT = .144 V. 66 5.14. Find the magnitude of the electric field intensity in a conductor if: a) the current density is 5 MA/m2 , the electron mobility is 3 10-3 m2 /V s, and the volume charge density is -2.4 1010 C/m3 : In magnitude, we have E= J 5 106 = 6.9 10-2 V/m = e v (2.4 1010 )(3 10-3 ) m: E = J = (3 106 )(3 10-8 ) = b) J = 3 MA/m2 and the resistivity is 3 10-8 9 10-2 V/m. 5.15. Let V = 10( + 1)z2 cos V in free space. a) Let the equipotential surface V = 20 V define a conductor surface. Find the equation of the conductor surface: Set the given potential function equal to 20, to find: ( + 1)z2 cos = 2 b) Find and E at that point on the conductor surface where = 0.2 and z = 1.5: At the given values of and z, we solve the equation of the surface found in part a for , obtaining = .10. Then 1 V V V E = -V = - a - a - az z +1 2 z sin a - 20( + 1)z cos az = -10z2 cos a + 10 Then E(.10, .2, 1.5) = -18.2 a + 145 a - 26.7 az V/m c) Find |s | at that point: Since E is at the perfectly-conducting surface, it will be normal to the surface, so we may write: s = 0E n surface = 0 EE = |E| 0 EE= 0 (18.2)2 + (145)2 + (26.7)2 = 1.32 nC/m2 5.16. A potential field in free space is given as V = (80 cos sin )/r 3 V. Point P (r = 2, = /3, = /2) lies on a conducting surface. a) Write the equation of the conducting surface: The surface will be an equipotential, where the value of the potential is VP : 80 cos(/3) sin(/2) VP = =5 (2)3 So the equation of the surface is 80 cos sin = 5 or 16 cos sin = r 3 r3 67 5.16c. (I will work parts b and c in reverse order) Find E at P : E = -V = - 1 V 1 V V ar - a - a r r r sin 80(3) cos sin 80 sin sin 80 cos cos a = ar + a - 4 4 r r r 4 sin Now 80(1/2)(1)(3) 80( 3/2)(1) EP = ar + a - 0 a = 7.5 ar + 4.3 a V/m 16 16 b) Find a unit vector directed outward to the surface, assuming the origin is inside the surface: Such a unit normal can be construced from the result of part c: aN = 7.5 ar + 4.3 a = 0.87 ar + 0.50 a 4.33 5.17. Given the potential field V = in free space: a) Find D at the surface z = 0: Use E = -V = -100z At z = 0, we use this to find D(z = 0) = 0 E(z 100xz V x2 + 4 x x x2 + 4 ax - 0 ay - 100x az V/m x2 + 4 = 0) = - 100 0 x az C/m2 x2 + 4 b) Show that the z = 0 surface is an equipotential surface: There are two reasons for this: 1) E at z = 0 is everywhere z-directed, and so moving a charge around on the surface involves doing no work; 2) When evaluating the given potential function at z = 0, the result is 0 for all x and y. c) Assume that the z = 0 surface is a conductor and find the total charge on that portion of the conductor defined by 0 < x < 2, -3 < y < 0: We have s = D az So Q= 0 -3 0 2 z=0 =- 100 0 x C/m2 x2 + 4 - 100 0 x dx dy = -(3)(100) x2 + 4 0 1 ln(x 2 + 4) 2 2 0 = -150 0 ln 2 = -0.92 nC 68 5.18. Let us assume a field E = 3y 2 z3 ax + 6xyz3 ay + 9xy 2 z2 az V/m in free space, and also assume that point P (2, 1, 0) lies on a conducting surface. a) Find v just adjacent to the surface at P : v = D = Then at P , v = 0, since z = 0. b) Find s at P : s = D n P 0 E = 6xz3 + 18xy 2 z C/m3 = 0 En P Note however, that this computation involves evaluating E at the surface, yielding a value of 0. Therefore the surface charge density at P is 0. c) Show that V = -3xy 2 z3 V: The simplest way to show this is just to take -V , which yields the given field: A more general method involves deriving the potential from the given field: We write Ex = - Ey = - Ez = - V = 3y 2 z3 V = -3xy 2 z3 + f (y, z) x V = 6xyz3 V = -3xy 2 z3 + f (x, z) y V = 9xy 2 z2 V = -3xy 2 z3 + f (x, y) z where the integration "constants" are functions of all variables other than the integration variable. The general procedure is to adjust the functions, f , such that the result for V is the same in all three integrations. In this case we see that f (x, y) = f (x, z) = f (y, z) = 0 accomplishes this, and the potential function is V = -3xy 2 z3 as given. d) Determine VP Q , given Q(1, 1, 1): Using the potential function of part c, we have VP Q = VP - VQ = 0 - (-3) = 3 V 5.19. Let V = 20x 2 yz - 10z2 V in free space. a) Determine the equations of the equipotential surfaces on which V = 0 and 60 V: Setting the given potential function equal to 0 and 60 and simplifying results in: At 0 V : 2x 2 y - z = 0 At 60 V : 2x 2 y - z = 6 z b) Assume these are conducting surfaces and find the surface charge density at that point on the V = 60 V surface where x = 2 and z = 1. It is known that 0 V 60 V is the field-containing region: First, on the 60 V surface, we have 2x 2 y - z - 6 7 = 0 2(2)2 y(1) - 1 - 6 = 0 y = z 8 69 5.19b. (continued) Now E = -V = -40xyz ax - 20x 2 z ay - [20xy - 20z] az Then, at the given point, we have D(2, 7/8, 1) = 0 E(2, 7/8, 1) = - 0 [70 ax + 80 ay + 50 az ] C/m2 We know that since this is the higher potential surface, D must be directed away from it, and so the charge density would be positive. Thus s = D D = 10 0 72 + 82 + 52 = 1.04 nC/m2 c) Give the unit vector at this point that is normal to the conducting surface and directed toward the V = 0 surface: This will be in the direction of E and D as found in part b, or an = - 7ax + 8ay + 5az = -[0.60ax + 0.68ay + 0.43az ] 72 + 8 2 + 5 2 5.20. A conducting plane is located at z = 0 in free space, and a 20 nC point charge is present at Q(2, 4, 6). a) If V = 0 at z = 0, find V at P (5, 3, 1): The plane can be replaced by an image charge of -20 nC at Q (2, 4, -6). Vectors R and R directed from Q and Q to P are R = (5, 3, 1) - (2, 4, 6) = (3, -1, -5) and R = (5, 3, 1) - (2, 4, -6) = (3, -1, 7). Their magnitudes are R = 35 and R = 59. The potential at P is given by VP = b) Find E at P : EP = qR qR (20 10-9 )(3, -1, -5) (20 10-9 )(3, -1, 7) - = - 4 0 R 3 4 0 (R )3 4 0 (35)3/2 4 0 (59)3/2 7 20 10-9 1 1 5 (3ax - ay ) - az = - + 3/2 3/2 3/2 4 0 (35) (59) (59) (35)3/2 = 1.4ax - 0.47ay - 7.1az V/m q 4 0 R - q 4 0 R = 20 10-9 20 10-9 - = 7.0 V 4 0 35 4 0 59 c) Find s at A(5, 3, 0): First, find the electric field there: EA = 20 10-9 4 0 surf ace (5, 3, 0) - (2, 4, 6) (5, 3, 0) - (2, 4, -6) = -6.9az V/m - (46)3/2 (46)3/2 = -6.9 0 az az = -61 pC/m2 . Then s = D n 70 5.21. Let the surface y = 0 be a perfect conductor in free space. Two uniform infinite line charges of 30 nC/m each are located at x = 0, y = 1, and x = 0, y = 2. a) Let V = 0 at the plane y = 0, and find V at P (1, 2, 0): The line charges will image across the plane, producing image line charges of -30 nC/m each at x = 0, y = -1, and x = 0, y = -2. We find the potential at P by evaluating the work done in moving a unit positive charge from the y = 0 plane (we choose the origin) to P : For each line charge, this will be: VP - V0,0,0 = - l final distance from charge ln 2 0 initial distance from charge where V0,0,0 = 0. Considering the four charges, we thus have 1 l VP = - ln 2 0 2 = 2 + ln 1 17 - ln 2 30 10-9 17 10 17 = 10 + ln ln + ln 2 2 0 2 10 - ln 1 l 1 ln (2) + ln 2 0 2 = 1.20 kV b) Find E at P : Use EP = l (1, 2, 0) - (0, 1, 0) (1, 2, 0) - (0, 2, 0) + 2 0 |(1, 1, 0)|2 |(1, 0, 0)|2 (1, 2, 0) - (0, -1, 0) (1, 2, 0) - (0, -2, 0) - - |(1, 3, 0)|2 |(1, 4, 0)|2 (1, 1, 0) (1, 0, 0) (1, 3, 0) (1, 4, 0) l + - - = 723 ax - 18.9 ay V/m = 2 0 2 1 10 17 5.22. Let the plane x = 0 be a perfect conductor in free space. Locate a point charge of 4nC at P1 (7, 1, -2) and a point charge of -3nC at P2 (4, 2, 1). a) Find E at A(5, 0, 0): Image charges will be located at P1 (-7, 1, -2) (-4nC) and at P2 (-4, 2, 1) (3nC). Vectors from all four charges to point A are: R1 = (5, 0, 0) - (7, 1, -2) = (-2, -1, 2) R1 = (5, 0, 0) - (-7, 1, -2) = (12, -1, 2) R2 = (5, 0, 0) - (4, 2, 1) = (1, -2, -1) and R2 = (5, 0, 0) - (-4, 2, 1) = (9, -2, -1) Replacing the plane by the image charges enables the field at A to be calculated through: EA = 10-9 (4)(-2, -1, 2) (3)(1, -2, -1) (4)(12, -1, 2) (3)(9, -2, -1) - - + 4 0 93/2 63/2 (149)3/2 (86)3/2 = -4.43ax + 2.23ay + 4.42az V/m 71 5.22b. Find |s | at B(0, 0, 0) (note error in problem statement): First, E at the origin is done as per the setup in part a, except the vectors are directed from the charges to the origin: EB = 10-9 4 0 (4)(-7, -1, 2) (3)(-4, -2, -1) (4)(7, -1, 2) (3)(4, -2, -1) - - + (54)3/2 (21)3/2 (54)3/2 (21)3/2 Now s = D n|surf ace = D ax in our case (note the other components cancel anyway as they must, but we still need to express s as a scalar): sB = 0 EB ax = 10-9 4 (4)(-7) (3)(-4) (4)(7) (3)(4) = 8.62 pC/m2 - - + 3/2 3/2 3/2 (54) (21) (54) (21)3/2 5.23. A dipole with p = 0.1az C m is located at A(1, 0, 0) in free space, and the x = 0 plane is perfectlyconducting. a) Find V at P (2, 0, 1). We use the far-field potential for a z-directed dipole: V = p cos p z = 2 2 + y 2 + z2 ]1.5 4 0 r 4 0 [x The dipole at x = 1 will image in the plane to produce a second dipole of the opposite orientation at x = -1. The potential at any point is now: V = p 4 - 1)2 z z - 2 + z2 ]1.5 2 + y 2 + z2 ]1.5 +y [(x + 1) 0 [(x Substituting P (2, 0, 1), we find V = .1 106 4 0 1 1 = 289.5 V - 2 2 10 10 b) Find the equation of the 200-V equipotential surface in cartesian coordinates: We just set the potential exression of part a equal to 200 V to obtain: - 1)2 z z = 0.222 - 2 + z2 ]1.5 2 + y 2 + z2 ]1.5 [(x + 1) +y [(x 5.24. The mobilities for intrinsic silicon at a certain temperature are e = 0.14 m2 /V s and h = 0.035 m2 /V s. The concentration of both holes and electrons is 2.2 1016 m-3 . Determine both the conductivity and the resistivity of this silicon sample: Use = -e e + h h = (1.6 10-19 C)(2.2 1016 m-3 )(0.14 m2 /V s + 0.035 m2 /V s) = 6.2 10-4 S/m Conductivity is = 1/ = 1.6 103 m. 72 5.25. Electron and hole concentrations increase with temperature. For pure silicon, suitable expressions are h = -e = 6200T 1.5 e-7000/T C/m3 . The functional dependence of the mobilities on temperature is given by h = 2.3 105 T -2.7 m2 /V s and e = 2.1 105 T -2.5 m2 /V s, where the temperature, T , is in degrees Kelvin. The conductivity will thus be = -e e + h h = 6200T 1.5 e-7000/T 2.1 105 T -2.5 + 2.3 105 T -2.7 = 1.30 109 -7000/T 1 + 1.095T -.2 S/m e T Find at: a) 0 C: With T = 273 K, the expression evaluates as (0) = 4.7 10-5 S/m. b) 40 C: With T = 273 + 40 = 313, we obtain (40) = 1.1 10-3 S/m. c) 80 C: With T = 273 + 80 = 353, we obtain (80) = 1.2 10-2 S/m. 5.26. A little donor impurity, such as arsenic, is added to pure silicon so that the electron concentration is 2 1017 conduction electrons per cubic meter while the number of holes per cubic meter is only 1.11015 . If e = 0.15 m2 /V s for this sample, and h = 0.045 m2 /V s, determine the conductivity and resistivity: = -e e + h h = (1.6 10-19 ) (2 1017 )(0.15) + (1.1 1015 )(0.045) = 4.8 10-3 S/m Then = 1/ = 2.1 102 m. 5.27. Atomic hydrogen contains 5.5 102 5 atoms/m3 at a certain temperature and pressure. When an electric field of 4 kV/m is applied, each dipole formed by the electron and positive nucleus has an effective length of 7.1 10-19 m. a) Find P: With all identical dipoles, we have P = Nqd = (5.5 1025 )(1.602 10-19 )(7.1 10-19 ) = 6.26 10-12 C/m2 = 6.26 pC/m2 b) Find R: We use P = 0 e E, and so e = Then R P 6.26 10-12 = = 1.76 10-4 -12 )(4 103 ) (8.85 10 0E = 1 + e = 1.000176. 5.28. In a certain region where the relative permittivity is 2.4, D = 2ax - 4ay + 5az nC/m2 . Find: a) b) E= D = (2ax - 4ay + 5az ) 10-9 = 94ax - 188ay + 235az V/m (2.4)(8.85 10-12 ) 0E = 0 E( P =D- (2ax - 4ay + 5az ) 10-9 (2.4 - 1) R 2.4 = 1.2ax - 2.3ay + 2.9az nC/m2 - 1) = |V | = |E| = [(94.1)2 + (188)2 + (235)2 ]1/2 = 315 V/m 73 c) 5.29. A coaxial conductor has radii a = 0.8 mm and b = 3 mm and a polystyrene dielectric for which 2 R = 2.56. If P = (2/)a nC/m in the dielectric, find: a) D and E as functions of : Use E= Then D= 0E + P = P 0( R - 1) = (2/) 10-9 a 144.9 a V/m = -12 )(1.56) (8.85 10 2 10-9 a 3.28 10-9 a 3.28a 1 +1 = C/m2 = nC/m2 1.56 b) Find Vab and e : Use Vab = - e = r 0.8 3 3 144.9 d = 144.9 ln 0.8 = 192 V - 1 = 1.56, as found in part a. c) If there are 4 1019 molecules per cubic meter in the dielectric, find p(): Use p= (2 10-9 /) 5.0 10-29 P = a C m a = N 4 1019 = 2.1, find: 5.30. Given the potential field V = 200 - 50x + 20y V in a dielectric material for which a) E = -V = 50ax - 20ay V/m. b) D = E = (2.1)(8.85 10-12 )(50ax - 20ay ) = 930ax - 372ay pC/m2 . c) P = 0 E( R R - 1) = (8.85 10-12 )(50ax - 20ay )(1.1) = 487ax - 195ay pC/m2 . d) v = D = 0. e) b = - P = 0 f) T = 0E =0 R 5.31. The surface x = 0 separates two perfect dielectrics. For x > 0, let where x < 0. If E1 = 80ax - 60ay - 30az V/m, find: a) EN 1 : This will be E1 ax = 80 V/m. = R1 = 3, while R2 =5 b) ET 1 . This consists of components of E1 not normal to the surface, or ET 1 = -60ay - 30az V/m. c) ET 1 = d) E1 = (60)2 + (30)2 = 67.1 V/m. (80)2 + (60)2 + (30)2 = 104.4 V/m. e) The angle 1 between E1 and a normal to the surface: Use cos 1 = E1 ax 80 1 = 40.0 = E1 104.4 74 5.31 (continued) f) DN 2 = DN 1 = g) DT 2 = h) D2 = R1 0 EN 1 = 3(8.85 10-12 )(80) = 2.12 nC/m2 . R2 0 ET 1 = 5(8.85 10-12 )(67.1) = 2.97 nC/m2 . + R2 0 ET 1 R1 0 EN 1 ax 0 E2 = 2.12ax - 2.66ay - 1.33az nC/m2 . R2 )] i) P2 = D2 - = D2 [1 - (1/ = (4/5)D2 = 1.70ax - 2.13ay - 1.06az nC/m2 . j) the angle 2 between E2 and a normal to the surface: Use cos 2 = E2 ax D2 ax = = E2 D2 2.12 (2.12)2 = (2.66)2 + (1.33)2 = .581 Thus 2 = cos-1 (.581) = 54.5 . 5.32. In Fig. 5.18, let D = 3ax - 4ay + 5az nC/m2 and find: a) D2 : First, the electric field in region 1 is E1 = 4 5 3 ax - ay + az 10-9 V/m 2 0 2 0 2 0 Since, at the dielectric interface, tangential electric field and normal electric flux density are continuous, we may write D2 = R2 0 ET 1 + DN 1 = 5 3ax - 2 5 4ay + 5az = 7.5ax - 10ay + 5az nC/m2 2 b) DN 2 = 5az , as explained above. c) DT 2 = R2 0 ET 2 = R2 0 ET 1 = 7.5ax - 10ay nC/m2 . d) the energy density in each region: we1 = 1 2 1 2 R1 0 E1 E1 = 1 (2) 2 1 (5) 2 0 3 2 0 3 2 0 2 + 2 4 2 0 4 2 0 2 + 2 5 2 0 5 5 0 2 10-18 = 1.41 J/m3 2 we2 = R2 0 E2 E2 = 0 + + 10-18 = 2.04 J/m3 e) the angle that D2 makes with az : Use D2 az = |D2 | cos = Dz = 5. where |D2 | = 1/2 = 13.5. So = cos-1 (5/13.5) = 68 . (7.5)2 + (10)2 + (5)2 f) D2 /D1 = (7.5)2 + (10)2 + (5)2 1/2 / (3)2 + (4)2 + (5)2 1/2 = 1.91. g) P2 /P1 : First P1 = 0 E1 ( R1 - 1) = 1.5ax - 2ay + 2.5az nC/m2 . Then P2 = 0 E2 ( R2 - 1) = 6ax - 8ay + 4az nC/m2 . So P2 [(6)2 + (8)2 + (4)2 ]1/2 = = 3.04 P1 [(1.5)2 + (2)2 + (2.5)2 ]1/2 75 5.33. Two perfect dielectrics have relative permittivities R1 = 2 and R2 = 8. The planar interface between them is the surface x - y + 2z = 5. The origin lies in region 1. If E1 = 100ax + 200ay - 50az V/m, find E2 : We need to find the components of E1 that are normal and tangent to the boundary, and then apply the appropriate boundary conditions. The normal component will be EN 1 = E1 n. Taking f = x - y + 2z, the unit vector that is normal to the surface is n= 1 f = ax - ay + 2az |f | 6 This normal will point in the direction of increasing f , which will be away from the origin, or into region 2 (you can visualize a portion of the surface as a triangle whose vertices are on the three coordinate axes at x = 5, y = -5, and z = 2.5). So EN 1 = (1/ 6)[100 - 200 - 100] = -81.7 V/m. Since the magnitude is negative, the normal component points into region 1 from the surface. Then 1 EN 1 = -81.65 [ax - ay + 2az ] = -33.33ax + 33.33ay - 66.67az V/m 6 Now, the tangential component will be ET 1 = E1 - EN 1 = 133.3ax + 166.7ay + 16.67az Our boundary conditions state that ET 2 = ET 1 and EN 2 = ( R1 / R2 )EN 1 = (1/4)EN 1 . Thus 1 E2 = ET 2 + EN 2 = ET 1 + EN 1 = 133.3ax + 166.7ay + 16.67az - 8.3ax + 8.3ay - 16.67az 4 = 125ax + 175ay V/m 5.34. Let the spherical surfaces r = 4 cm and r = 9 cm be separated by two perfect dielectric shells, R1 = 2 for 4 < r < 6 cm and R2 = 5 for 6 < r < 9 cm. If E1 = (2000/r 2 )ar V/m, find: a) E2 : Since E is normal to the interface between R1 and R2 , D will be continuous across the boundary, and so 2 0 (2000) ar = D2 D1 = r2 Then D2 2 2000 800 E2 = = ar = 2 ar V/m 2 5 0 5 r r b) the total electrostatic energy stored in each region: In region 1, the energy density is we1 = In region 2: we2 = 1 2 R2 2 0 |E2 | = 1 2 2 R1 0 |E1 | = (2000)2 1 (2) 0 J/m3 2 r4 (800)2 1 (5) 0 J/m3 2 r4 76 5.34. (continued) The energies in each region are then Region 1 : We1 = (2000)2 2 0 0 0 .06 .04 1 2 r sin dr d d r2 = 4 0 (2000)2 We2 = (800)2 5 2 1 1 - = 3.7 mJ .04 .06 2 0 .09 .06 Region 2 : 0 0 1 2 r sin dr d d r2 = 4 0 (800)2 5 2 1 1 - = 0.99 mJ .06 .09 5.35. Let the cylindrical surfaces = 4 cm and = 9 cm enclose two wedges of perfect dielectrics, R1 = 2 for 0 < < /2, and R2 = 5 for /2 < < 2. If E1 = (2000/)a V/m, find: a) E2 : The interfaces between the two media will lie on planes of constant , to which E1 is parallel. Thus the field is the same on either side of the boundaries, and so E2 = E1 . b) the total electrostatic energy stored in a 1m length of each region: In general we have wE = (1/2) R 0 E 2 . So in region 1: WE1 = 1 0 0 /2 4 9 1 (2000)2 d d dz = (2) 0 2 2 2 2 0 (2000) ln 9 4 = 45.1 J In region 2, we have WE2 = 1 0 2 /2 4 9 1 (2000)2 15 (5) 0 d d dz = 2 2 4 0 (2000) 2 ln 9 4 = 338 J 5.36. Let S = 120 cm2 , d = 4 mm, and R = 12 for a parallel-plate capacitor. a) Calculate the capacitance: C = R 0 S/d = [12 0 (120 10-4 )]/[4 10-3 ] = 3.19 10-10 = 319 pF. b) After connecting a 40 V battery across the capacitor, calculate E, D, Q, and the total stored electrostatic energy: E = V /d = 40/(4 10-3 ) = 104 V/m. D = R 0 E = 12 0 104 = 1.06 C/m2 . Then Q = D n|surf ace S = 1.06 10-6 (120 10-4 ) = 1.27 10-8 C = 2 12.7 nC. Finally We = (1/2)CV0 = (1/2)(319 10-12 )(40)2 = 255 nJ. c) The source is now removed and the dielectric is carefully withdrawn from between the plates. Again calculate E, D, Q, and the energy: With the source disconnected, the charge is constant, and thus so is D: Therefore, Q = 12.7 nC, D = 1.06 C/m2 , and E = D/ 0 = 104 /8.85 10-12 = 1.2 105 V/m. The energy is then We = 1 1 D E S = (1.06 10-6 )(1.2 105 )(120 10-4 )(4 10-3 ) = 3.05 J 2 2 d) What is the voltage between the plates? V = E d = (1.2 105 )(4 10-3 ) = 480 V. 77 5.37. Capacitors tend to be more expensive as their capacitance and maximum voltage, Vmax , increase. The voltage Vmax is limited by the field strength at which the dielectric breaks down, EBD . Which of these dielectrics will give the largest CVmax product for equal plate areas: (a) air: R = 1, EBD = 3 MV/m; (b) barium titanate: R = 1200, EBD = 3 MV/m; (c) silicon dioxide: R = 3.78, EBD = 16 MV/m; (d) polyethylene: R = 2.26, EBD = 4.7 MV/m? Note that Vmax = EBDd, where d is the plate separation. Also, C = R 0 A/d, and so Vmax C = R 0 AEBD , where A is the plate area. The maximum CVmax product is found through the maximum R EBD product. Trying this with the given materials yields the winner, which is barium titanate. 5.38. A dielectric circular cylinder used between the plates of a capacitor has a thickness of 0.2 mm and a radius of 1.4 cm. The dielectric properties are R = 400 and = 10-5 S/m. a) Calculate C: C= R 0S d = (400)(8.854 10-12 )(1.4 10-2 )2 = 1.09 10-8 = 10.9 nF 2 10-4 b) Find the quality factor QQF (QQ F = RC) of the capacitor at f = 10 kHz: Use the relation RC = / to write QQF = RC = 2f = (2 104 )(400)(8.854 10-12 ) = 22.3 10-5 c) If the maximum field strength permitted in the dielectric is 2 MV/m, what is the maximum permissible voltage across the capacitor? Vmax = EBD d = (2 106 )(2 10-4 ) = 400 V. d) What energy is stored when this voltage is applied? We,max = 1 1 2 CVmax = (10.9 10-9 )(400)2 = 8.7 10-4 = 0.87 mJ 2 2 5.39. A parallel plate capacitor is filled with a nonuniform dielectric characterized by R = 2 + 2 106 x 2 , where x is the distance from one plate. If S = 0.02 m2 , and d = 1 mm, find C: Start by assuming charge density s on the top plate. D will, as usual, be x-directed, originating at the top plate and terminating on the bottom plate. The key here is that D will be constant over the distance between plates. This can be understood by considering the x-varying dielectric as constructed of many thin layers, each having constant permittivity. The permittivity changes from layer to layer to approximate the given function of x. The approximation becomes exact as the layer thicknesses approach zero. We know that D, which is normal to the layers, will be continuous across each boundary, and so D is constant over the plate separation distance, and will be given in magnitude by s . The electric field magnitude is now s D = E= 6 2 0 R 0 (2 + 2 10 x ) The voltage beween plates is then 10-3 6 10-3 s dx 1 1 s s -1 x 4 10 tan = V0 = = 6x2) 3 4 0 2 0 (2 + 2 10 0 0 2 10 4 106 0 Now Q = s (.02), and so C= s (.02) 0 (2 103 )(4) Q = = 4.51 10-10 F = 451 pF V0 s 78 5.40a. The width of the region containing R1 in Fig. 5.19 is 1.2 m. Find R1 if R2 = 2.5 and the total capacitance is 60 nF: The plate areas associated with each capacitor are A1 = 1.2(2) = 2.4 m2 and A2 = 0.8(2) = 1.6 m2 . Having parallel capacitors, the capacitances will add, so C = C1 + C2 60 10-9 = Solve this to obtain R1 2.5 0 (1.6) R1 0 (2.4) + 2 10-3 2 10-3 =3 and C1 = 2C2 : = 4.0. R2 R1 , b) Find the width of each region (containing R1 and R2 ) if Ctotal = 80 nF, Let w1 be the width of region 1. The above conditions enable us to write: R1 0 w1 (2) 2 10-3 =2 3 R1 0 (2 - w1 )(2) 2 10-3 w1 = 6(2 - w1 ) So that w1 = 12/7 = 1.7 m and w2 = 0.3 m. 5.41. Let R1 = 2.5 for 0 < y < 1 mm, R2 = 4 for 1 < y < 3 mm, and R3 for 3 < y < 5 mm. Conducting surfaces are present at y = 0 and y = 5 mm. Calculate the capacitance per square meter of surface area if: a) R3 is that of air; b) R3 = R1 ; c) R3 = R2 ; d) R3 is silver: The combination will be three capacitors in series, for which 1 1 1 1 = + + = C C1 C2 C3 So that C= d1 R1 0 (1) + d2 R2 0 (1) + d3 R3 0 (1) = 10-3 0 2 2 1 + + 2.5 4 R3 (5 10-3 ) 10 + 4.5 0 R3 R3 Evaluating this for the four cases, we find a) C = 3.05 nF for R3 = 1, b) C = 5.21 nF for R3 = 2.5, c) C = 6.32 nF for R3 = 4, and d) C = 9.83 nF if silver (taken as a perfect conductor) forms region 3; this has the effect of removing the term involving R3 from the original formula (first equation line), or equivalently, allowing R3 to approach infinity. 5.42. Cylindrical conducting surfaces are located at = 0.8 cm and 3.6 cm. The region 0.8 < < a contains a dielectric for which R = 4, while R = 2 for a < < 3.6. a) Find a so that the voltage across each dielectric layer is the same: Assuming charge density s on the inner cylinder, we have D = s (0.8)/ a , which gives E(0.8 < < a) = (0.8s )/(4 0 )a and E(a < < 3.6) = (0.8s )/(2 0 )a . The voltage between conductors is now V0 = - We require ln 3.6 a = 1 a ln 2 0.8 3.6 = a a a = 2.2 cm 0.8 a 3.6 0.8s d - 2 0 0.8 a 0.8s 0.8s 3.6 d = ln 4 0 2 0 a + 1 a ln 2 0.8 b) Find the total capacitance per meter: Using the part a result, have V0 = 3.6 0.8s ln 2 0 2.2 + 1 2.2 ln 2 0.8 = 0.4s 0 79 5.42b. (continued) The charge on a unit length of the inner conductor is Q = 2(0.8)(1)s . The capacitance is now 2(0.8)(1)s Q = = 4 0 = 111 pF/m C= V0 0.4s / 0 Note that throughout this problem, I left all dimensions in cm, knowing that all cm units would cancel, leaving the units of capacitance to be those used for 0 . 5.43. Two coaxial conducting cylinders of radius 2 cm and 4 cm have a length of 1m. The region between the cylinders contains a layer of dielectric from = c to = d with R = 4. Find the capacitance if a) c = 2 cm, d = 3 cm: This is two capacitors in series, and so 1 1 1 1 = + = C C1 C2 2 3 1 ln 4 2 + ln 4 3 C = 143 pF 0 b) d = 4 cm, and the volume of the dielectric is the same as in part a: Having equal volumes requires that 32 - 22 = 42 - c2 , from which c = 3.32 cm. Now 1 1 1 1 + = = C C1 C2 2 ln 0 3.32 2 + 1 4 ln 4 3.32 C = 101 pF 5.44. Conducting cylinders lie at = 3 and = 12 mm; both extend from z = 0 to z = 1 m. Perfect dielectrics occupy the interior region: R = 1 for 3 < < 6 mm, R = 4 for 6 < < 9 mm, and R = 8 for 9 < < 12 mm. a) Calculate C: First we know that D = (3s /)a C/m2 , with expressed in mm. Then, with in mm, 3s a V/m (3 < < 6) E1 = 0 E2 = and E3 = 3s a V/m (6 < < 9) 4 0 3s a V/m (9 < < 12) 8 0 The voltage between conductors will be: V0 = - 6 3 3 3 3s s s d - d - d 10-3 (m/mm) 8 0 4 0 0 12 9 6 .003s 1 12 1 9 6 .003s (0.830) V = ln + ln + ln = 8 9 4 6 3 0 0 9 Now, the charge on the 1 m length of the inner conductor is Q = 2(.003)(1)s . The capacitance is then Q 2(.003)(1)s 2 0 C= = 67 pF = = V0 (.003)s (.830)/ 0 .830 80 5.44b. If the voltage between the cylinders is 100 V, plot |E | vs. : Have Q = CV0 = (67 10-12 )(100) = 6.7nC. Then s = 6.7 10-9 = 355 nC/m2 2(.003)(1) Then, using the electric field expressions from part a, we find E1 = 3 355 10-9 120 12 104 V/m = kV/m (3 < < 6) = -12 8.854 10 where is expressed in mm. Similarly, we find E2 = E1 /4 = 30/ kV/m (6 < < 9) and E3 = E1 /8 = 15 kV/m (9 < < 12). These fields are plotted below. 81 5.45. Two conducting spherical shells have radii a = 3 cm and b = 6 cm. The interior is a perfect dielectric for which R = 8. a) Find C: For a spherical capacitor, we know that: C= 4 1 a R 0 1 -b = 4(8) 1 3 0 - 1 6 (100) = 1.92 0 = 53.3 pF b) A portion of the dielectric is now removed so that R = 1.0, 0 < < /2, and R = 8, /2 < < 2. Again, find C: We recognize here that removing that portion leaves us with two capacitors in parallel (whose C's will add). We use the fact that with the dielectric completely removed, the capacitance would be C( R = 1) = 53.3/8 = 6.67 pF. With one-fourth the dielectric removed, the total capacitance will be C= 3 1 (6.67) + (53.4) = 41.7 pF 4 4 5.46. (see Problem 5.44). 5.47. With reference to Fig. 5.17, let b = 6 m, h = 15 m, and the conductor potential be 250 V. Take = Find values for K1 , L , a, and C: We have K1 = We then have L = Next, a = h2 - b2 = h+ h2 + b2 b 2 0. = 15 + (15)2 + (6)2 6 2 = 23.0 4 0 V0 4 0 (250) = 8.87 nC/m = ln K1 ln(23) (15)2 - (6)2 = 13.8 m. Finally, C= 2 0 2 = = 35.5 pF -1 (h/b) cosh cosh-1 (15/6) 82 5.48. A potential function in free space is given by V = -20 + 10 ln (5 + y)2 + x 2 (5 - y)2 + x 2 a) Describe the 0-V equipotential surface: Setting the given expression equal to zero, we find (5 + y)2 + x 2 = e2 = 7.39 (5 - y)2 + x 2 So 6.39x 2 + 6.39y 2 - 83.9y + 160 = 0. Completing the square in the y trinomial leads to x 2 + (y - 6.56)2 = 18.1 = (4.25)2 , which we recognize as a right circular cylinder whose axis is located at x = 0, y = 6.56, and whose radius is 4.25. b) Describe the 10-V equipotential surface: In this case, the given expression is set equal to ten, leading to (5 + y)2 + x 2 = e3 = 20.1 (5 - y)2 + x 2 So 19.1x 2 + 19.1y 2 - 211y + 477 = 0. Following the same procedure as in part a, this becomes x 2 + (y - 5.52)2 = 5.51 = (2.35)2 , which we recognize again as a right circular cylinder with axis at x = 0, y = 5.52, and of radius 2.35. 5.49. A 2 cm diameter conductor is suspended in air with its axis 5 cm from a conducting plane. Let the potential of the cylinder be 100 V and that of the plane be 0 V. Find the surface charge density on the: a) cylinder at a point nearest the plane: The cylinder will image across the plane, producing an equivalent two-cylinder problem, with the second one at location 5 cm below the plane. We will take the plane as the zy plane, with the cylinder positions at x = 5. Now b = 1 cm, h = 5 cm, and V0 = 100 V. Thus a = h2 - b2 = 4.90 cm. Then K1 = [(h + a)/b]2 = 98.0, and L = (4 0 V0 )/ ln K1 = 2.43 nC/m. Now D= and s, max = D (-ax ) x=h-b,y=0 0E =- L 2 (x + a)ax + yay (x - a)ax + yay - 2 + y2 (x + a) (x - a)2 + y 2 = L 2 h-b+a h-b-a = 473 nC/m2 - (h - b + a)2 (h - b - a)2 b) plane at a point nearest the cylinder: At x = y = 0, D(0, 0) = - from which s = D(0, 0) ax = - L 2 aax -aax - a2 a2 =- L 2 ax 2 a L = -15.8 nC/m2 a 83 CHAPTER 6. 6.1 Construct a curvilinear square map for a coaxial capacitor of 3-cm inner radius and 8-cm outer radius. These dimensions are suitable for the drawing. a) Use your sketch to calculate the capacitance per meter length, assuming R = 1: The sketch is shown below. Note that only a 9 sector was drawn, since this would then be duplicated 40 times around the circumference to complete the drawing. The capacitance is thus . C= 0 NQ = NV 0 40 = 59 pF/m 6 b) Calculate an exact value for the capacitance per unit length: This will be C= 2 0 = 57 pF/m ln(8/3) 84 6.2 Construct a curvilinear-square map of the potential field about two parallel circular cylinders, each of 2.5 cm radius, separated by a center-to-center distance of 13cm. These dimensions are suitable for the actual sketch if symmetry is considered. As a check, compute the capacitance per meter both from your sketch and from the exact formula. Assume R = 1. Symmetry allows us to plot the field lines and equipotentials over just the first quadrant, as is done in the sketch below (shown to one-half scale). The capacitance is found from the formula C = (NQ /NV ) 0 , where NQ is twice the number of squares around the perimeter of the half-circle and NV is twice the number of squares between the half-circle and the left vertical plane. The result is C= NQ NV 0 = 32 16 0 =2 0 = 17.7 pF/m We check this result with that using the exact formula: C= 0 -1 (d/2a) cosh = 0 -1 (13/5) cosh = 1.95 0 = 17.3 pF/m 85 6.3. Construct a curvilinear square map of the potential field between two parallel circular cylinders, one of 4-cm radius inside one of 8-cm radius. The two axes are displaced by 2.5 cm. These dimensions are suitable for the drawing. As a check on the accuracy, compute the capacitance per meter from the sketch and from the exact expression: C= cosh-1 (a 2 2 + b2 - D 2 )/(2ab) where a and b are the conductor radii and D is the axis separation. The drawing is shown below. Use of the exact expression above yields a capacitance value of C = 11.5 0 F/m. Use of the drawing produces: . 22 2 C= 4 0 = 11 0 F/m 86 6.4. A solid conducting cylinder of 4-cm radius is centered within a rectangular conducting cylinder with a 12-cm by 20-cm cross-section. a) Make a full-size sketch of one quadrant of this configuration and construct a curvilinear-square map for its interior: The result below could still be improved a little, but is nevertheless sufficient for a reasonable capacitance estimate. Note that the five-sided region in the upper right corner has been partially subdivided (dashed line) in anticipation of how it would look when the next-level subdivision is done (doubling the number of field lines and equipotentials). b) Assume = 0 and estimate C per meter length: In this case NQ is the number of squares around the full perimeter of the circular conductor, or four times the number of squares shown in the drawing. NV is the number of squares between the circle and the rectangle, or 5. The capacitance is estimated to be NQ 4 13 . C= 0 = 0 = 10.4 0 = 90 pF/m NV 5 87 6.5. The inner conductor of the transmission line shown in Fig. 6.12 has a square cross-section 2a 2a, while the outer square is 5a 5a. The axes are displaced as shown. (a) Construct a good-sized drawing of the transmission line, say with a = 2.5 cm, and then prepare a curvilinear-square plot of the electrostatic field between the conductors. (b) Use the map to calculate the capacitance per meter length if = 1.6 0 . (c) How would your result to part b change if a = 0.6 cm? a) The plot is shown below. Some improvement is possible, depending on how much time one wishes to spend. b) From the plot, the capacitance is found to be . 16 2 C= (1.6) 4 0 = 12.8 0 . = 110 pF/m c) If a is changed, the result of part b would not change, since all dimensions retain the same relative scale. 88 6.6. Let the inner conductor of the transmission line shown in Fig. 6.12 be at a potential of 100V, while the outer is at zero potential. Construct a grid, 0.5a on a side, and use iteration to find V at a point that is a units above the upper right corner of the inner conductor. Work to the nearest volt: The drawing is shown below, and we identify the requested voltage as 38 V. 89 6.7. Use the iteration method to estimate the potentials at points x and y in the triangular trough of Fig. 6.13. Work only to the nearest volt: The result is shown below. The mirror image of the values shown occur at the points on the other side of the line of symmetry (dashed line). Note that Vx = 78 V and Vy = 26 V. 90 6.8. Use iteration methods to estimate the potential at point x in the trough shown in Fig. 6.14. Working to the nearest volt is sufficient. The result is shown below, where we identify the voltage at x to be 40 V. Note that the potentials in the gaps are 50 V. 6.9. Using the grid indicated in Fig. 6.15, work to the nearest volt to estimate the potential at point A: The voltages at the grid points are shown below, where VA is found to be 19 V. Half the figure is drawn since mirror images of all values occur across the line of symmetry (dashed line). 91 6.10. Conductors having boundaries that are curved or skewed usually do not permit every grid point to coincide with the actual boundary. Figure 6.16a illustrates the situation where the potential at V0 is to be estimated in terms of V1 , V2 , V3 , and V4 , and the unequal distances h1 , h2 , h3 , and h4 . a) Show that . V0 = + V1 1+ 1+ h1 h3 h4 h2 h1 h3 h4 h2 h4 h2 h3 h1 1+ V4 1+ + V2 1+ h2 h4 1+ h2 h4 h1 h3 + V3 1+ h3 h1 1+ h1 h3 h4 h2 note error, corrected here, in the equation (second term) Referring to the figure, we write: V x Then 2V x 2 2V1 2V3 2V0 . (V1 - V0 )/ h1 - (V0 - V3 )/ h3 = = + - (h1 + h3 )/2 h1 (h1 + h3 ) h3 (h1 + h3 ) h1 h3 . V1 - V0 = h1 V x . V0 - V3 = h3 M1 M3 V0 We perform the same procedure along the y axis to obtain: 2V y 2 2V2 2V4 2V0 . (V2 - V0 )/ h2 - (V0 - V4 )/ h4 = = + - (h2 + h4 )/2 h2 (h2 + h4 ) h4 (h2 + h4 ) h2 h4 V0 Then, knowing that 2V x 2 V0 + 2V y 2 V0 =0 the two equations for the second derivatives are added to give 2V1 2V2 2V3 2V4 + + + = V0 h1 (h1 + h3 ) h2 (h2 + h4 ) h3 (h1 + h3 ) h4 (h2 + h4 ) Solve for V0 to obtain the given equation. b) Determine V0 in Fig. 6.16b: Referring to the figure, we note that h1 = h2 = a. The other two distances are found by writing equations for the circles: (0.5a + h3 )2 + a 2 = (1.5a)2 and (a + h4 )2 + (0.5a)2 = (1.5a)2 These are solved to find h3 = 0.618a and h4 = 0.414a. The four distances and potentials are now substituted into the given equation: . V0 = + 80 1+ 1 .618 h1 h3 + h2 h4 h1 h2 h3 h4 1+ 100 .618 .414 .414 .618 + 60 1+ 1 .414 1+ .414 .618 + 100 (1 + .618) 1 + .618 .414 (1 + .414) 1 + = 90 V 92 6.11. Consider the configuration of conductors and potentials shown in Fig. 6.17. Using the method described in Problem 10, write an expression for Vx (not V0 ): The result is shown below, where Vx = 70 V. 6.12a) After estimating potentials for the configuation of Fig. 6.18, use the iteration method with a square grid 1 cm on a side to find better estimates at the seven grid points. Work to the nearest volt: 25 0 0 0 0 50 48 42 19 0 75 50 25 0 0 0 0 100 48 100 42 34 0 19 0 b) Construct a 0.5 cm grid, establish new rough estimates, and then use the iteration method on the 0.5 cm grid. Again, work to the nearest volt: The result is shown below, with values for the original grid points underlined: 25 0 0 0 0 0 0 0 0 50 32 26 23 20 15 10 5 0 50 50 48 45 40 30 19 9 0 50 68 72 70 64 44 26 12 0 75 50 50 50 48 45 40 30 19 9 0 50 32 26 23 20 15 10 5 0 25 0 0 0 0 0 0 0 0 100 68 100 72 100 70 100 64 54 30 14 0 44 26 12 0 93 6.12c. Use the computer to obtain values for a 0.25 cm grid. Work to the nearest 0.1 V: Values for the left half of the configuration are shown in the table below. Values along the vertical line of symmetry are included, and the original grid values are underlined. 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 50 26.5 18.0 14.5 12.8 11.7 10.8 10.0 9.0 7.9 6.8 5.6 4.4 3.3 2.2 1.1 0 50 38.0 31.0 27.1 24.8 23.1 21.6 20.0 18.1 15.9 13.6 11.2 8.8 6.6 4.4 2.2 0 50 44.6 40.7 38.1 36.2 34.4 32.5 30.2 27.4 24.0 20.4 16.8 13.2 9.8 6.4 3.2 0 50 49.6 49.0 48.3 47.3 45.8 43.8 40.9 37.1 32.4 27.3 22.2 17.4 12.8 8.4 4.2 0 50 54.6 57.5 58.8 58.8 57.8 55.8 52.5 47.6 41.2 34.2 27.4 21.2 15.4 10.0 5.0 0 50 61.4 67.7 70.6 71.4 70.8 69.0 65.6 59.7 50.4 40.7 32.0 24.4 17.6 11.4 5.6 0 50 73.2 81.3 84.3 85.2 85.0 83.8 81.2 75.2 59.8 46.3 35.4 26.6 19.0 12.2 6.0 0 75 100 100 100 100 100 100 100 100 67.2 49.2 36.8 27.4 19.5 12.5 6.1 0 94 6.13. Perfectly-conducting concentric spheres have radii of 2 and 6 cm. The region 2 < r < 3 cm is filled with a solid conducting material for which = 100 S/m, while the portion for which 3 < r < 6 cm has = 25 S/m. The inner sphere is held at 1 V while the outer is at V = 0. a. Find E and J everywhere: From symmetry, E and J will be radially-directed, and we note the fact that the current, I , must be constant at any cross-section; i.e., through any spherical surface at radius r between the spheres. Thus we require that in both regions, J= The fields will thus be E1 = I I a (2 < r < 3) and E2 = ar (3 < r < 6) 2 r 41 r 42 r 2 I ar 4r 2 where 1 = 100 S/m and 2 = 25 S/m. Since we know the voltage between spheres (1V), we can find the value of I through: 1V = - and so I= .03 .06 I dr - 42 r 2 .02 .03 I I dr = 2 41 r 0.24 1 1 + 1 2 0.24 = 15.08 A (1/1 + 1/2 ) Then finally, with I = 15.08 A substituted into the field expressions above, we find E1 = and E2 = The current density is now J = 1 E1 = 2 E2 = 1.2 A/m (2 < r < 6) r2 .012 ar V/m (2 < r < 3) r2 .048 ar V/m (3 < r < 6) r2 b) What resistance would be measured between the two spheres? We use R= V 1V = = 6.63 10-2 I 15.08 A c) What is V at r = 3 cm? This we find through V =- .03 .06 1 .048 1 - dr = .048 2 r .03 .06 = 0.8 V 95 6.14. The cross-section of the transmission line shown in Fig. 6.12 is drawn on a sheet of conducting paper with metallic paint. The sheet resistance is 2000 /sq and the dimension a is 2 cm. a) Assuming a result for Prob. 6b of 110 pF/m, what total resistance would be measured between the metallic conductors drawn on the conducting paper? We assume a paper thickness of t m, so that the capacitance is C = 110t pF, and the surface resistance is Rs = 1/( t) = 2000 /sq. We now use RC = R= C = Rs t (1.6 8.854 10-12 )(2000) = = 257.6 110 10-12 t 110 10-12 b) What would the total resistance be if a = 2 cm? The result is independent of a, provided the proportions are maintained. So again, R = 257.6 . 6.15. two concentric annular rings are painted on a sheet of conducting paper with a highly conducting metal paint. The four radii are 1, 1.2, 3.5, and 3.7 cm. Connections made to the two rings show a resistance of 215 ohms between them. a) What is Rs for the conducting paper? Using the two radii (1.2 and 3.5 cm) at which the rings are at their closest separation, we first evaluate the capacitance: C= 2 0 t = 5.19 10-11 t F ln(3.5/1.2) where t is the unknown paper coating thickness. Now use RC = Thus Rs = 0 R= 8.85 10-12 = 215 5.19 10-11 t 1 (51.9)(215) = = 1.26 k /sq t 8.85 b) If the conductivity of the material used as the surface of the paper is 2 S/m, what is the thickness of the coating? We use t= 1 1 = = 3.97 10-4 m = 0.397 mm Rs 2 1.26 103 96 6.16. The square washer shown in Fig. 6.19 is 2.4 mm thick and has outer dimensions of 2.5 2.5 cm and inner dimensions of 1.25 1.25 cm. The inside and outside surfaces are perfectly-conducting. If the material has a conductivity of 6 S/m, estimate the resistance offered between the inner and outer surfaces (shown shaded in Fig. 6.19). A few curvilinear squares are suggested: First we find the surface resistance, Rs = 1/( t) = 1/(6 2.4 10-3 ) = 69.4 /sq. Having found this, we can construct the total resistance by using the fundamental square as a building block. Specifically, R = Rs (Nl /Nw ) where Nl is the number of squares between the inner and outer surfaces and Nw is the number of squares around the perimeter of the washer. These numbers are found from the curvilinear square plot shown . . below, which covers one-eighth the washer. The resistance is thus R = 69.4[4/(8 5)] = 6.9 . 6.17. A two-wire transmission line consists of two parallel perfectly-conducting cylinders, each having a radius of 0.2 mm, separated by center-to-center distance of 2 mm. The medium surrounding the wires has R = 3 and = 1.5 mS/m. A 100-V battery is connected between the wires. Calculate: a) the magnitude of the charge per meter length on each wire: Use C= cosh-1 (h/b) = 3 8.85 10-12 = 3.64 10-9 C/m cosh-1 (1/0.2) Then the charge per unit length will be Q = CV0 = (3.64 10-11 )(100) = 3.64 10-9 C/m = 3.64 nC/m b) the battery current: Use RC = Then I= R= 3 8.85 10-12 = 486 (1.5 10-3 )(3.64 10-11 ) V0 100 = = 0.206 A = 206 mA R 486 97 6.18. A coaxial transmission line is modelled by the use of a rubber sheet having horizontal dimensions that are 100 times those of the actual line. Let the radial coordinate of the model be m . For the line itself, let the radial dimension be designated by as usual; also, let a = 0.6 mm and b = 4.8 mm. The model is 8 cm in height at the inner conductor and zero at the outer. If the potential of the inner conductor is 100 V: a) Find the expression for V (): Assuming charge density s on the inner conductor, we use Gauss' Law to find 2D = 2as , from which E = D/ = as /( ) in the radial direction. The potential difference between inner and outer conductors is Vab = V0 = - from which s = a b as b as d = ln a V0 V0 E= a ln(b/a) ln(b/a) Now, as a function of radius, and assuming zero potential on the outer conductor, the potential function will be: V () = - b ln(.0048/) .0048 V0 ln(b/) d = V0 = 100 = 48.1 ln ln(b/a) ln(b/a) ln(.0048/.0006) V b) Write the model height as a function of m (not ): We use the part a result, since the gravitational function must be the same as that for the electric potential. We replace V0 by the maximum height, and multiply all dimensions by 100 to obtain: h(m ) = 0.08 .48 ln(.48/m ) = 0.038 ln ln(.48/.06) m m 98 CHAPTER 7 7.1. Let V = 2xy 2 z3 and = 0 . Given point P (1, 2, -1), find: a) V at P : Substituting the coordinates into V , find VP = -8 V. b) E at P : We use E = -V = -2y 2 z3 ax - 4xyz3 ay - 6xy 2 z2 az , which, when evaluated at P , becomes EP = 8ax + 8ay - 24az V/m c) v at P : This is v = D = - 0 2 V = -4xz(z2 + 3y 2 ) C/m3 d) the equation of the equipotential surface passing through P : At P , we know V = -8 V, so the equation will be xy 2 z3 = -4. e) the equation of the streamline passing through P : First, Ey 4xyz3 dy 2x = = = 2 z3 Ex dx 2y y Thus 1 ydy = 2xdx, and so y 2 = x 2 + C1 2 Evaluating at P , we find C1 = 1. Next, Ez 6xy 2 z2 dz 3x = = = 2 z3 Ex dx 2y z Thus 3 1 3xdx = zdz, and so x 2 = z2 + C2 2 2 Evaluating at P , we find C2 = 1. The streamline is now specified by the equations: y 2 - 2x 2 = 2 and 3x 2 - z2 = 2 f) Does V satisfy Laplace's equation? No, since the charge density is not zero. 7.2. A potential field V exists in a region where = f (x). Find 2 V if v = 0. First, D = (x)E = -f (x)V . Then D = v = 0 = (-f (x)V ). So 0 = (-f (x)V ) = - =- Therefore, 2V = - df V + f (x) 2 V dx x 1 df V f (x) dx x df V 2V + f (x) 2 dx x x f (x) 2V 2V + f (x) 2 y 2 z 99 7.3. Let V (x, y) = 4e2x + f (x) - 3y 2 in a region of free space where v = 0. It is known that both Ex and V are zero at the origin. Find f (x) and V (x, y): Since v = 0, we know that 2 V = 0, and so 2V = Therefore 2V 2V d 2f + = 16e2x + -6=0 x 2 y 2 dx 2 d 2f df = -8e2x + 6x + C1 = -16e2x + 6 2 dx dx Ex = V df = 8e2x + x dx df dx Now and at the origin, this becomes Ex (0) = 8 + = 0(as given) x=0 Thus df/dx |x=0 = -8, and so it follows that C1 = 0. Integrating again, we find f (x, y) = -4e2x + 3x 2 + C2 which at the origin becomes f (0, 0) = -4 + C2 . However, V (0, 0) = 0 = 4 + f (0, 0). So f (0, 0) = -4 and C2 = 0. Finally, f (x, y) = -4e2x + 3x 2 , and V (x, y) = 4e2x -4e2x +3x 2 -3y 2 = 3(x 2 - y 2 ). 7.4. Given the potential field V = A ln tan2 (/2) + B: a) Show that 2 V = 0: Since V is a function only of , 2V = where dV d A 2A d = A ln tan2 (/2) + B = = (2A ln tan(/2)) = d d d sin(/2) cos(/2) sin Then 2V = 1 r 2 sin ) d d sin 2A sin =0 1 r 2 sin ) d d sin dV d b) Select A and B so that V = 100 V and E = 500 V/m at P (r = 5, = 60 , = 45 ): First, E = -V = - So A = -1082.5 V. Then VP = -(1082.5) ln tan2 (30 ) + B = 100 B = -1089.3 V Summarizing, V ( ) = -1082.5 ln tan2 (/2) - 1089.3. 100 1 dV 2A 2A =- =- = -0.462A = 500 r d r sin 5 sin 60 7.5. Given the potential field V = (A 4 + B -4 ) sin 4: a) Show that 2 V = 0: In cylindrical coordinates, 2V = V 1 2V 1 + 2 2 1 1 (4A 3 - 4B -5 ) sin 4 - 2 16(A 4 + B -4 ) sin 4 = 16 16 (A 3 + B -5 ) sin 4 - 2 (A 4 + B -4 ) sin 4 = 0 = b) Select A and B so that V = 100 V and |E| = 500 V/m at P ( = 1, = 22.5 , z = 2): First, E = -V = - 1 V V a - a = -4 (A 3 - B -5 ) sin 4 a + (A 3 + B -5 ) cos 4 a and at P , EP = -4(A - B) a . Thus |EP | = 4(A - B). Also, VP = A + B. Our two equations are: 4(A - B) = 500 and A + B = 100 We thus have two pairs of values for A and B: A = 112.5, B = -12.5 or A = -12.5, B = 112.5 7.6. If V = 20 sin /r 3 V in free space, find: a) v at P (r = 2, = 30 , = 0): We use Poisson's equation in free space, 2 V = -v / 0 , where, with no variation: 2V = Substituting: 2V = 1 r 2 r 1 = 2 r r -r 2 20 cos 60 sin 1 sin + 2 r4 r sin r3 -60 sin 1 10 sin 2 + 2 2 r r sin r3 1 r 2 r r2 V r + 1 r 2 sin sin V = So vP = - 0 20(4 sin2 + 1) v 20 cos 2 120 sin = =- + 5 5 5 sin r r sin r 0 20(4 sin2 + 1) r 5 sin = -2.5 = -22.1 pC/m3 r=2, =30 0 101 7.6b. the total charge within the spherical shell 1 < r < 2 m: We integrate the charge density found in part a over the specified volume: Q=- 2 0 0 0 0 0 1 2 = -2(20) = -3.9 nC 20(4 sin2 + 1) 2 r sin dr d d r 5 sin 2 (4 sin2 + 1) dr d = -40 0 r3 1 2 1 3 dr = 60 2 r3 1 2 0 2 r 1 = -45 2 0 7.7. Let V = (cos 2)/ in free space. a) Find the volume charge density at point A(0.5, 60 , 1): Use Poisson's equation: v = - 0 2 V = - =- So at A we find: vA = 0 1 3 V 1 2V 1 + 2 0 2 - cos 2 4 cos 2 3 - 2 = ) 0 cos 2 3 0 cos(120 0.53 = -12 0 = -106 pC/m3 b) Find the surface charge density on a conductor surface passing through B(2, 30 , 1): First, we find E: V 1 V E = -V = - a - a cos 2 2 sin 2 = a + a 2 2 At point B the field becomes EB = cos 60 2 sin 60 a + a = 0.125 a + 0.433 a 4 4 The surface charge density will now be sB = |DB | = 0 |EB | = 0.451 0 = 0.399 pC/m2 The charge is positive or negative depending on which side of the surface we are considering. The problem did not provide information necessary to determine this. 7.8. Let V1 (r, , ) = 20/r and V2 (r, , ) = (4/r) + 4. a) State whether V1 and V2 satisfy Laplace's equation: 2 V1 = 2 V2 = 1 d r 2 dr 1 d r 2 dr r2 dV1 dr dV2 dr 102 = = 1 d r2 r 2 dr 1 d r2 r 2 dr -20 r2 -4 r2 =0 =0 r2 7.8b. Evaluate V1 and V2 on the closed surface r = 4: V1 (r = 4) = 20 4 = 5 V2 (r = 4) = + 4 = 5 4 4 c) Conciliate your results with the uniqueness theorem: Uniqueness specifies that there is only one potential that will satisfy all the given boundary conditions. While both potentials have the same value at r = 4, they do not as r . So they apply to different situations. 7.9. The functions V1 (, , z) and V2 (, , z) both satisfy Laplace's equation in the region a < < b, 0 < 2, -L < z < L; each is zero on the surfaces = b for -L < z < L; z = -L for a < < b; and z = L for a < < b; and each is 100 V on the surface = a for -L < z < L. a) In the region specified above, is Laplace's equation satisfied by the functions V1 + V2 , V1 - V2 , V1 + 3, and V1 V2 ? Yes for the first three, since Laplace's equation is linear. No for V1 V2 . b) On the boundary surfaces specified, are the potential values given above obtained from the functions V1 + V2 , V1 - V2 , V1 + 3, and V1 V2 ? At the 100 V surface ( = a), No for all. At the 0 V surfaces, yes, except for V1 + 3. c) Are the functions V1 + V2 , V1 - V2 , V1 + 3, and V1 V2 identical with V1 ? Only V2 is, since it is given as satisfying all the boundary conditions that V1 does. Therefore, by the uniqueness theorem, V2 = V1 . The others, not satisfying the boundary conditions, are not the same as V1 . 7.10. Conducting planes at z = 2cm and z = 8cm are held at potentials of -3V and 9V, respectively. The region between the plates is filled with a perfect dielectric with = 5 0 . Find and sketch: a) V (z): We begin with the general solution of the one-dimensional Laplace equation in rectangular coordinates: V (z) = Az + B. Applying the boundary conditions, we write -3 = A(2) + B and 9 = A(8) + B. Subtracting the former equation from the latter, we find 12 = 6A or A = 2 V/cm. Using this we find B = -7 V. Finally, V (z) = 2z - 7 V (z in cm) or V (z) = 200z - 7 V (z in m). b) Ez (z): We use E = -V = -(dV /dz)az = -2 V/cm = -200 V/m. c) Dz (z): Working in meters, have Dz = Ez = -200 = -1000 0 C/m2 7.11. The conducting planes 2x + 3y = 12 and 2x + 3y = 18 are at potentials of 100 V and 0, respectively. Let = 0 and find: a) V at P (5, 2, 6): The planes are parallel, and so we expect variation in potential in the direction normal to them. Using the two boundary condtions, our general potential function can be written: V (x, y) = A(2x + 3y - 12) + 100 = A(2x + 3y - 18) + 0 and so A = -100/6. We then write V (x, y) = - 100 100 (2x + 3y - 18) = - x - 50y + 300 6 3 and VP = - 100 (5) - 100 + 300 = 33.33 V. 3 b) Find E at P : Use E = -V = 100 ax + 50 ay V/m 3 103 7.12. Conducting cylinders at = 2 cm and = 8 cm in free space are held at potentials of 60mV and -30mV, respectively. a) Find V (): Working in volts and meters, we write the general one-dimensional solution to the Laplace equation in cylindrical coordinates, assuming radial variation: V () = A ln() + B. Applying the given boundary conditions, this becomes V (2cm) = .060 = A ln(.02) + B and V (8cm) = -.030 = A ln(.08) + B. Subtracting the former equation from the latter, we find -.090 = A ln(.08/.02) = A ln 4 A = -.0649. B is then found through either equation; e.g., B = .060 + .0649 ln(.02) = -.1940. Finally, V () = -.0649 ln - .1940. b) Find E (): E = -V = -(dV /d)a = (.0649/)a V/m. c) Find the surface on which V = 30 mV: Use .03 = -.0649 ln - .1940 = .0317 m = 3.17 cm. 7.13. Coaxial conducting cylinders are located at = 0.5 cm and = 1.2 cm. The region between the cylinders is filled with a homogeneous perfect dielectric. If the inner cylinder is at 100V and the outer at 0V, find: a) the location of the 20V equipotential surface: From Eq. (16) we have V () = 100 ln(.012/) V ln(.012/.005) We seek at which V = 20 V, and thus we need to solve: 20 = 100 b) E max : We have E = - .012 ln(.012/) = = 1.01 cm ln(2.4) (2.4)0.2 dV 100 V =- = d ln(2.4) whose maximum value will occur at the inner cylinder, or at = .5 cm: E max = c) 100 = 2.28 104 V/m = 22.8 kV/m .005 ln(2.4) if the charge per meter length on the inner cylinder is 20 nC/m: The capacitance per meter length is Q 2 0 R = C= ln(2.4) V0 R We solve for R: R = (20 10-9 ) ln(2.4) = 3.15 2 0 (100) 104 7.14. Two semi-infinite planes are located at = - and = , where < /2. A narrow insulating strip separates them along the z axis. The potential at = - is V0 , while V = 0 at = . a) Find V () in terms of and V0 : We use the one-dimensional solution form for Laplace's equation assuming variation along : V () = A + B. The boundary conditions are then substituted: V0 = -A + B and 0 = A + B. Subtract the latter equation from the former to obtain: V0 = -2A A = -V0 /(2). Then 0 = -V0 /(2) + B B = V0 /2. Finally V () = V0 2 1- V b) Find E at = 20 , = 2 cm, if V0 = 100 V and = 30 : E = - V0 1 dV 100 = V/m Then E(2cm, 20 ) = = 4.8 kV/m d 2 2(30 2/360)(.02) 7.15. The two conducting planes illustrated in Fig. 7.8 are defined by 0.001 < < 0.120 m, 0 < z < 0.1 m, = 0.179 and 0.188 rad. The medium surrounding the planes is air. For region 1, 0.179 < < 0.188, neglect fringing and find: a) V (): The general solution to Laplace's equation will be V = C1 + C2 , and so 20 = C1 (.188) + C2 and 200 = C1 (.179) + C2 Subtracting one equation from the other, we find -180 = C1 (.188 - .179) C1 = -2.00 104 Then 20 = -2.00 104 (.188) + C2 C2 = 3.78 103 Finally, V () = (-2.00 104 ) + 3.78 103 V. b) E(): Use E() = -V = - c) D() = 0 E() 1 dV 2.00 104 = a V/m d = (2.00 104 0 /) a C/m2 . 2.00 104 2.00 104 a a = C/m2 d) s on the upper surface of the lower plane: We use s = D n surf ace = e) Q on the upper surface of the lower plane: This will be Qt = .1 0 .120 .001 2.00 104 0 d dz = 2.00 104 0 (.1) ln(120) = 8.47 10-8 C = 84.7 nC f) Repeat a) to c) for region 2 by letting the location of the upper plane be = .188 - 2 , and then find s and Q on the lower surface of the lower plane. Back to the beginning, we use 20 = C1 (.188 - 2) + C2 and 200 = C1 (.179) + C2 105 7.15f (continued) Subtracting one from the other, we find -180 = C1 (.009 - 2) C1 = 28.7 Then 200 = 28.7(.179) + C2 C2 = 194.9. Thus V () = 28.7 + 194.9 in region 2. Then E=- 28.7 28.7 a V/m and D = - 0 a C/m2 s on the lower surface of the lower plane will now be s = - 28.7 0 a (-a ) = 28.7 0 C/m2 The charge on that surface will then be Qb = 28.7 0 (.1) ln(120) = 122 pC. g) Find the total charge on the lower plane and the capacitance between the planes: Total charge will be Qnet = Qt + Qb = 84.7 nC + 0.122 nC = 84.8 nC. The capacitance will be C= 84.8 Qnet = = 0.471 nF = 471 pF V 200 - 20 7.16. a) Solve Laplace's equation for the potential field in the homogeneous region between two concentric conducting spheres with radii a and b, b > a, if V = 0 at r = b and V = V0 at r = a. With radial variation only, we have dV 1 d 2V = 2 r2 =0 r dr dr Multiply by r 2 : d dr r2 dV dr = 0 or r 2 dV =A dr dV A A = 2 V = +B dr r r Note that in the last integration step, I dropped the minus sign that would have otherwise occurred in front of A, since we can choose A as we wish. Next, apply the boundary conditions: 0= V0 = Finally, V (r) = V0 r 1 a Divide by r 2 : A A +B B =- b b V0 1 a A A - A= a b V0 b 1 a - 1 b - 1 b - - 1 b = 1 r V0 1 a - - 1 b 1 b 106 7.16b. Find the capacitance between them: Assume permittivity . First, the electric field will be E = -V = - V0 dV ar = 1 dr r2 a - ar V/m 1 b Next, on the inner sphere, the charge density will be s = D The capacitance is now C= Q 4a 2 s = = V0 V0 4 1 a r=a ar = V0 a2 1 a - 1 b C/m2 - 1 b F 7.17. Concentric conducting spheres are located at r = 5 mm and r = 20 mm. The region between the spheres is filled with a perfect dielectric. If the inner sphere is at 100 V and the outer sphere at 0 V: a) Find the location of the 20 V equipotential surface: Solving Laplace's equation gives us V (r) = V0 1 r 1 a - - 1 b 1 b where V0 = 100, a = 5 and b = 20. Setting V (r) = 20, and solving for r produces r = 12.5 mm. b) Find Er,max : Use E = -V = - Then Er,max = E(r = a) = 100 V0 = = 26.7 V/mm = 26.7 kV/m a(1 - (a/b)) 5(1 - (5/20)) dV V0 ar ar = 2 1 - dr r a 1 b c) Find R if the surface charge density on the inner sphere is 100 C/m2 : s will be equal in magnitude to the electric flux density at r = a. So s = (2.67 104 V/m) R 0 = 10-4 C/m2 . Thus R = 423 ! (obviously a bad choice of numbers here possibly a misprint. A more reasonable charge on the inner sphere would have been 1 C/m2 , leading to R = 4.23). 7.18. Concentric conducting spheres have radii of 1 and 5 cm. There is a perfect dielectric for which R = 3 between them. The potential of the inner sphere is 2V and that of the outer is -2V. Find: a) V (r): We use the general expression derived in Problem 7.16: V (r) = (A/r) + B. At the inner sphere, 2 = (A/.01) + B, and at the outer sphere, -2 = (A/.05) + B. Subtracting the latter equation from the former gives 4=A 1 1 - .01 .05 = 80A so A = .05. Substitute A into either of the two potential equations at the boundaries to find B = -3. Finally, V (r) = (.05/r) - 3. 107 7.18b. E(r) = -(dV /dr)ar = (.05/r 2 )ar V/m. c) V at r = 3 cm: V (.03) = (.05/.03) - 3 = -1.33 V. d) the location of the 0-V equipotential surface: Use 0 = (.05/r0 ) - 3 r0 = (.05/3) = .0167 m = 1.67 cm e) the capacitance between the spheres: C= 4 1 a - 1 b = 4(3) 1 .01 - 0 1 .05 = 12 80 0 = 4.2 pF 7.19. Two coaxial conducting cones have their vertices at the origin and the z axis as their axis. Cone A has the point A(1, 0, 2) on its surface, while cone B has the point B(0, 3, 2) on its surface. Let VA = 100 V and VB = 20 V. Find: a) for each cone: Have A = tan-1 (1/2) = 26.57 and B = tan-1 (3/2) = 56.31 . b) V at P (1, 1, 1): The potential function between cones can be written as V () = C1 ln tan(/2) + C2 Then Solving these two equations, we find C1 = -97.7 and C2 = -41.1. Now at P , = tan-1 ( 2) = 54.7 . Thus VP = -97.7 ln tan(54.7/2) - 41.1 = 23.3 V 20 = C1 ln tan(56.31/2) + C2 and 100 = C1 ln tan(26.57/2) + C2 7.20. A potential field in free space is given as V = 100 ln tan(/2) + 50 V. a) Find the maximum value of |E | on the surface = 40 for 0.1 < r < 0.8 m, 60 < < 90 . First E=- 1 dV 100 100 100 a = - a = - a a =- 2 (/2) r d 2r tan(/2) cos 2r sin(/2) cos(/2) r sin This will maximize at the smallest value of r, or 0.1: Emax ( = 40 ) = E(r = 0.1, = 40 ) = - 100 a = 1.56 a kV/m 0.1 sin(40) b) Describe the surface V = 80 V: Set 100 ln tan /2 + 50 = 80 and solve for : Obtain ln tan /2 = 0.3 tan /2 = e.3 = 1.35 = 107 (the cone surface at = 107 degrees). 108 7.21. In free space, let v = 200 0 /r 2.4 . a) Use Poisson's equation to find V (r) if it is assumed that r 2 Er 0 when r 0, and also that V 0 as r : With r variation only, we have 2V = or 1 d r 2 dr d dr r2 or dV dr =- r2 dV dr =- v = -200r -2.4 r2 dV dr = -200r -.4 Integrate once: 200 .6 r + C1 = -333.3r .6 + C1 .6 dV C1 = -333.3r -1.4 + 2 = V (in this case) = -Er dr r Our first boundary condition states that r 2 Er 0 when r 0 Therefore C1 = 0. Integrate again to find: 333.3 -.4 r + C2 V (r) = .4 From our second boundary condition, V 0 as r , we see that C2 = 0. Finally, V (r) = 833.3r -.4 V b) Now find V (r) by using Gauss' Law and a line integral: Gauss' law applied to a spherical surface of radius r gives: r 200 r .6 0 4r 2 Dr = 4 (r )2 dr = 800 0 2.4 .6 0 (r ) Thus Er = Now V (r) = - Dr 0 = r 800 0 r .6 = 333.3r -1.4 V/m .6(4) 0 r 2 333.3(r )-1.4 dr = 833.3r -.4 V 7.22. Let the volume charge density in Fig. 7.3a be given by v = v0 (x/a)e-|x|/a (note error in the exponent in the formula stated in the book). a) Determine v,max and v,min and their locations: Let x = x/a. Then v = x e-|x | . Differentiate with respect to x to obtain: dv = v0 e-|x | (1 - |x |) dx This derivative is zero at x = 1, or the minimum and maximum occur at x = a respectively. The values of v at these points will be v,max = v0 e-1 = 0.368v0 , occurring at x = a. v,min = -v0 e-1 = -0.368v0 , occurring at x = -a. 109 7.22b. Find Ex and V (x) if V (0) = 0 and Ex 0 as x : We use Poisson's equation: 2V = - For x > 0, this becomes v d 2V v0 =- dx 2 x -x/a e a x -|x|/a e a d 2V v0 =- 2 dx Integrate once over x: v0 dV (x > 0) = - dx x -x/a av0 -x/a x e e + 1 + C1 dx + C1 = a a Noting that Ex = -dV /dx, we use the first boundary condition, Ex 0 as x , to establish that C1 = 0. Over the range x < 0, we have v0 dV (x < 0) = - dx av0 x/a x x/a e dx + C1 = e a -x + 1 + C1 a where C1 = 0, since, by symmetry, Ex 0 as x -. These two equations can be unified to cover the entire range of x; the final expression for the electric field becomes: Ex = - av0 dV =- dx |x| + 1 e-|x|/a V/m a The potential function is now found by a second integration. For x > 0, this is V (x) = av0 x -x/a a 2 v0 + e-x/a dx + C2 = e a -x -x/a - 2e-x/a + C2 e a We use the second boundary condition, V (0) = 0, from which C2 = 2a 2 v0 / . Substituting this yields V (x) (x > 0) = a 2 v0 -x -x/a + 2(1 - e-x/a ) e a We repeat the procedure for x < 0 to obtain V (x) = av0 a 2 v0 x x/a e + ex/a dx + C2 = a -x x/a e - 2ex/a + C2 a Again, with the V (0) = 0 boundary condition, we find C2 = -2a 2 v0 / , which when substituted leads to a 2 v0 -x x/a V (x) (x < 0) = e - 2(1 - ex/a ) a Combining the results for both ranges of x, we write V (x) = -a 2 v0 x -|x|/a 2x - e 1 - e-|x|/a a |x| 110 7.22c. Use a development similar to that of Sec. 7.4 to show that C = dQ/dV0 = S/8a ( note error in problem statement): First, the overall potential difference is V0 = Vx - Vx- = 2 From this we find a = Q=S 0 2a 2 v0 = 4a 2 v0 ( V0 )/(4v0 ). Then the total charge on one side will be -x x -x/a e -1 dx = Sv0 ae-x/a a a C= dQ d = dV0 dV0 1 S 2 V0 v0 = 0 v0 = Sv0 a = v0 V0 1 S 2 V0 v0 Now But a = ( V0 )/(4v0 ), from which (v0 /V0 ) = /(4a 2 ). Substituting this into the capacitance expression gives C= S 4 2 S 4 4a 2 = S 8a 7.23. A rectangular trough is formed by four conducting planes located at x = 0 and 8 cm and y = 0 and 5 cm in air. The surface at y = 5 cm is at a potential of 100 V, the other three are at zero potential, and the necessary gaps are placed at two corners. Find the potential at x = 3 cm, y = 4 cm: This situation is the same as that of Fig. 7.6, except the non-zero boundary potential appears on the top surface, rather than the right side. The solution is found from Eq. (39) by simply interchanging x and y, and b and d, obtaining: 4V0 mx 1 sinh(my/d) V (x, y) = sin m sinh(mb/d) d 1,odd where V0 = 100 V, d = 8 cm, and b = 5 cm. We will use the first three terms to evaluate the potential at (3,4): 1 sinh(3/2) 1 sinh(5/2) . 400 sinh(/2) V (3, 4) = sin(3/8) + sin(9/8) + sin(15/8) sinh(5/8) 3 sinh(15/8) 5 sinh(25/8) 400 = [.609 - .040 - .011] = 71.1 V Additional accuracy is found by including more terms in the expansion. Using thirteen terms, and . using six significant figure accuracy, the result becomes V (3, 4) = 71.9173 V. The series converges rapidly enough so that terms after the sixth one produce no change in the third digit. Thus, quoting three significant figures, 71.9 V requires six terms, with subsequent terms having no effect. 111 7.24. The four sides of a square trough are held at potentials of 0, 20, -30, and 60 V; the highest and lowest potentials are on opposite sides. Find the potential at the center of the trough: Here we can make good use of symmetry. The solution for a single potential on the right side, for example, with all other sides at 0V is given by Eq. (39): 4V0 V (x, y) = 1,odd my 1 sinh(mx/b) sin m sinh(md/b) b In the current problem, we can account for the three voltages by superposing three solutions of the above form, suitably modified to account for the different locations of the boundary potentials. Since we want V at the center of a square trough, it no longer matters on what boundary each of the given potentials is, and we can simply write: 4(0 + 20 - 30 + 60) V (center) = 1,odd 1 sinh(m/2) sin(m/2) = 12.5 V m sinh(m) The series converges to this value in three terms. 7.25. In Fig. 7.7, change the right side so that the potential varies linearly from 0 at the bottom of that side to 100 V at the top. Solve for the potential at the center of the trough: Since the potential reaches zero periodically in y and also is zero at x = 0, we use the form: V (x, y) = m=1 Vm sinh mx my sin b b Now, at x = d, V = 100(y/b). Thus y 100 = b Vm sinh m=1 md b sin my b We then multiply by sin(ny/b), where n is a fixed integer, and integrate over y from 0 to b: b 0 y ny 100 sin b b dy = m=1 Vm sinh md b b sin 0 my ny sin b b dy =b/2 if m=n, zero if m=n The integral on the right hand side picks the nth term out of the series, enabling the coefficients, Vn , to be solved for individually as we vary n. We find in general, Vm = The integral evaluates as b 0 2 b sinh(m/d) b 0 ny y 100 sin b b dy ny y 100 sin b b dy = -100/m (m even) 100/m (m odd) = (-1)m+1 100 m 112 7.25 (continued) Thus Vm = So that finally, V (x, y) = 200 b m=1 200(-1)m+1 mb sinh(md/b) my (-1)m+1 sinh (mx/b) sin m sinh (md/b) b Now, with a square trough, set b = d = 1, and so 0 < x < 1 and 0 < y < 1. The potential becomes 200 V (x, y) = m=1 (-1)m+1 sinh (mx) sin (my) m sinh (m) Now at the center of the trough, x = y = 0.5, and, using four terms, we have . 200 sinh(/2) 1 sinh(3/2) 1 sinh(5/2) 1 sinh(7/2) V (.5, .5) = - + - = 12.5 V sinh() 3 sinh(3) 5 sinh(5) 7 sinh(7) where additional terms do not affect the three-significant-figure answer. 7.26. If X is a function of x and X + (x - 1)X - 2X = 0, assume a solution in the form of an infinite power series and determine numerical values for a2 to a8 if a0 = 1 and a1 = -1: The series solution will be of the form: X= m=0 am x m The first 8 terms of this are substituted into the given equation to give: (2a2 - a1 - 2a0 ) + (6a3 + a1 - 2a2 - 2a1 )x + (12a4 + 2a2 - 3a3 - 2a2 )x 2 + (3a3 - 4a4 - 2a3 + 20a5 )x 3 + (30a6 + 4a4 - 5a5 - 2a4 )x 4 + (42a7 + 5a5 - 6a6 - 2a5 )x 5 + (56a8 + 6a6 - 7a7 - 2a6 )x 6 + (7a7 - 8a8 - 2a7 )x 7 + (8a8 - 2a8 )x 8 = 0 For this equation to be zero, each coefficient term (in parenthesis) must be zero. The first of these is 2a2 - a1 - 2a0 = 2a2 + 1 - 2 = 0 a2 = 1/2 The second coefficient is 6a3 + a1 - 2a2 - 2a1 = 6a3 - 1 - 1 + 2 = 0 a3 = 0 Third coefficient: 12a4 + 2a2 - 3a3 - 2a2 = 12a4 + 1 - 0 - 1 = 0 a4 = 0 Fourth coefficient: 3a3 - 4a4 - 2a3 + 20a5 = 0 - 0 - 0 + 20a5 = 0 a5 = 0 In a similar manner, we find a6 = a7 = a8 = 0. 113 7.27. It is known that V = XY is a solution of Laplace's equation, where X is a function of x alone, and Y is a function of y alone. Determine which of the following potential function are also solutions of Laplace's equation: a) V = 100X: We know that 2 XY = 0, or 2 Y 2 X =- = 2 XY + 2 XY = 0 Y X + XY = 0 x 2 y X Y Therefore, 2 X = 100X = 0 No. b) V = 50XY : Would have 2 V = 50 2 XY = 0 Yes. c) V = 2XY + x - 3y: 2 V = 2 2 XY + 0 - 0 = 0 Yes. d) V = xXY : 2 V = x 2 XY + XY 2 x = 0 Yes. e) V = X2 Y : 2 V = X 2 XY + XY 2 X = 0 + XY 2 X No. 7.28. Assume a product solution of Laplace's equation in cylindrical coordinates, V = P F , where V is not a function of z, P is a function only of , and F is a function only of . a) Obtain the two separated equations if the separation constant is n2 . Select the sign of n2 so that the solution of the equation leads to trigonometric functions: Begin with Laplace's equation in cylindrical coordinates, in which there is no z variation: 2V = 1 V + 1 2V =0 2 2 We substitute the product solution V = P F to obtain: F d d dP d + F dP P d 2F P d 2F d 2P = =0 +F 2 + 2 2 d 2 d d d 2 Next, multiply by 2 and divide by F P to obtain dP 1 d 2F 2 d 2P + + =0 P d P d 2 F d 2 n2 -n2 The equation is now grouped into two parts as shown, each a function of only one of the two variables; each is set equal to plus or minus n2 , as indicated. The equation now becomes d 2F + n2 F = 0 F = Cn cos(n) + Dn sin(n) (n 1) d 2 Note that n is required to be an integer, since physically, the solution must repeat itself every 2 radians in . If n = 0, then d 2F = 0 F = C0 + D0 d 2 114 7.28b. Show that P = A n + B -n satisfies the equation: From part a, the radial equation is: 2 Substituting A n , we find 2 n(n - 1) n-2 + n n-1 - n2 n = n2 n - n n + n n - n2 n = 0 Substituting B -n : 2 n(n + 1) -(n+2) - n -(n+1) - n2 -n = n2 -n + n -n - n -n - n2 -n = 0 So it works. c) Construct the solution V (, ). Functions of this form are called circular harmonics. To assemble the complete solution, we need the radial solution for the case in which n = 0. The equation to solve is d 2P dP 2 + =0 d d Let S = dP /d, and so dS/d = d 2 P /d 2 . The equation becomes Integrating, find - ln + ln A0 = ln S ln S = ln where A0 is a constant. So now d dP = A0 Pn=0 = A0 ln + B0 A0 S= A0 dP = d d dS dS +S =0 - = d S d 2P dP - n2 P = 0 + 2 d d We may now construct the solution in its complete form, encompassing n 0: V (, ) = (A0 ln + B0 )(C0 + D0 ) + n=0 solution n=1 [An n + Bn -n ][Cn cos(n) + Dn sin(n)] 115 CHAPTER 8 8.1a. Find H in cartesian components at P (2, 3, 4) if there is a current filament on the z axis carrying 8 mA in the az direction: Applying the Biot-Savart Law, we obtain Ha = - I dL aR = 4R 2 - I dz az [2ax + 3ay + (4 - z)az ] = 4(z2 - 8z + 29)3/2 - I dz[2ay - 3ax ] 4(z2 - 8z + 29)3/2 Using integral tables, this evaluates as Ha = I 4 2(2z - 8)(2ay - 3ax ) 52(z2 - 8z + 29)1/2 - = I (2ay - 3ax ) 26 Then with I = 8 mA, we finally obtain Ha = -294ax + 196ay A/m b. Repeat if the filament is located at x = -1, y = 2: In this case the Biot-Savart integral becomes Hb = - I dz az [(2 + 1)ax + (3 - 2)ay + (4 - z)az ] = 4(z2 - 8z + 26)3/2 - I dz[3ay - ax ] 4(z2 - 8z + 26)3/2 Evaluating as before, we obtain with I = 8 mA: Hb = I 4 2(2z - 8)(3ay - ax ) 40(z2 - 8z + 26)1/2 - = I (3ay - ax ) = -127ax + 382ay A/m 20 c. Find H if both filaments are present: This will be just the sum of the results of parts a and b, or HT = Ha + Hb = -421ax + 578ay A/m This problem can also be done (somewhat more simply) by using the known result for H from an infinitely-long wire in cylindrical components, and transforming to cartesian components. The BiotSavart method was used here for the sake of illustration. 8.2. A current filament of 3ax A lies along the x axis. Find H in cartesian components at P (-1, 3, 2): We use the Biot-Savart law, I dL aR H= 4R 2 where I dL = 3dxax , aR = [-(1 + x)ax + 3ay + 2az ]/R, and R = x 2 + 2x + 14. Thus HP = 3dxax [-(1 + x)ax + 3ay + 2az ] (9az - 6ay ) dx = 2 + 2x + 14)3/2 2 3/2 4(x - - 4(x + 2x + 14) (9az - 6ay )(x + 1) 2(9az - 6ay ) = = = 0.110az - 0.073ay A/m 4(13) 4(13) x 2 + 2x + 14 - 116 8.3. Two semi-infinite filaments on the z axis lie in the regions - < z < -a (note typographical error in problem statement) and a < z < . Each carries a current I in the az direction. a) Calculate H as a function of and at z = 0: One way to do this is to use the field from an infinite line and subtract from it that portion of the field that would arise from the current segment at -a < z < a, found from the Biot-Savart law. Thus, H= I a - 2 a -a I dz az [ a - z az ] 4[ 2 + z2 ]3/2 The integral part simplifies and is evaluated: a -a I dz a I z a = 2 + z2 ]3/2 4[ 4 2 2 + z2 I 1- 2 a 2 + a2 a -a = Ia 2 2 + a 2 a Finally, H= a A/m b) What value of a will cause the magnitude of H at = 1, z = 0, to be one-half the value obtained for an infinite filament? We require 1- a 2 + a2 =1 = a 1 1 a = 1/ 3 = 2 2 1 + a2 8.4a.) A filament is formed into a circle of radius a, centered at the origin in the plane z = 0. It carries a current I in the a direction. Find H at the origin: We use the Biot-Savart law, which in this case becomes: 2 I a d a (-a ) I dL aR I = = 0.50 az A/m H= 2 2 4R 4a a loop 0 b.) A filament of the same length is shaped into a square in the z = 0 plane. The sides are parallel to the coordinate axes and a current I flows in the general a direction. Again, find H at the origin: Since the loop is the same length, its perimeter is 2a, and so each of the four sides is of length a/2. Using symmetry, we can find the magnetic field at the origin associated with each of the 8 half-sides (extending from 0 to a/4 along each coordinate direction) and multiply the result by 8: Taking one of the segments in the y direction, the Biot-Savart law becomes H= = I dL aR =8 4R 2 a/4 0 a/4 0 I dy ay -(a/4) ax - y ay 4 y 2 + (a/4)2 = 3/2 a/4 0 loop aI 2 dy az 3/2 y 2 + (a/4)2 y az aI 2 y 2 + (a/4)2 2 (a/4) = 0.57 I az A/m a 117 8.5. The parallel filamentary conductors shown in Fig. 8.21 lie in free space. Plot |H| versus y, -4 < y < 4, along the line x = 0, z = 2: We need an expression for H in cartesian coordinates. We can start with the known H in cylindrical for an infinite filament along the z axis: H = I /(2) a , which we transform to cartesian to obtain: Ix -Iy a + ay H= 2 + y2) x 2(x 2(x 2 + y 2 ) If we now rotate the filament so that it lies along the x axis, with current flowing in positive x, we obtain the field from the above expression by replacing x with y and y with z: H= -I z Iy ay + az 2(y 2 + z2 ) 2(y 2 + z2 ) Now, with two filaments, displaced from the x axis to lie at y = 1, and with the current directions as shown in the figure, we use the previous expression to write Iz Iz I (y + 1) I (y - 1) - ay + - az 2 + z2 ] 2 + z2 ] 2 + z2 ] 2[(y + 1) 2[(y - 1) 2[(y - 1) 2[(y + 1)2 + z2 ] We now evaluate this at z = 2, and find the magnitude ( H H), resulting in H= I |H| = 2 2 2 - 2 2 + 2y + 5 y y - 2y + 5 2 + (y + 1) (y - 1) - 2 2 - 2y + 5 y y + 2y + 5 2 1/2 This function is plotted below 8.6a. A current filament I is formed into circle, = a, in the z = z plane. Find Hz at P (0, 0, z) if I flows in the a direction: Use the Biot-Savart law, H= I dL aR 4R 2 a 2 + (z - z )2 . The where in this case I dL = I da , aR = [-aa + (z - z )az ]/R, and R = setup becomes H= 0 2 I ad a [-aa + (z - z )az ] = 4[a 2 + (z - z )2 ]3/2 118 2 0 I a[aaz + (z - z )a ] d 4[a 2 + (z - z )2 ]3/2 At this point we need to be especially careful. Note that we are integrating a vector with an a component around a complete circle, where the vector has no dependence. This sum of all a components will be zero even though this doesn't happen when we go ahead with the integration without this knowledge. The problem is that the integral "interprets" a as a constant direction, when in fact as we know a continually changes direction as varies. We drop the a component in the integral to give H= 0 2 a 2 I az m I a 2 az d = = A/m 2 + (z - z )2 ]3/2 2 + (z - z )2 ]3/2 2 + (z - z )2 ]3/2 4[a 2[a 2[a where m = a 2 I az is the magnetic moment of the loop. b) Find Hz at P caused by a uniform surface current density K = K0 a , flowing on the cylindrical surface, = a, 0 < z < h. The results of part a should help: Using part a, we can write down the differential field at P arising from a circular current ribbon of differential height, dz , at location z . The ribbon is of radius a and carries current K0 dz a A: dH = a 2 K0 dz az A/m 2[a 2 + (z - z )2 ]3/2 The total magnetic field at P is now the sum of the contributions of all differential rings that comprise the cylinder: Hz = a 2 K0 dz dz a 2 K0 h = 2[a 2 + (z - z )2 ]3/2 2 [a 2 + z2 - 2zz + (z )2 ]3/2 0 0 h h 2(2z - 2z) a 2 K0 K0 (z - z) = = 2 4a 2 a 2 + z2 - 2zz + (z )2 0 2 a 2 + (z - z)2 0 = K0 2 (h - z) a2 + (h - z)2 z + 2 + z2 a A/m h 8.7. Given points C(5, -2, 3) and P (4, -1, 2); a current element I dL = 10-4 (4, -3, 1) A m at C produces a field dH at P . a) Specify the direction of dH by a unit vector aH : Using the Biot-Savart law, we find dH = 10-4 [4ax - 3ay + az ] [-ax + ay - az ] [2ax + 3ay + az ] 10-4 I dL aCP = = 2 433/2 65.3 4RCP aH = b) Find |dH|. |dH| = 2ax + 3ay + az = 0.53ax + 0.80ay + 0.27az 14 14 10-4 = 5.73 10-6 A/m = 5.73 A/m 65.3 from which c) What direction al should I dL have at C so that dH = 0? I dL should be collinear with aCP , thus rendering the cross product in the Biot-Savart law equal to zero. Thus the answer is al = (-ax + ay - az )/ 3 119 8.8. For the finite-length current element on the z axis, as shown in Fig. 8.5, use the Biot-Savart law to derive Eq. (9) of Sec. 8.1: The Biot-Savart law reads: H= z2 z1 I dL aR = 4R 2 tan 2 tan 1 I dzaz (a - zaz ) = 4( 2 + z2 )3/2 tan 2 tan 1 Ia dz 4( 2 + z2 )3/2 The integral is evaluated (using tables) and gives the desired result: H= I za 4 2 + z2 tan 2 tan 1 = I 4 tan 2 1 + tan2 2 - tan 1 1 + tan2 1 a = I (sin 2 - sin 1 )a 4 8.9. A current sheet K = 8ax A/m flows in the region -2 < y < 2 in the plane z = 0. Calculate H at P (0, 0, 3): Using the Biot-Savart law, we write HP = K aR dx dy = 4R 2 2 -2 - 8ax (-xax - yay + 3az ) dx dy 4(x 2 + y 2 + 9)3/2 Taking the cross product gives: HP = 2 -2 - 8(-yaz - 3ay ) dx dy 4(x 2 + y 2 + 9)3/2 We note that the z component is anti-symmetric in y about the origin (odd parity). Since the limits are symmetric, the integral of the z component over y is zero. We are left with HP = 2 -2 - -24 ay dx dy 6 = - ay 4(x 2 + y 2 + 9)3/2 2 2 -2 x (y 2 + 9) x 2 + y 2 + 9 2 -2 - dy 6 = - ay -2 y2 2 y 12 1 dy = - ay tan-1 +9 3 3 4 = - (2)(0.59) ay = -1.50 ay A/m 8.10. Let a filamentary current of 5 mA be directed from infinity to the origin on the positive z axis and then back out to infinity on the positive x axis. Find H at P (0, 1, 0): The Biot-Savart law is applied to the two wire segments using the following setup: HP = -I dza (-za + a ) I dL aR z z y = + 4(z2 + 1)3/2 4R 2 0 I dzax I dxaz I = + = 2 + 1)3/2 2 + 1)3/2 4(z 4(x 4 0 0 I (ax + az ) = 0.40(ax + az ) mA/m = 4 I dxax (-xax + ay ) 4(x 2 + 1)3/2 0 zax xaz + 2+1 0 2+1 0 z x 120 8.11. An infinite filament on the z axis carries 20 mA in the az direction. Three uniform cylindrical current sheets are also present: 400 mA/m at = 1 cm, -250 mA/m at = 2 cm, and -300 mA/m at = 3 cm. Calculate H at = 0.5, 1.5, 2.5, and 3.5 cm: We find H at each of the required radii by applying Ampere's circuital law to circular paths of those radii; the paths are centered on the z axis. So, at 1 = 0.5 cm: H dL = 21 H1 = Iencl = 20 10-3 A 10 10-3 10 10-3 = = 2.0 A/m 1 0.5 10-2 At = 2 = 1.5 cm, we enclose the first of the current cylinders at = 1 cm. Ampere's law becomes: H1 = 22 H2 = 20 + 2(10-2 )(400) mA H2 = Following this method, at 2.5 cm: H3 = and at 3.5 cm, H4 = 10 + 4.00 - (2 10-2 )(250) = 360 mA/m 2.5 10-2 10 + 4.00 - 5.00 - (3 10-2 )(300) =0 3.5 10-2 10 + 4.00 = 933 mA/m 1.5 10-2 Thus 8.12. In Fig. 8.22, let the regions 0 < z < 0.3 m and 0.7 < z < 1.0 m be conducting slabs carrying uniform current densities of 10 A/m2 in opposite directions as shown. The problem asks you to find H at various positions. Before continuing, we need to know how to find H for this type of current configuration. The sketch below shows one of the slabs (of thickness D) oriented with the current coming out of the page. The problem statement implies that both slabs are of infinite length and width. To find the magnetic field inside a slab, we apply Ampere's circuital law to the rectangular path of height d and width w, as shown, since by symmetry, H should be oriented horizontally. For example, if the sketch below shows the upper slab in Fig. 8.22, current will be in the positive y direction. Thus H will be in the positive x direction above the slab midpoint, and will be in the negative x direction below the midpoint. Hout d w D Hout 121 8.12 (continued) In taking the line integral in Ampere's law, the two vertical path segments will cancel each other. Ampere's circuital law for the interior loop becomes H dL = 2Hin w = Iencl = J w d Hin = Jd 2 The field outside the slab is found similarly, but with the enclosed current now bounded by the slab thickness, rather than the integration path height: 2Hout w = J w D Hout = JD 2 where Hout is directed from right to left below the slab and from left to right above the slab (right hand rule). Reverse the current, and the fields, of course, reverse direction. We are now in a position to solve the problem. Find H at: a) z = -0.2m: Here the fields from the top and bottom slabs (carrying opposite currents) will cancel, and so H = 0. b) z = 0.2m. This point lies within the lower slab above its midpoint. Thus the field will be oriented in the negative x direction. Referring to Fig. 8.22 and to the sketch on the previous page, we find that d = 0.1. The total field will be this field plus the contribution from the upper slab current: H= -10(0.1) 10(0.3) ax - ax = -2ax A/m 2 2 lower slab upper slab c) z = 0.4m: Here the fields from both slabs will add constructively in the negative x direction: H = -2 10(0.3) ax = -3ax A/m 2 d) z = 0.75m: This is in the interior of the upper slab, whose midpoint lies at z = 0.85. Therefore d = 0.2. Since 0.75 lies below the midpoint, magnetic field from the upper slab will lie in the negative x direction. The field from the lower slab will be negative x-directed as well, leading to: H= -10(0.2) 10(0.3) ax = -2.5ax A/m ax - 2 2 upper slab lower slab e) z = 1.2m: This point lies above both slabs, where again fields cancel completely: Thus H = 0. 122 8.13. A hollow cylindrical shell of radius a is centered on the z axis and carries a uniform surface current density of Ka a . a) Show that H is not a function of or z: Consider this situation as illustrated in Fig. 8.11. There (sec. 8.2) it was stated that the field will be entirely z-directed. We can see this by applying Ampere's circuital law to a closed loop path whose orientation we choose such that current is enclosed by the path. The only way to enclose current is to set up the loop (which we choose to be rectangular) such that it is oriented with two parallel opposing segments lying in the z direction; one of these lies inside the cylinder, the other outside. The other two parallel segments lie in the direction. The loop is now cut by the current sheet, and if we assume a length of the loop in z of d, then the enclosed current will be given by Kd A. There will be no variation in the field because where we position the loop around the circumference of the cylinder does not affect the result of Ampere's law. If we assume an infinite cylinder length, there will be no z dependence in the field, since as we lengthen the loop in the z direction, the path length (over which the integral is taken) increases, but then so does the enclosed current by the same factor. Thus H would not change with z. There would also be no change if the loop was simply moved along the z direction. b) Show that H and H are everywhere zero. First, if H were to exist, then we should be able to find a closed loop path that encloses current, in which all or or portion of the path lies in the direction. This we cannot do, and so H must be zero. Another argument is that when applying the Biot-Savart law, there is no current element that would produce a component. Again, using the Biot-Savart law, we note that radial field components will be produced by individual current elements, but such components will cancel from two elements that lie at symmetric distances in z on either side of the observation point. c) Show that Hz = 0 for > a: Suppose the rectangular loop was drawn such that the outside z-directed segment is moved further and further away from the cylinder. We would expect Hz outside to decrease (as the Biot-Savart law would imply) but the same amount of current is always enclosed no matter how far away the outer segment is. We therefore must conclude that the field outside is zero. d) Show that Hz = Ka for < a: With our rectangular path set up as in part a, we have no path integral contributions from the two radial segments, and no contribution from the outside z-directed segment. Therefore, Ampere's circuital law would state that H dL = Hz d = Iencl = Ka d Hz = Ka where d is the length of the loop in the z direction. e) A second shell, = b, carries a current Kb a . Find H everywhere: For < a we would have both cylinders contributing, or Hz ( < a) = Ka + Kb . Between the cylinders, we are outside the inner one, so its field will not contribute. Thus Hz (a < < b) = Kb . Outside ( > b) the field will be zero. 123 8.14. A toroid having a cross section of rectangular shape is defined by the following surfaces: the cylinders = 2 and = 3 cm, and the planes z = 1 and z = 2.5 cm. The toroid carries a surface current density of -50az A/m on the surface = 3 cm. Find H at the point P (, , z): The construction is similar to that of the toroid of round cross section as done on p.239. Again, magnetic field exists only inside the toroid cross section, and is given by H= Iencl a 2 (2 < < 3) cm, (1 < z < 2.5) cm where Iencl is found from the given current density: On the outer radius, the current is Iouter = -50(2 3 10-2 ) = -3 A This current is directed along negative z, which means that the current on the inner radius ( = 2) is directed along positive z. Inner and outer currents have the same magnitude. It is the inner current that is enclosed by the circular integration path in a within the toroid that is used in Ampere's law. So Iencl = +3 A. We can now proceed with what is requested: a) PA (1.5cm, 0, 2cm): The radius, = 1.5 cm, lies outside the cross section, and so HA = 0. b) PB (2.1cm, 0, 2cm): This point does lie inside the cross section, and the and z values do not matter. We find 3a Iencl a = = 71.4 a A/m HB = 2 2(2.1 10-2 ) c) PC (2.7cm, /2, 2cm): again, and z values make no difference, so HC = 3a = 55.6 a A/m 2(2.7 10-2 ) d) PD (3.5cm, /2, 2cm). This point lies outside the cross section, and so HD = 0. 8.15. Assume that there is a region with cylindrical symmetry in which the conductivity is given by = 1.5e-150 kS/m. An electric field of 30 az V/m is present. a) Find J: Use J = E = 45e-150 az kA/m2 b) Find the total current crossing the surface < 0 , z = 0, all : I= J dS = 0 2 0 0 45e-150 d d = 2(45) -150 e [-150 - 1] (150)2 0 0 kA = 12.6 1 - (1 + 1500 )e-1500 A c) Make use of Ampere's circuital law to find H: Symmetry suggests that H will be -directed only, and so we consider a circular path of integration, centered on and perpendicular to the z axis. Ampere's law becomes: 2H = Iencl , where Iencl is the current found in part b, except with 0 replaced by the variable, . We obtain H = 2.00 1 - (1 + 150)e-150 A/m 124 8.16. The cylindrical shell, 2mm < < 3mm, carries a uniformly-distributed total current of 8A in the -az direction, and a filament on the z axis carries 8A in the az direction. Find H everywhere: We use Ampere's circuital law, noting that from symmetry, H will be a directed. Inside the shell ( < 2mm), A circular integration path centered on the z axis encloses only the filament current along z: Therefore H( < 2mm) = 4 8 a = a A/m ( in m) 2 With the circular integration path within (2 < < 3mm), the enclosed current will consist of the filament plus that portion of the shell current that lies inside . Ampere's circuital law applied to a loop of radius is: H dL = If ilament + where the current density is J=- So 2H = 8 + Solve for H to find: H(2 < < 3 mm) = 4 1 - (2 105 )( 2 - 4 10-6 ) a A/m ( in m) 2 0 210-3 shell area J dS 8 8 106 az A/m2 az = - -3 )2 - (2 10-3 )2 (3 10 5 -8 106 az az d d = 8 - 1.6 106 ( )2 5 210-3 Outside ( > 3mm), the total enclosed current is zero, and so H( > 3mm) = 0. 125 8.17. A current filament on the z axis carries a current of 7 mA in the az direction, and current sheets of 0.5 az A/m and -0.2 az A/m are located at = 1 cm and = 0.5 cm, respectively. Calculate H at: a) = 0.5 cm: Here, we are either just inside or just outside the first current sheet, so both we will calculate H for both cases. Just inside, applying Ampere's circuital law to a circular path centered on the z axis produces: 2H = 7 10-3 H(just inside) = 7 10-3 a = 2.2 10-1 a A/m 2(0.5 10-2 Just outside the current sheet at .5 cm, Ampere's law becomes 2H = 7 10-3 - 2(0.5 10-2 )(0.2) H(just outside) = 7.2 10-4 a = 2.3 10-2 a A/m 2(0.5 10-2 ) b) = 1.5 cm: Here, all three currents are enclosed, so Ampere's law becomes 2(1.5 10-2 )H = 7 10-3 - 6.28 10-3 + 2(10-2 )(0.5) H( = 1.5) = 3.4 10-1 a A/m c) = 4 cm: Ampere's law as used in part b applies here, except we replace = 1.5 cm with = 4 cm on the left hand side. The result is H( = 4) = 1.3 10-1 a A/m. d) What current sheet should be located at = 4 cm so that H = 0 for all > 4 cm? We require that the total enclosed current be zero, and so the net current in the proposed cylinder at 4 cm must be negative the right hand side of the first equation in part b. This will be -3.2 10-2 , so that the surface current density at 4 cm must be K= -3.2 10-2 az = -1.3 10-1 az A/m 2(4 10-2 ) 8.18. Current density is distributed as follows: J = 0 for |y| > 2 m, J = 8yaz A/m2 for |y| < 1 m, J = 8(2 - y) az A/m2 for 1 < y < 2 m, J = -8(2 + y) az A/m2 for -2 < y < -1 m. Use symmetry and Ampere's law to find H everywhere. Symmetry does help significantly in this problem. The current densities in the regions 0 < y < 1 and -1 < y < 0 are mirror images of each other across the plane y = 0 this in addition to being of opposite sign. This is also true of the current densities in the regions 1 < y < 2 and -2 < y < -1. As a consequence of this, we find that the net current in region 1, I1 (see the diagram on the next page), is equal and opposite to the net current in region 4, I4 . Also, I2 is equal and opposite to I3 . This means that when applying Ampere's law to the path a - b - c - d - a, as shown in the figure, zero current is enclosed, so that H dL = 0 over the path. In addition, the symmetry of the current configuration implies that H = 0 outside the slabs along the vertical paths a - b and c - d. H from all sources should completely cancel along the two vertical paths, as well as along the two horizontal paths. 126 8.18. (continued) To find the magnetic field in region 1, we apply Ampere's circuital law to the path c - d - e - f - c, again noting that H will be zero along the two horizontal segments and along the right vertical segment. This leaves only the left vertical segment, e - f , pointing in the +x direction, and along which is field, Hx1 . The counter-clockwise direction of the path integral is chosen using the right-hand convention, where we take the normal to the path in the +z direction, which is the same as the current direction. Assuming the height of the path is x, we find Hx1 x = x 2 y1 8(2 - y)dy = x 16y - 4y 2 2 y1 = 2 x 16(2 - y1 ) - 4(4 - y1 ) Replacing y1 with y, we find Hx1 = 4[8 - 4y - 4 + y 2 ] H1 (1 < y < 2) = 4(y - 2)2 ax A/m H1 lies in the positive x direction, since the result of the integration is net positive. H in region 2 is now found through the line integral over the path d - g - h - c, enclosing all of region 1 within x and part of region 2 from y = y2 to 1: Hx2 x = x 1 2 8(2 - y) dy + x 1 y2 8y dy = 2 2 x 4(1 - 2)2 + 4(1 - y2 ) = 4(2 - y2 ) x so that in terms of y, H2 (0 < y < 1) = 4(2 - y 2 )ax A/m 4 a 3 2 g e 1 d -2 -1 0 y2 1 y1 2 y b h f c x I4 x I3 x . I2 . I1 127 8.18. (continued) The procedure is repeated for the remaining two regions, -2 < y < -1 and -1 < y < 0, by taking the integration path with its right vertical segment within each of these two regions, while the left vertical path is a - b. Again the integral is taken counter-clockwise, which means that the right vertical path will be directed along -x. But the current is now in the opposite direction of that for y > 0, making the enclosed current net negative. Therefore, H will be in the opposite direction from that of the right vertical path, which is the positive x direction. The magnetic field will therefore be symmetric about the y = 0 plane. We can use the results for regions 1 and 2 to construct the field everywhere: H = 0 (y > 2) and (y < -2) H = 4(2 - |y|2 )ax A/m (0 < |y| < 1) H = 4(|y| - 2)2 ax A/m (1 < |y| < 2) 8.19. Calculate [( G)] if G = 2x 2 yz ax - 20y ay + (x 2 - z2 ) az : Proceding, we first find G = 4xyz - 20 - 2z. Then ( G) = 4yz ax + 4xz ay + (4xy - 2) az . Then [( G)] = (4x - 4x) ax - (4y - 4y) ay + (4z - 4z) az = 0 8.20. The magnetic field intensity is given in the square region x = 0, 0.5 < y < 1, 1 < z < 1.5 by H = z2 ax + x 3 ay + y 4 az A/m. a) evaluate H dL about the perimeter of the square region: Using dL = dxax + dyay + dzaz , and using the given field, we find, in the x = 0 plane: H dL = b) Find H: H= Hy Hz - y z ax + Hz Hx - z x ay + Hy Hx - x y az 1 .5 0 dy + 1 1.5 (1)4 dz + 1 .5 0 dy + 1 1.5 (.5)4 dz = 0.46875 = 4y 3 ax + 2zay + 3x 2 az c) Calculate ( H)x at the center of the region: Here, y = 0.75 and so ( H)x = 4(.75)3 = 1.68750. d) Does ( H)x = [ HdL]/Area Enclosed? Using the part a result, [ HdL]/Area Enclosed = 0.46875/0.25 = 1.8750, which is off the value found in part c. Answer: No. Reason: the limit of the area shrinking to zero must be taken before the results will be equal. 8.21. Points A, B, C, D, E, and F are each 2 mm from the origin on the coordinate axes indicated in Fig. 8.23. The value of H at each point is given. Calculate an approximate value for H at the origin: We use the approximation: . H dL curl H = a where no limit as a 0 is taken (hence the approximation), and where a = 4 mm2 . Each curl component is found by integrating H over a square path that is normal to the component in question. 128 8.21. (continued) Each of the four segments of the contour passes through one of the given points. Along each segment, the field is assumed constant, and so the integral is evaluated by summing the products of the field and segment length (4 mm) over the four segments. The x component of the curl is thus: -3 . (Hz,C - Hy,E - Hz,D + Hy,F )(4 10 ) ( H)x = (4 10-3 )2 = (15.69 + 13.88 - 14.35 - 13.10)(250) = 530 A/m2 The other components are: . (Hz,B + Hx,E - Hz,A - Hx,F )(4 10-3 ) ( H)y = (4 10-3 )2 = (15.82 + 11.11 - 14.21 - 10.88)(250) = 460 A/m2 and -3 . (Hy,A - Hx,C - Hy,B Hx,D )(4 10 ) ( H)z = (4 10-3 )2 = (-13.78 - 10.49 + 12.19 + 11.49)(250) = -148 A/m2 Finally we assemble the results and write: . H = 530 ax + 460 ay - 148 az 8.22. In the cylindrical region 0.6 mm, H = (2/) + (/2) A/m, while H = (3/) A/m for > 0.6 mm. a) Determine J for < 0.6mm: We have only a component that varies with . Therefore H= 1 d(H ) 2 1 d az = 2+ az = J = 1az A/m2 d d 2 b) Determine J for > 0.6 mm: In this case J= 1 d 3 az = 0 d c) Is there a filamentary current at = 0? If so, what is its value? As 0, H , which implies the existence of a current filament along the z axis: So, YES. The value is found by through Ampere's circuital law, by integrating H around a circular path of vanishingly-small radius. The current enclosed is therefore I = 2(2/) = 4 A. d) What is J at = 0? Since a filament current lies along z at = 0, this forms a singularity, and so the current density there is infinite. 8.23. Given the field H = 20 2 a A/m: a) Determine the current density J: This is found through the curl of H, which simplifies to a single term, since H varies only with and has only a component: J= H= 1 d(H ) 1 d az = 20 3 az = 60 az A/m2 d d 129 8.23. (continued) b) Integrate J over the circular surface = 1, 0 < < 2, z = 0, to determine the total current passing through that surface in the az direction: The integral is: I= J dS = 0 2 0 1 60az d daz = 40 A c) Find the total current once more, this time by a line integral around the circular path = 1, 0 < < 2, z = 0: I= H dL = 0 2 20 2 a (1)da = =1 2 0 20 d = 40 A 8.24. Evaluate both sides of Stokes'theorem for the field G = 10 sin a and the surface r = 3, 0 90 , 0 90 . Let the surface have the ar direction: Stokes' theorem reads: G dL = ( G) n da C S Considering the given surface, the contour, C, that forms its perimeter consists of three joined arcs of radius 3 that sweep out 90 in the xy, xz, and zy planes. Their centers are at the origin. Of these three, only the arc in the xy plane (which lies along a ) is in the direction of G; the other two (in the -a and a directions respectively) are perpendicular to it, and so will not contribute to the path integral. The left-hand side therefore consists of only the xy plane portion of the closed path, and evaluates as G dL = 0 /2 10 sin a /2 a 3 sin /2 d = 15 To evaluate the right-hand side, we first find G= 1 d 20 cos ar [(sin )10 sin ] ar = r sin d r The surface over which we integrate this is the one-eighth spherical shell of radius 3 in the first octant, bounded by the three arcs described earlier. The right-hand side becomes ( G) n da = 0 /2 0 /2 S 20 cos ar ar (3)2 sin d d = 15 3 It would appear that the theorem works. 130 8.25. (This problem was discovered to be flawed I will proceed with it and show how). Given the field H= 1 cos a - sin a A/m 2 2 2 evaluate both sides of Stokes' theorem for the path formed by the intersection of the cylinder = 3 and the plane z = 2, and for the surface defined by = 3, 0 2 , and z = 0, 0 3: This surface resembles that of an open tin can whose bottom lies in the z = 0 plane, and whose open circular edge, at z = 2, defines the line integral contour. We first evaluate H dL over the circular contour, where we take the integration direction as clockwise, looking down on the can. We do this because the outward normal from the bottom of the can will be in the -az direction. H dL = 0 2 H 3d(-a ) = 2 3 sin 0 d = 12 A 2 With our choice of contour direction, this indicates that the current will flow in the negative z direction. Note for future reference that only the component of the given field contributed here. Next, we evalute H dS, over the surface of the tin can. We find H=J= 1 (H ) H - az = 1 - sin 1 + sin 2 4 2 az = - 3 sin az A/m 4 2 Note that both field components contribute here. The integral over the tin can is now only over the bottom surface, since H has only a z component. We use the outward normal, -az , and find H dS = - 3 4 2 0 0 3 1 9 sin az (-az ) d d = 2 4 2 sin 0 d = 9 A 2 Note that if the radial component of H were not included in the computation of H, then the factor of 3/4 in front of the above integral would change to a factor of 1, and the result would have been 12 A. What would appear to be a violation of Stokes' theorem is likely the result of a missing term in the component of H, having zero curl, which would have enabled the original line integral to have a value of 9A. The reader is invited to explore this further. 8.26. Let G = 15ra . a) Determine G dL for the circular path r = 5, = 25 , 0 2 : G dL = 0 2 15(5)a a (5) sin(25 ) d = 2(375) sin(25 ) = 995.8 b) Evaluate S ( G)dS over the spherical cap r = 5, 0 25 , 0 2: When evaluating the curl of G using the formula in spherical coordinates, only one of the six terms survives: G= Then ( G) dS = 0 2 25 0 25 0 1 (G sin ) 1 ar = 15r cos ar = 15 cot ar r sin r sin S 15 cot ar ar (5)2 sin d d = 2 15 cos (25) d = 2(15)(25) sin(25 ) = 995.8 131 8.27. The magnetic field intensity is given in a certain region of space as H= x + 2y 2 ay + az A/m 2 z z a) Find H: For this field, the general curl expression in rectangular coordinates simplifies to H=- Hy Hy 2(x + 2y) 1 ax + az = ax + 2 az A/m 3 z x z z b) Find J: This will be the answer of part a, since H = J. c) Use J to find the total current passing through the surface z = 4, 1 < x < 2, 3 < y < 5, in the az direction: This will be I= J az dx dy = z=4 5 3 1 2 1 dx dy = 1/8 A 42 d) Show that the same result is obtained using the other side of Stokes' theorem: We take H dL over the square path at z = 4 as defined in part c. This involves two integrals of the y component of H over the range 3 < y < 5. Integrals over x, to complete the loop, do not exist since there is no x component of H. We have I= H dL = z=4 5 3 2 + 2y dy + 16 3 5 1 1 1 + 2y dy = (2) - (2) = 1/8 A 16 8 16 8.28. Given H = (3r 2 / sin )a + 54r cos a A/m in free space: a) find the total current in the a direction through the conical surface = 20 , 0 2 , 0 r 5, by whatever side of Stokes' theorem you like best. I chose the line integral side, where the integration path is the circular path in around the top edge of the cone, at r = 5. The path direction is chosen to be clockwise looking down on the xy plane. This, by convention, leads to the normal from the cone surface that points in the positive a direction (right hand rule). We find H dL = 0 2 (3r 2 / sin )a + 54r cos a r=5, =20 5 sin(20 ) d (-a ) = -2(54)(25) cos(20 ) sin(20 ) = -2.73 103 A This result means that there is a component of current that enters the cone surface in the -a direction, to which is associated a component of H in the positive a direction. b) Check the result by using the other side of Stokes' theorem: We first find the current density through the curl of the magnetic field, where three of the six terms in the spherical coordinate formula survive: H= Thus J = 54 cot ar - 108 cos a + 1 1 1 54r 2 cos a + (54r cos sin )) ar - r sin r r r r 9r a sin 3r 3 sin a = J 132 8.28b. (continued) The calculation of the other side of Stokes' theorem now involves integrating J over the surface of the cone, where the outward normal is positive a , as defined in part a: ( H) dS = 0 2 0 2 0 0 5 S 54 cot ar - 108 cos a + 5 9r a sin =20 a r sin(20 ) dr d =- 108 cos(20 ) sin(20 )rdrd = -2(54)(25) cos(20 ) sin(20 ) = -2.73 103 A 8.29. A long straight non-magnetic conductor of 0.2 mm radius carries a uniformly-distributed current of 2 A dc. a) Find J within the conductor: Assuming the current is +z directed, J= 2 az = 1.59 107 az A/m2 (0.2 10-3 )2 b) Use Ampere's circuital law to find H and B within the conductor: Inside, at radius , we have 2H = 2 J H = J a = 7.96 106 a A/m 2 Then B = 0 H = (4 10-7 )(7.96 106 )a = 10 a Wb/m2 . c) Show that H = J within the conductor: Using the result of part b, we find, H= 1 d 1 d (H ) az = d d 1.59 107 2 2 az = 1.59 107 az A/m2 = J d) Find H and B outside the conductor (note typo in book): Outside, the entire current is enclosed by a closed path at radius , and so H= Now B = 0 H = 0 /() a Wb/m2 . e) Show that H = J outside the conductor: Here we use H outside the conductor and write: H= 1 d 1 d (H ) az = d d 1 az = 0 (as expected) I 1 a = a A/m 2 133 8.30. A solid nonmagnetic conductor of circular cross-section has a radius of 2mm. The conductor is inhomogeneous, with = 106 (1 + 106 2 ) S/m. If the conductor is 1m in length and has a voltage of 1mV between its ends, find: a) H inside: With current along the cylinder length (along az , and with symmetry, H will be directed only. We find E = (V0 /d)az = 10-3 az V/m. Then J = E = 103 (1 + 106 2 )az A/m2 . Next we apply Ampere's circuital law to a circular path of radius , centered on the z axis and normal to the axis: H dL = 2H = Thus H = Finally, H = 103 0 S J dS = 0 2 0 103 (1 + 106 ( )2 )az az d d 103 2 106 4 + 2 4 + 106 ( )3 d = 500(1 + 5 105 3 )a A/m (0 < < 2mm). b) the total magnetic flux inside the conductor: With field in the direction, a plane normal to B will be that in the region 0 < < 2 mm, 0 < z < 1 m. The flux will be = B dS = 0 0 1 0 210-3 S 500 + 2.5 108 3 ddz = 8 10-10 Wb = 2.5 nWb 8.31. The cylindrical shell defined by 1 cm < < 1.4 cm consists of a non-magnetic conducting material and carries a total current of 50 A in the az direction. Find the total magnetic flux crossing the plane = 0, 0 < z < 1: a) 0 < < 1.2 cm: We first need to find J, H, and B: The current density will be: J= [(1.4 10-2 )2 50 az = 1.66 105 az A/m2 - (1.0 10-2 )2 ] Next we find H at radius between 1.0 and 1.4 cm, by applying Ampere's circuital law, and noting that the current density is zero at radii less than 1 cm: 2H = Iencl = 2 0 10-2 1.66 105 d d ( 2 - 10-4 ) A/m (10-2 m < < 1.4 10-2 m) ( 2 - 10-4 ) a Wb/m2 10-4 1.2 1.0 H = 8.30 104 Then B = 0 H, or B = 0.104 Now, a = = 0.104 B dS = 1 1.210-2 10-2 0 -2 )2 - 10-4 (1.2 10 0.104 - - 10-4 ln d dz = 3.92 10-7 Wb = 0.392 Wb 2 134 8.31b) 1.0 cm < < 1.4 cm (note typo in book): This is part a over again, except we change the upper limit of the radial integration: b = = 0.104 B dS = 1 1.410-2 10-2 0 -2 )2 - 10-4 (1.4 10 0.104 - - 10-4 ln 10-4 1.4 1.0 d dz = 1.49 10-6 Wb = 1.49 Wb 2 c) 1.4 cm < < 20 cm: This is entirely outside the current distribution, so we need B there: We modify the Ampere's circuital law result of part a to find: Bout = 0.104 We now find c [(1.4 10-2 )2 - 10-4 ] 10-5 a = a Wb/m2 = 0 1 2010-2 1.410-2 20 10-5 d dz = 10-5 ln 1.4 = 2.7 10-5 Wb = 27 Wb 8.32. The free space region defined by 1 < z < 4 cm and 2 < < 3 cm is a toroid of rectangular cross-section. Let the surface at = 3 cm carry a surface current K = 2az kA/m. a) Specify the current densities on the surfaces at = 2 cm, z = 1cm, and z = 4cm. All surfaces must carry equal currents. With this requirement, we find: K( = 2) = -3 az kA/m. Next, the current densities on the z = 1 and z = 4 surfaces must transistion between the current density values at = 2 and = 3. Knowing the the radial current density will vary as 1/, we find K(z = 1) = (60/)a A/m with in meters. Similarly, K(z = 4) = -(60/)a A/m. b) Find H everywhere: Outside the toroid, H = 0. Inside, we apply Ampere's circuital law in the manner of Problem 8.14: H dL = 2H = H=- 2 0 K( = 2) az (2 10-2 ) d 2(3000)(.02) a = -60/ a A/m (inside) c) Calculate the total flux within the toriod: We have B = -(600 /)a Wb/m2 . Then = .04 .01 .03 .02 -600 3 a (-a ) d dz = (.03)(60)0 ln 2 = 0.92 Wb 8.33. Use an expansion in cartesian coordinates to show that the curl of the gradient of any scalar field G is identically equal to zero. We begin with G = and y + x G G G ax + ay + az x y z G y G x 135 ax + z G x - x G z ay G = G - z z G - y y az = 0 for any G 8.34. A filamentary conductor on the z axis carries a current of 16A in the az direction, a conducting shell at = 6 carries a total current of 12A in the -az direction, and another shell at = 10 carries a total current of 4A in the -az direction. a) Find H for 0 < < 12: Ampere's circuital law states that H dL = Iencl , where the line integral and current direction are related in the usual way through the right hand rule. Therefore, if I is in the positive z direction, H is in the a direction. We proceed as follows: 0 < < 6 : 2H = 16 H = 16/(2)a 6 < < 10 : 2H = 16 - 12 H = 4/(2)a > 10 : 2H = 16 - 12 - 4 = 0 H = 0 b) Plot H vs. : c) Find the total flux = 0 1 1 6 crossing the surface 1 < < 7, 0 < z < 1: This will be 160 d dz + 2 1 0 6 7 20 40 d dz = [4 ln 6 + ln(7/6)] = 5.9 Wb 2 8.35. A current sheet, K = 20 az A/m, is located at = 2, and a second sheet, K = -10 az A/m is located at = 4. a.) Let Vm = 0 at P ( = 3, = 0, z = 5) and place a barrier at = . Find Vm (, , z) for - < < : Since the current is cylindrically-symmetric, we know that H = I /(2) a , where I is the current enclosed, equal in this case to 2(2)K = 80 A. Thus, using the result of Section 8.6, we find 80 I =- = -40 A Vm = - 2 2 which is valid over the region 2 < < 4, - < < , and - < z < . For > 4, the outer current contributes, leading to a total enclosed current of Inet = 2(2)(20) - 2(4)(10) = 0 With zero enclosed current, H = 0, and the magnetic potential is zero as well. 136 8.35b. Let A = 0 at P and find A(, , z) for 2 < < 4: Again, we know that H = H (), since the current is cylindrically symmetric. With the current only in the z direction, and again using symmmetry, we expect only a z component of A which varies only with . We can then write: A=- Thus 0 I dAz a = B = a d 2 0 I 0 I dAz =- Az = - ln() + C d 2 2 We require that Az = 0 at = 3. Therefore C = [(0 I )/(2)] ln(3), Then, with I = 80, we finally obtain 3 0 (80) az Wb/m A=- [ln() - ln(3)] az = 400 ln 2 8.36. Let A = (3y - z)ax + 2xzay Wb/m in a certain region of free space. a) Show that A = 0: (3y - z) + 2xz = 0 A= x y b) At P (2, -1, 3), find A, B, H, and J: First AP = -6ax + 12ay . Then, using the curl formula in cartesian coordinates, B = A = -2xax - ay + (2z - 3)az BP = -4ax - ay + 3az Wb/m2 Now HP = (1/0 )BP = -3.2 106 ax - 8.0 105 ay + 2.4 106 az A/m Then J = H = (1/0 ) B = 0, as the curl formula in cartesian coordinates shows. 8.37. Let N = 1000, I = 0.8 A, 0 = 2 cm, and a = 0.8 cm for the toroid shown in Fig. 8.12b. Find Vm in the interior of the toroid if Vm = 0 at = 2.5 cm, = 0.3 . Keep within the range 0 < < 2: Well-within the toroid, we have H= Thus Vm = - Then, 0=- or C = 120. Finally Vm = 120 - 400 A (0 < < 2) 1000(0.8)(0.3) +C 2 NI 1 dVm a = -Vm = - a 2 d NI +C 2 137 8.38. The solenoid shown in Fig. 8.11b contains 400 turns, carries a current I = 5 A, has a length of 8cm, and a radius a = 1.2 cm (hope it doesn't blow up!). a) Find H within the solenoid. Assuming the current flows in the a direction, H will then be along the positive z direction, and will be given by H= (400)(5) NI az = az = 2.5 104 A/m d .08 b) If Vm = 0 at the origin, specify Vm (, , z) inside the solenoid: Since H is only in the z direction, Vm should vary with z only. Use H = -Vm = - dVm az Vm = -Hz z + C dz At z = 0, Vm = 0, so C = 0. Therefore Vm (z) = -2.5 104 z A c) Let A = 0 at the origin, and specify A(, , z) inside the solenoid if the medium is free space. A should be in the same direction as the current, and so would have a component only. Furthermore, since A = B, the curl will be z-directed only. Therefore A= Then 1 (A )az = 0 Hz az 0 Hz (A ) = 0 Hz A = +C 2 (4 10-7 )(2.5 104 ) a = 15.7a mWb/m 2 A = 0 at the origin, so C = 0. Finally, A= 8.39. Planar current sheets of K = 30az A/m and -30az A/m are located in free space at x = 0.2 and x = -0.2 respectively. For the region -0.2 < x < 0.2: a) Find H: Since we have parallel current sheets carrying equal and opposite currents, we use Eq. (12), H = K aN , where aN is the unit normal directed into the region between currents, and where either one of the two currents are used. Choosing the sheet at x = 0.2, we find H = 30az -ax = -30ay A/m b) Obtain and expression for Vm if Vm = 0 at P (0.1, 0.2, 0.3): Use H = -30ay = -Vm = - So dVm ay dy dVm = 30 Vm = 30y + C1 dy 0 = 30(0.2) + C1 C1 = -6 Vm = 30y - 6 A 138 Then 8.39c) Find B: B = 0 H = -300 ay Wb/m2 . d) Obtain an expression for A if A = 0 at P : We expect A to be z-directed (with the current), and so from A = B, where B is y-directed, we set up - Then 0 = 300 (0.1) + C2 C2 = -30 So finally A = 0 (30x - 3)az Wb/m 8.40. Let A = (3y 2 - 2z)ax - 2x 2 zay + (x + 2y)az Wb/m in free space. Find A at P (-2, 3, -1): First A = (x + 2y) (-2x 2 z) - ax + y z (3y 2 - 2z) (x + 2y) - ay + z x (-2x 2 z) (3y 2 - 2z) - az x y dAz = -300 Az = 300 x + C2 dx = (2 + 2x 2 )ax - 3ay - (4xz + 6y)az Then A= (4xz + 6y) (4xz + 6y) ay - ax = -6ax + 4zay x y At P this becomes A|P = -6ax - 4ay Wb/m3 . 8.41. Assume that A = 50 2 az Wb/m in a certain region of free space. a) Find H and B: Use Az a = -100 a Wb/m2 B= A=- Then H = B/0 = -100/0 a A/m. b) Find J: Use J= H= 1 1 (H )az = -100 2 0 az = - 200 az A/m2 0 c) Use J to find the total current crossing the surface 0 1, 0 < 2 , z = 0: The current is I= J dS = 0 2 0 1 -200 -200 az az d d = A = -500 kA 0 0 H dL for = 1, z = 0: Have d) Use the value of H at = 1 to calculate H dL = I = 0 2 -100 -200 a a (1)d = A = -500 kA 0 0 139 3 8.42. Show that 2 (1/R12 ) = -1 (1/R12 ) = R21 /R12 . First 2 1 R12 = 2 (x2 - x1 )2 + (y2 - y1 )2 + (z2 - z1 )2 =- -1/2 -R12 1 2(x2 - x1 )ax + 2(y2 - y1 )ay + 2(z2 - z1 )az R21 = = 3 3 2 + (y - y )2 + (z - z )2 ]3/2 2 [(x2 - x1 ) R12 R12 2 1 2 1 Also note that 1 (1/R12 ) would give the same result, but of opposite sign. 8.43. Compute the vector magnetic potential within the outer conductor for the coaxial line whose vector magnetic potential is shown in Fig. 8.20 if the outer radius of the outer conductor is 7a. Select the proper zero reference and sketch the results on the figure: We do this by first finding B within the outer conductor and then "uncurling" the result to find A. With -z-directed current I in the outer conductor, the current density is I I Jout = - az = - az (7a)2 - (5a)2 24a 2 Since current I flows in both conductors, but in opposite directions, Ampere's circuital law inside the outer conductor gives: 2H = I - 2 0 5a I I d d H = 2 24a 2 49a 2 - 2 24a 2 Now, with B = 0 H, we note that A will have a component only, and from the direction and symmetry of the current, we expect A to be z-directed, and to vary only with . Therefore A=- and so dAz a = 0 H d 49a 2 - 2 24a 2 dAz 0 I =- d 2 Then by direct integration, Az = -0 I (49) d + 48 0 I 0 I d + C = 2 48a 96 2 - 98 ln + C a2 As per Fig. 8.20, we establish a zero reference at = 5a, enabling the evaluation of the integration constant: 0 I C=- [25 - 98 ln(5a)] 96 Finally, 2 0 I 5a Az = Wb/m - 25 + 98 ln 2 96 a A plot of this continues the plot of Fig. 8.20, in which the curve goes negative at = 5a, and then approaches a minimum of -.090 I / at = 7a, at which point the slope becomes zero. 140 8.44. By expanding Eq.(58), Sec. 8.7 in cartesian coordinates, show that (59) is correct. Eq. (58) can be rewritten as 2 A = ( A) - A We begin with A= Then the x component of ( A) is [( A)]x = Now A= 2 Ay 2 Az 2 Ax + + x 2 xy xz Ax Az - z x Ay Ax - x y Ay Ax Az + + x y z Ay Az - y z ax + ay + az and the x component of A is [ A]x = Then, using the underlined results [( A) - A]x = 2 Ax 2 Ax 2 Ax + + = 2 Ax 2 2 2 x y z 2 Ay 2 Ax 2 Ax 2 Az - - + xy y 2 z2 zy Similar results will be found for the other two components, leading to ( A) - A = 2 Ax ax + 2 Ay ay + 2 Az az 2 A QED 141 CHAPTER 9 9.1. A point charge, Q = -0.3 C and m = 3 10-16 kg, is moving through the field E = 30 az V/m. Use Eq. (1) and Newton's laws to develop the appropriate differential equations and solve them, subject to the initial conditions at t = 0: v = 3 105 ax m/s at the origin. At t = 3 s, find: a) the position P (x, y, z) of the charge: The force on the charge is given by F = qE, and Newton's second law becomes: F = ma = m d 2z = qE = (-0.3 10-6 )(30 az ) dt 2 describing motion of the charge in the z direction. The initial velocity in x is constant, and so no force is applied in that direction. We integrate once: dz qE = vz = t + C1 dt m The initial velocity along z, vz (0) is zero, and so C1 = 0. Integrating a second time yields the z coordinate: qE 2 t + C2 z= 2m The charge lies at the origin at t = 0, and so C2 = 0. Introducing the given values, we find z= (-0.3 10-6 )(30) 2 t = -1.5 1010 t 2 m 2 3 10-16 At t = 3 s, z = -(1.5 1010 )(3 10-6 )2 = -.135 cm. Now, considering the initial constant velocity in x, the charge in 3 s attains an x coordinate of x = vt = (3 105 )(3 10-6 ) = .90 m. In summary, at t = 3 s we have P (x, y, z) = (.90, 0, -.135). b) the velocity, v: After the first integration in part a, we find vz = qE t = -(3 1010 )(3 10-6 ) = -9 104 m/s m Including the intial x-directed velocity, we finally obtain v = 3 105 ax - 9 104 az m/s. c) the kinetic energy of the charge: Have K.E. = 1 1 m|v|2 = (3 10-16 )(1.13 105 )2 = 1.5 10-5 J 2 2 9.2. A point charge, Q = -0.3 C and m = 3 10-16 kg, is moving through the field B = 30az mT. Make use of Eq. (2) and Newton's laws to develop the appropriate differential equations, and solve them, subject to the initial condition at t = 0, v = 3 105 m/s at the origin. Solve these equations (perhaps with the help of an example given in Section 7.5) to evaluate at t = 3s: a) the position P (x, y, z) of the charge; b) its velocity; c) and its kinetic energy: We begin by visualizing the problem. Using F = qv B, we find that a positive charge moving along positive ax , would encounter the z-directed B field and be deflected into the negative y direction. 142 9.2 (continued) Motion along negative y through the field would cause further deflection into the negative x direction. We can construct the differential equations for the forces in x and in y as follows: Fx ax = m Fy ay = m or dvx ax = qvy ay Baz = qBvy ax dt dvy ay = qvx ax Baz = -qBvx ay dt dvx qB = vy dt m (1) (2) dvy qB =- vx dt m To solve these equations, we first differentiate (2) with time and substitute (1), obtaining: d 2 vy qB qB dvx =- =- dt 2 m dt m 2 and vy Therefore, vy = A sin(qBt/m) + A cos(qBt/m). However, at t = 0, vy = 0, and so A = 0, leaving vy = A sin(qBt/m). Then, using (2), vx = - qBt m dvy = -A cos qB dt m Now at t = 0, vx = vx0 = 3 105 . Therefore A = -vx0 , and so vx = vx0 cos(qBt/m), and vy = -vx0 sin(qBt/m). The positions are then found by integrating vx and vy over time: x(t) = vx0 cos qBt m dt + C = mvx0 qBt sin qB m +C where C = 0, since x(0) = 0. Then y(t) = -vx0 sin qBt m dt + D = mvx0 qBt cos qB m +D We require that y(0) = 0, so D = -(mvx0 )/(qB), and finally y(t) = -mvx0 /qB [1 - cos (qBt/m)]. Summarizing, we have, using q = -310-7 C, m = 310-16 kg, B = 3010-3 T, and vx0 = 3105 m/s: qBt mvx0 sin x(t) = = -10-2 sin(-3 10-7 t) m qB m y(t) = - mvx0 qBt 1 - cos qB m qBt m qBt m = 10-2 [1 - cos(-3 107 t)] m vx (t) = vx0 cos vy (t) = -vx0 sin = 3 105 cos(-3 107 t) m/s = -3 105 sin(-3 107 t) m/s 143 9.2 (continued) The answers are now: a) At t = 3 10-6 s, x = 8.9 mm, y = 14.5 mm, and z = 0. b) At t = 3 10-6 s, vx = -1.3 105 m/s, vy = 2.7 105 m/s, and so v(t = 3 s) = -1.3 105 ax + 2.7 105 ay m/s whose magnitude is v = 3 105 m/s as would be expected. c) Kinetic energy is K.E. = (1/2)mv 2 = 1.35 J at all times. 9.3. A point charge for which Q = 2 10-16 C and m = 5 10-26 kg is moving in the combined fields E = 100ax - 200ay + 300az V/m and B = -3ax + 2ay - az mT. If the charge velocity at t = 0 is v(0) = (2ax - 3ay - 4az ) 105 m/s: a) give the unit vector showing the direction in which the charge is accelerating at t = 0: Use F(t = 0) = q[E + (v(0) B)], where v(0) B = (2ax - 3ay - 4az )105 (-3ax + 2ay - az )10-3 = 1100ax + 1400ay - 500az So the force in newtons becomes F(0) = (210-16 )[(100+1100)ax +(1400-200)ay +(300-500)az ] = 410-14 [6ax +6ay -az ] The unit vector that gives the acceleration direction is found from the force to be aF = 6ax + 6ay - az = .70ax + .70ay - .12az 73 b) find the kinetic energy of the charge at t = 0: K.E. = 1 1 m|v(0)|2 = (5 10-26 kg)(5.39 105 m/s)2 = 7.25 10-15 J = 7.25 fJ 2 2 9.4. An electron (qe = -1.60219 10-19 C, m = 9.10956 10-31 kg) is moving at a constant velocity v = 4.5 107 ay m/s along the negative y axis. At the origin it encounters the uniform magnetic field B = 2.5az mT, and remains in it up to y = 2.5 cm. If we assume (with good accuracy) that the electron remains on the y axis while it is in the magnetic field, find its x-, y-, and z-coordinate values when y = 50 cm: The procedure is to find the electron velocity as it leaves the field, and then determine its coordinates at the time corresponding to y = 50 cm. The force it encounters while in the field is F = qv B = (-1.60219 10-19 )(4.5 107 )(2.5 10-3 )(ay az ) = -1.80 10-14 ax N This force will be constant during the time the electron traverses the field. It establishes a negative x-directed velocity as it leaves the field, given by the acceleration times the transit time, tt : vx = F tt = m -1.80 1014 N 9.10956 10-31 kg 2.5 10-2 m 4.5 107 m/s = -1.09 107 m/s 144 9.4 (continued) The time for the electron to travel along y between 2.5 and 50 cm is t50 = (50 - 2.5) 10-2 = 1.06 10-8 s 4.5 107 In that time, the electron moves to an x coordinate given by x = vx t50 = -(1.09 107 )(1.06 10-8 ) = -.115 m The coordinates at the time the electron reaches y = 50 cm are then: x = -11.5 cm, y = 50 cm, z = 0 9.5. A rectangular loop of wire in free space joins points A(1, 0, 1) to B(3, 0, 1) to C(3, 0, 4) to D(1, 0, 4) to A. The wire carries a current of 6 mA, flowing in the az direction from B to C. A filamentary current of 15 A flows along the entire z axis in the az direction. a) Find F on side BC: FBC = Thus FBC = 1 4 C B Iloop dL Bfrom wire at BC 150 ay = -1.8 10-8 ax N = -18ax nN 2(3) (6 10-3 ) dz az b) Find F on side AB: The field from the long wire now varies with position along the loop segment. We include that dependence and write FAB = 3 1 (6 10-3 ) dx ax 150 45 10-3 ay = 0 ln 3 az = 19.8az nN 2x c) Find Ftotal on the loop: This will be the vector sum of the forces on the four sides. Note that by symmetry, the forces on sides AB and CD will be equal and opposite, and so will cancel. This leaves the sum of forces on sides BC (part a) and DA, where FDA = 4 1 -(6 10-3 ) dz az 150 ay = 54ax nN 2(1) The total force is then Ftotal = FDA + FBC = (54 - 18)ax = 36 ax nN 9.6 The magnetic flux density in a region of free space is given by B = -3xax + 5yay - 2zaz T. Find the total force on the rectangular loop shown in Fig. 9.15 if it lies in the plane z = 0 and is bounded by x = 1, x = 3, y = 2, and y = 5, all dimensions in cm: First, note that in the plane z = 0, the z component of the given field is zero, so will not contribute to the force. We use F= which in our case becomes, with I = 30 A: F= + .03 .01 .01 .03 loop I dL B 30dxax (-3xax + 5y|y=.02 ay ) + 30dxax (-3xax + 5y|y=.05 ay ) + .05 .02 .02 .05 30dyay (-3x|x=.03 ax + 5yay ) 30dyay (-3x|x=.01 ax + 5yay ) 145 9.6. (continued) Simplifying, this becomes F= + .03 .01 .01 .03 30(5)(.02) az dx + 30(5)(.05) az dx + .05 .02 .02 .05 -30(3)(.03)(-az ) dy -30(3)(.01)(-az ) dy = (.060 + .081 - .150 - .027)az N = -36 az mN 9.7. Uniform current sheets are located in free space as follows: 8az A/m at y = 0, -4az A/m at y = 1, and -4az A/m at y = -1. Find the vector force per meter length exerted on a current filament carrying 7 mA in the aL direction if the filament is located at: a) x = 0, y = 0.5, and aL = az : We first note that within the region -1 < y < 1, the magnetic fields from the two outer sheets (carrying -4az A/m) cancel, leaving only the field from the center sheet. Therefore, H = -4ax A/m (0 < y < 1) and H = 4ax A/m (-1 < y < 0). Outside (y > 1 and y < -1) the fields from all three sheets cancel, leaving H = 0 (y > 1, y < -1). So at x = 0, y = .5, the force per meter length will be F/m = I az B = (7 10-3 )az -40 ax = -35.2ay nN/m b.) y = 0.5, z = 0, and aL = ax : F/m = I ax -40 ax = 0. c) x = 0, y = 1.5, aL = az : Since y = 1.5, we are in the region in which B = 0, and so the force is zero. 9.8. Filamentary currents of -25az and 25az A are located in the x = 0 plane in free space at y = -1 and y = 1m respectively. A third filamentary current of 10-3 az A is located at x = k, y = 0. Find the vector force on a 1-m length of the 1-mA filament and plot |F| versus k: The total B field arising from the two 25A filaments evaluated at the location of the 1-mA filament is, in cartesian components: 250 250 250 ax B= (kay + ax ) + (-kay + ax ) = 2) 2) 2(1 + k 2(1 + k (1 + k 2 ) line at y=+1 line at y=-1 The force on the 1m length of 1-mA line is now F = 10-3 (1)az 10-8 ay 10ay 250 ax (2.5 10-2 )(4 10-7 ) a N= nN = ay = 2) 2) 2) y (1 + k (1 + k (1 + k (1 + k 2 ) 146 9.9. A current of -100az A/m flows on the conducting cylinder = 5 mm and +500az A/m is present on the conducting cylinder = 1 mm. Find the magnitude of the total force acting to split the outer cylinder apart along its length: The differential force acting on the outer cylinder arising from the field of the inner cylinder is dF = Kouter B, where B is the field from the inner cylinder, evaluated at the outer cylinder location: 2(1)(500)0 a = 1000 a T B= 2(5) Thus dF = -100az 1000 a = 104 0 a N/m2 . We wish to find the force acting to split the outer cylinder, which means we need to evaluate the net force in one cartesian direction on one half of the cylinder. We choose the "upper" half (0 < < ), and integrate the y component of dF over this range, and over a unit length in the z direction: Fy = 1 0 0 104 0 a ay (5 10-3 ) d dz = 0 500 sin d = 1000 = 4 10-5 N/m Note that we did not include the "self force" arising from the outer cylinder's B field on itself. Since the outer cylinder is a two-dimensional current sheet, its field exists only just outside the cylinder, and so no force exists. If this cylinder possessed a finite thickness, then we would need to include its self-force, since there would be an interior field and a volume current density that would spatially overlap. 9.10. Two infinitely-long parallel filaments each carry 50 A in the az direction. If the filaments lie in the plane y = 0 at x = 0 and x = 5mm (note bad wording in problem statement in book), find the vector force per meter length on the filament passing through the origin: The force will be F= 0 1 I dL B where I dL is that of the filament at the origin, and B is that arising from the filament at x = 5mm evaluated at the location of the other filament (along the z axis). We obtain F= 0 1 50 dzaz -500 ay = 0.10 ax N/m 2(5 10-3 ) 9.11. a) Use Eq. (14), Sec. 9.3, to show that the force of attraction per unit length between two filamentary conductors in free space with currents I1 az at x = 0, y = d/2, and I2 az at x = 0, y = -d/2, is 0 I1 I2 /(2d): The force on I2 is given by F2 = 0 I1 I2 4 aR12 dL1 2 R12 dL2 Let z1 indicate the z coordinate along I1 , and z2 indicate the z coordinate along I2 . We then have R12 = (z2 - z1 )2 + d 2 and (z2 - z1 )az - day aR12 = (z2 - z1 )2 + d 2 Also, dL1 = dz1 az and dL2 = dz2 az The "inside" integral becomes: aR12 dL1 = 2 R12 [(z2 - z1 )az - day ] dz1 az = [(z2 - z1 )2 + d 2 ]1.5 - -d dz1 ax [(z2 - z1 )2 + d 2 ]1.5 147 9.11a. (continued) The force expression now becomes F2 = 0 I1 I2 4 - -d dz1 ax I1 I2 dz2 az = 0 2 + d 2 ]1.5 4 [(z2 - z1 ) 1 0 - d dz1 dz2 ay [(z2 - z1 )2 + d 2 ]1.5 Note that the "outside" integral is taken over a unit length of current I2 . Evaluating, obtain, F2 = 0 as expected. b) Show how a simpler method can be used to check your result: We use dF2 = I2 dL2 B12 , where the field from current 1 at the location of current 2 is B12 = so over a unit length of I2 , we obtain F2 = I2 az I1 I2 0 I1 ax = 0 ay N/m 2d 2d 0 I1 ax T 2d I1 I2 d ay (2) 4d 2 1 0 dz2 = 0 I1 I2 ay N/m 2d This second method is really just the first over again, since we recognize the inside integral of the first method as the Biot-Savart law, used to find the field from current 1 at the current 2 location. 9.12. A conducting current strip carrying K = 12az A/m lies in the x = 0 plane between y = 0.5 and y = 1.5 m. There is also a current filament of I = 5 A in the az direction on the z axis. Find the force exerted on the: a) filament by the current strip: We first need to find the field from the current strip at the filament location. Consider the strip as made up of many adjacent strips of width dy, each carrying current dI az = Kdy. The field along the z axis from each differential strip will be dB = [(Kdy0 )/(2y)]ax . The total B field from the strip evaluated along the z axis is therefore B= Now F= 0 1.5 0.5 60 1.5 120 ax dy = ln ax = 2.64 10-6 ax Wb/m2 2y 0.5 1 0 1 I dL B = 5dz az 2.64 10-6 ax dz = 13.2 ay N/m b) strip by the filament: In this case we integrate K B over a unit length in z of the strip area, where B is the field from the filament evaluated on the strip surface: F= K B da = 0 1 1.5 0.5 Area 12az -300 -50 ax dy = ln(3) ay = -13.2 ay N/m 2y 148 9.13. A current of 6A flows from M(2, 0, 5) to N(5, 0, 5) in a straight solid conductor in free space. An infinite current filament lies along the z axis and carries 50A in the az direction. Compute the vector torque on the wire segment using: a) an origin at (0, 0, 5): The B field from the long wire at the short wire is B = (0 Iz ay )/(2 x) T. Then the force acting on a differential length of the wire segment is dF = Iw dL B = Iw dx ax Now the differential torque about (0, 0, 5) will be dT = RT dF = xax 0 Iw Iz 0 Iw Iz dx az = - dx ay 2x 2 0 Iz 0 Iw Iz ay = dx az N 2x 2x The net torque is now found by integrating the differential torque over the length of the wire segment: T= 2 5 - 0 Iw Iz 30 (6)(50) dx ay = - ay = -1.8 10-4 ay N m 2 2 b) an origin at (0, 0, 0): Here, the only modification is in RT , which is now RT = x ax + 5 az So now 0 Iw Iz 0 Iw Iz dx az = - dx ay dT = RT dF = xax + 5az 2x 2 Everything from here is the same as in part a, so again, T = -1.8 10-4 ay N m. c) an origin at (3, 0, 0): In this case, RT = (x - 3)ax + 5az , and the differential torque is dT = (x - 3)ax + 5az Thus T= 2 5 0 Iw Iz 0 Iw Iz (x - 3) dx az = - dx ay 2x 2x - 5 0 Iw Iz (x - 3) dx ay = -6.0 10-5 3 - 3 ln 2x 2 ay = -1.5 10-5 ay N m 9.14. The rectangular loop of Prob. 6 is now subjected to the B field produced by two current sheets, K1 = 400 ay A/m at z = 2, and K2 = 300 az A/m at y = 0 in free space. Find the vector torque on the loop, referred to an origin: a) at (0,0,0): The fields from both current sheets, at the loop location, will be negative x-directed. They will add together to give, in the loop plane: B = -0 K1 K2 + 2 2 ax = -0 (200 + 150) ax = -3500 ax Wb/m2 With this field, forces will be acting only on the wire segments that are parallel to the y axis. The force on the segment nearer to the y axis will be F1 = I L B = -30(3 10-2 )ay -3500 ax = -3150 az N 149 9.14a (continued) The force acting on the segment farther from the y axis will be F2 = I L B = 30(3 10-2 )ay -3500 ax = 3150 az N The torque about the origin is now T = R1 F1 + R2 F2 , where R1 is the vector directed from the origin to the midpoint of the nearer y-directed segment, and R2 is the vector joining the origin to the midpoint of the farther y-directed segment. So R1 (cm) = ax + 3.5ay and R2 (cm) = 3ax + 3.5ay . Therefore T0,0,0 = [(ax + 3.5ay ) 10-2 ] -3150 az + [(3ax + 3.5ay ) 10-2 ] 3150 az = -6.300 ay = -7.92 10-6 ay N-m b) at the center of the loop: Use T = I S B where S = (2 3) 10-4 az m2 . So T = 30(6 10-4 az ) (-3500 ax ) = -7.92 10-6 ay N-m 9.15. A solid conducting filament extends from x = -b to x = b along the line y = 2, z = 0. This filament carries a current of 3 A in the ax direction. An infinite filament on the z axis carries 5 A in the az direction. Obtain an expression for the torque exerted on the finite conductor about an origin located at (0, 2, 0): The differential force on the wire segment arising from the field from the infinite wire is dF = 3 dx ax 50 150 cos dx 150 x dx a = - az az = - 2 2(x 2 + 4) 2 x 2 + 4 So now the differential torque about the (0, 2, 0) origin is dT = RT dF = x ax - The torque is then T= 150 x 150 x 2 dx a = ay x - 2 tan-1 2 + 4) y 2 2 -b 2(x b = (6 10-6 ) b - 2 tan-1 ay N m 2 b b -b 150 x dx 150 x 2 dx az = ay 2(x 2 + 4) 2(x 2 + 4) 9.16. Assume that an electron is describing a circular orbit of radius a about a positively-charged nucleus. a) By selecting an appropriate current and area, show that the equivalent orbital dipole moment is e ea 2 /2, where is the electron's angular velocity: The current magnitude will be I = T , where e is the electron charge and T is the orbital period. The latter is T = 2/, and so I = e/(2 ). Now the dipole moment magnitude will be m = I A, where A is the loop area. Thus m= 1 e a 2 = ea 2 // 2 2 b) Show that the torque produced by a magnetic field parallel to the plane of the orbit is ea 2 B/2: With B assumed constant over the loop area, we would have T = m B. With B parallel to the loop plane, m and B are orthogonal, and so T = mB. So, using part a, T = ea 2 B/2. 150 9.16. (continued) c) by equating the Coulomb and centrifugal forces, show that is (4 0 me a 3 /e2 )-1/2 , where me is the electron mass: The force balance is written as e2 = me 2 a = 4 0 a 2 4 0 me a 3 e2 -1/2 // d) Find values for the angular velocity, torque, and the orbital magnetic moment for a hydrogen atom, where a is about 6 10-11 m; let B = 0.5 T: First (1.60 10-19 )2 = 4(8.85 10-12 )(9.1 10-31 )(6 10-11 )3 T = Finally, m= 1/2 = 3.42 1016 rad/s 1 (3.42 1016 )(1.60 10-19 )(0.5)(6 10-11 )2 = 4.93 10-24 N m 2 T = 9.86 10-24 A m2 B 9.17. The hydrogen atom described in Problem 16 is now subjected to a magnetic field having the same direction as that of the atom. Show that the forces caused by B result in a decrease of the angular velocity by eB/(2me ) and a decrease in the orbital moment by e2 a 2 B/(4me ). What are these decreases for the hydrogen atom in parts per million for an external magnetic flux density of 0.5 T? We first write down all forces on the electron, in which we equate its coulomb force toward the nucleus to the sum of the centrifugal force and the force associated with the applied B field. With the field applied in the same direction as that of the atom, this would yield a Lorentz force that is radially outward in the same direction as the centrifugal force. Fe = Fcent + FB With B = 0, we solve for to find: = 0 = Then with B present, we find 2 = Therefore = 0 1 - . But = 0 , and so eB eB . = 0 1 - 2m 2 0 e 20 me eB 20 me 151 = 0 - eB // 2me e2 eB eB 2 - = 0 - 4 0 me a 3 me me e2 4 0 me a 3 e2 = me 2 a + eaB 4 0 a 2 QvB . = 0 1 - 9.17. (continued) As for the magnetic moment, we have m = IS = e 2 1 eB . 1 a = ea 2 = ea 2 0 - 2 2 2 2me = 1 1 e2 a 2 B 0 ea 2 - // 2 4 me Finally, for a = 6 10-11 m, B = 0.5 T, we have eB 1 . eB 1 1.60 10-19 0.5 = = = = 1.3 10-6 2me 2me 0 2 9.1 10-31 3.4 1016 where 0 = 3.4 1016 sec-1 is found from Problem 16. Finally, m e2 a 2 B 2 . eB = = = 1.3 10-6 m 4me ea 2 2me 0 9.18. Calculate the vector torque on the square loop shown in Fig. 9.16 about an origin at A in the field B, given: a) A(0, 0, 0) and B = 100ay mT: The field is uniform and so does not produce any translation of the loop. Therefore, we may use T = I S B about any origin, where I = 0.6 A and S = 16az m2 . We find T = 0.6(16)az 0.100ay = -0.96 ax N-m. b) A(0, 0, 0) and B = 200ax + 100ay mT: Using the same reasoning as in part a, we find T = 0.6(16)az (0.200ax + 0.100ay ) = -0.96ax + 1.92ay N-m c) A(1, 2, 3) and B = 200ax + 100ay - 300az mT: We observe two things here: 1) The field is again uniform and so again the torque is independent of the origin chosen, and 2) The field differs from that of part b only by the addition of a z component. With S in the z direction, this new component of B will produce no torque, so the answer is the same as part b, or T = -0.96ax + 1.92ay N-m. d) A(1, 2, 3) and B = 200ax + 100ay - 300az mT for x 2 and B = 0 elsewhere: Now, force is acting only on the y-directed segment at x = +2, so we need to be careful, since translation will occur. So we must use the given origin. The differential torque acting on the differential wire segment at location (2,y) is dT = R(y) dF, where dF = I dL B = 0.6 dy ay [0.2ax + 0.1ay - 0.3az ] = [-0.18ax - 0.12az ] dy and R(y) = (2, y, 0) - (1, 2, 3) = ax + (y - 2)ay - 3az . We thus find dT = R(y) dF = ax + (y - 2)ay - 3az [-0.18ax - 0.12az ] dy = -0.12(y - 2)ax + 0.66ay + 0.18(y - 2)az dy The net torque is now T= 2 -2 -0.12(y - 2)ax + 0.66ay + 0.18(y - 2)az dy = 0.96ax + 2.64ay - 1.44az N-m 152 9.19. Given a material for which m = 3.1 and within which B = 0.4yaz T, find: a) H: We use B = 0 (1 + m )H, or 0.4yay = 77.6yaz kA/m H= (1 + 3.1)0 b) = (1 + 3.1)0 = 5.15 10-6 H/m. c) R = (1 + 3.1) = 4.1. d) M = m H = (3.1)(77.6yay ) = 241yaz kA/m e) J = H = (dHz )/(dy) ax = 77.6 ax kA/m2 . f) Jb = M = (dMz )/(dy) ax = 241 ax kA/m2 . g) JT = B/0 = 318ax kA/m2 . 9.20. Find H in a material where: a) R = 4.2, there are 2.7 1029 atoms/m3 , and each atom has a dipole moment of 2.6 10-30 ay A m2 . Since all dipoles are identical, we may write M = Nm = (2.7 1029 )(2.6 10-30 ay ) = 0.70ay A/m. Then 0.70 ay M H= = = 0.22 ay A/m R - 1 4.2 - 1 b) M = 270 az A/m and = 2 H/m: Have R = /0 = (2 10-6 )/(4 10-7 ) = 1.59. Then H = 270az /(1.59 - 1) = 456 az A/m. c) m = 0.7 and B = 2az T: Use H= B 2az = = 936 az kA/m 0 (1 + m ) (4 10-7 )(1.7) d) Find M in a material where bound surface current densities of 12 az A/m and -9 az A/m exist at = 0.3 m and = 0.4 m, respectively: We use M dL = Ib , where, since currents are in the z direction and are symmetric about the z axis, we chose the path integrals to be circular loops centered on and normal to z. From the symmetry, M will be -directed and will vary only with radius. Note first that for < 0.3 m, no bound current will be enclosed by a path integral, so we conclude that M = 0 for < 0.3m. At radii between the currents the path integral will enclose only the inner current so, 3.6 a A/m (0.3 < < 0.4m) M dL = 2M = 2(0.3)12 M = Finally, for > 0.4 m, the total enclosed bound current is Ib,tot = 2(0.3)(12)-2(0.4)(9) = 0, so therefore M = 0 ( > 0.4m). 9.21. Find the magnitude of the magnetization in a material for which: a) the magnetic flux density is 0.02 Wb/m2 and the magnetic susceptibility is 0.003 (note that this latter quantity is missing in the original problem statement): From B = 0 (H + M) and from M = m H, we write B M= 0 1 +1 m -1 = 0.02 B = = 47.7 A/m 0 (334) (4 10-7 )(334) 153 9.21b) the magnetic field intensity is 1200 A/m and the relative permeability is 1.005: From B = 0 (H+M) = 0 R H, we write M = (R - 1)H = (.005)(1200) = 6.0 A/m c) there are 7.2 1028 atoms per cubic meter, each having a dipole moment of 4 10-30 A m2 in the same direction, and the magnetic susceptibility is 0.0003: With all dipoles identical the dipole moment density becomes M = n m = (7.2 1028 )(4 10-30 ) = 0.288 A/m 9.22. Three current sheets are located as follows: 160az A/m at x = 1cm, -40az A/m at x = 5cm, and 50az A/m at x = 8cm. Let = 0 for x < 1cm and x > 8cm; for 1 < x < 5 cm, = 30 , and for 5 < x < 8cm, = 20 . Find B everywhere: We know that the H field from an infinite current sheet will be given in magnitude by H = K/2, and will be directed parallel to the sheet and perpendicular to the current, with the directions on either side of the sheet determined by the right hand rule. With this in mind, we can construct the following expressions for the B field in all four regions: B(x < 1) = 1 0 (-160 + 40 - 50) = -1.07 10-4 ay T 2 1 (30 )(160 + 40 - 50) = 2.83 10-4 ay T 2 1 B(5 < x < 8) = (20 )(160 - 40 - 50) = 8.80 10-5 ay T 2 1 B(x > 8) = 0 (160 - 40 + 50) = 1.07 10-4 ay T 2 B(1 < x < 5) = 9.23. Calculate values for H , B , and M at = c for a coaxial cable with a = 2.5 mm and b = 6 mm if it carries current I = 12 A in the center conductor, and = 3 H/m for 2.5 < < 3.5 mm, = 5 H/m for 3.5 < < 4.5 mm, and = 10 H/m for 4.5 < < 6 mm. Compute for: a) c = 3 mm: Have I 12 H = = = 637 A/m 2 2(3 10-3 ) Then B = H = (3 10-6 )(637) = 1.91 10-3 Wb/m2 . Finally, M = (1/0 )B - H = 884 A/m. b) c = 4 mm: Have H = 12 I = = 478 A/m 2 2(4 10-3 ) Then B = H = (5 10-6 )(478) = 2.39 10-3 Wb/m2 . Finally, M = (1/0 )B - H = 1.42 103 A/m. c) c = 5 mm: Have H = 12 I = = 382 A/m 2 2(5 10-3 ) Then B = H = (10 10-6 )(382) = 3.82 10-3 Wb/m2 . Finally, M = (1/0 )B - H = 2.66 103 A/m. 154 9.24. A coaxial transmission line has a = 5 mm and b = 20 mm. Let its center lie on the z axis and let a dc current I flow in the az direction in the center conductor. The volume between the conductors contains a magnetic material for which R = 2.5, as well as air. Find H, B, and M everywhere between conductors if H = 600/ A/m at = 10 mm, = /2, and the magnetic material is located where: a) a < < 3a; First, we know that H = I /2, from which we construct: I 600 = I = 12 A 2(10-2 ) Since the interface between the two media lies in the a direction, we use the boundary condition of continuity of tangential H and write H(5 < < 20) = In the magnetic material, we find B(5 < < 15) = H = (2.5)(4 10-7 )(12) a = (6/)a T 2 12 6 a = a A/m 2 Then, in the free space region, B(15 < < 20) = 0 H = (2.4/)a T. b) 0 < < ; Again, we are given H = 600/ a A/m at = 10 and at = /2. Now, since the interface between media lies in the a direction, and noting that magnetic field will be normal to this (a directed), we use the boundary condition of continuity of B normal to an interface, and write B(0 < < ) = B1 = B( < < 2) = B2 , or 2.50 H1 = 0 H2 . Now, using Ampere's circuital law, we write H dL = H1 + H2 = 3.5H1 = I Using the given value for H1 at = 10 mm, I = 3.5(600/)( 10-2 ) = 21 A. Therefore, H1 = 21/(3.5) = 6/(), or H(0 < < ) = 6/() a A/m. Then H2 = 2.5H1 , or H( < < 2) = 15/() a A/m. Now B(0 < < 2) = 2.50 (6/())a = 6/ a T. Now, in general, M = (R -1)H, and so M(0 < < ) = (2.5-1)6/()a = 9/() a A/m and M( < < 2) = 0. 9.25. A conducting filament at z = 0 carries 12 A in the az direction. Let R = 1 for < 1 cm, R = 6 for 1 < < 2 cm, and R = 1 for > 2 cm. Find a) H everywhere: This result will depend on the current and not the materials, and is: H= I 1.91 a = A/m (0 < < ) 2 b) B everywhere: We use B = R 0 H to find: B( < 1 cm) = (1)0 (1.91/) = (2.4 10-6 /)a T B(1 < < 2 cm) = (6)0 (1.91/) = (1.4 10-5 /)a T B( > 2 cm) = (1)0 (1.91/) = (2.4 10-6 /)a T where is in meters. 155 9.26. Point P (2, 3, 1) lies on the planar boundary boundary separating region 1 from region 2. The unit vector aN 12 = 0.6ax +0.48ay +0.64az is directed from region 1 to region 2. Let R1 = 2, R2 = 8, and H1 = 100ax - 300ay + 200az A/m. Find H2 : First B1 = 2000 ax - 6000 ay + 4000 az . Then its normal component at the boundary will be B1N = (B1 aN 12 )aN 12 = (52.8ax +42.24ay +56.32az )0 = B2N . Then H2N = B2N /(80 ) = 6.60ax + 5.28ay + 7.04az , and H1N = B1N /20 = 26.40ax + 21.12ay + 28.16az . Now H1T = H1 - H1N = (100ax - 300ay + 200az ) - (26.40ax + 21.12ay + 28.16az ) = 73.60ax - 321.12ay + 171.84az = H2T . Finally, H2 = H2N + H2T = 80.2ax - 315.8ay + 178.9az A/m. 9.27. Let R1 = 2 in region 1, defined by 2x+3y-4z > 1, while R2 = 5 in region 2 where 2x+3y-4z < 1. In region 1, H1 = 50ax - 30ay + 20az A/m. Find: a) HN 1 (normal component of H1 at the boundary): We first need a unit vector normal to the surface, found through aN = 2ax + 3ay - 4az (2x + 3y - 4z) = = .37ax + .56ay - .74az | (2x + 3y - 4z)| 29 Since this vector is found through the gradient, it will point in the direction of increasing values of 2x + 3y - 4z, and so will be directed into region 1. Thus we write aN = aN 21 . The normal component of H1 will now be: HN 1 = (H1 aN 21 )aN 21 = (50ax - 30ay + 20az ) (.37ax + .56ay - .74az ) (.37ax + .56ay - .74az ) = -4.83ax - 7.24ay + 9.66az A/m b) HT 1 (tangential component of H1 at the boundary): HT 1 = H1 - HN 1 = (50ax - 30ay + 20az ) - (-4.83ax - 7.24ay + 9.66az ) = 54.83ax - 22.76ay + 10.34az A/m c) HT 2 (tangential component of H2 at the boundary): Since tangential components of H are continuous across a boundary between two media of different permeabilities, we have HT 2 = HT 1 = 54.83ax - 22.76ay + 10.34az A/m d) HN 2 (normal component of H2 at the boundary): Since normal components of B are continuous across a boundary between media of different permeabilities, we write 1 HN 1 = 2 HN 2 or HN 2 = R1 2 HN 1 = (-4.83ax - 7.24ay + 9.66az ) = -1.93ax - 2.90ay + 3.86az A/m R 2 5 e) 1 , the angle between H1 and aN 21 : This will be cos 1 = 50ax - 30ay + 20az H1 (.37ax + .56ay - .74az ) = -0.21 aN 21 = |H1 | (502 + 302 + 202 )1/2 Therefore 1 = cos-1 (-.21) = 102 . 156 9.27f) 2 , the angle between H2 and aN 21 : First, H2 = HT 2 + HN 2 = (54.83ax - 22.76ay + 10.34az ) + (-1.93ax - 2.90ay + 3.86az ) = 52.90ax - 25.66ay + 14.20az A/m Now cos 2 = 52.90ax - 25.66ay + 14.20az H2 aN 21 = (.37ax + .56ay - .74az ) = -0.09 |H2 | 60.49 Therefore 2 = cos-1 (-.09) = 95 . 9.28. For values of B below the knee on the magnetization curve for silicon steel, approximate the curve by a straight line with = 5 mH/m. The core shown in Fig. 9.17 has areas of 1.6 cm2 and lengths of 10 cm in each outer leg, and an area of 2.5 cm2 and a length of 3 cm in the central leg. A coil of 1200 turns carrying 12 mA is placed around the central leg. Find B in the: a) center leg: We use mmf = R, where, in the central leg, Rc = Lin 3 10-2 = 2.4 104 H = Ain (5 10-3 )(2.5 10-4 ) In each outer leg, the reluctance is Ro = Lout 10 10-2 = 1.25 105 H = Aout (5 10-3 )(1.6 10-4 ) The magnetic circuit is formed by the center leg in series with the parallel combination of the two outer legs. The total reluctance seen at the coil location is RT = Rc + (1/2)Ro = 8.65 104 H. We now have 14.4 mmf = = 1.66 10-4 Wb = RT 8.65 104 The flux density in the center leg is now B= A = 1.66 10-4 = 0.666 T 2.5 10-4 b) center leg, if a 0.3-mm air gap is present in the center leg: The air gap reluctance adds to the total reluctance already calculated, where Rair = 0.3 10-3 = 9.55 105 H (4 10-7 )(2.5 10-4 ) Now the total reluctance is Rnet = RT + Rair = 8.56 104 + 9.55 105 = 1.04 106 . The flux in the center leg is now = and B= 14.4 = 1.38 10-5 Wb 1.04 106 1.38 10-5 = 55.3 mT 2.5 10-4 157 9.29. In Problem 9.28, the linear approximation suggested in the statement of the problem leads to a flux density of 0.666 T in the center leg. Using this value of B and the magnetization curve for silicon steel, . what current is required in the 1200-turn coil? With B = 0.666 T, we read Hin = 120 A t/m in Fig. 9.11. The flux in the center leg is = 0.666(2.5 10-4 ) = 1.66 10-4 Wb. This divides equally in the two outer legs, so that the flux density in each outer leg is Bout = 1 2 1.66 10-4 = 0.52 Wb/m2 1.6 10-4 . Using Fig. 9.11 with this result, we find Hout = 90 A t/m We now use H dL = NI to find I= 1 (120)(3 10-2 ) + (90)(10 10-2 ) = 10.5 mA (Hin Lin + Hout Lout ) = N 1200 9.30. A toroidal core has a circular cross section of 4 cm2 area. The mean radius of the toroid is 6 cm. The core is composed of two semi-circular segments, one of silicon steel and the other of a linear material with R = 200. There is a 4mm air gap at each of the two joints, and the core is wrapped by a 4000-turn coil carrying a dc current I1 . a) Find I1 if the flux density in the core is 1.2 T: I will use the reluctance method here. Reluctances of the steel and linear materials are respectively, Rs = Rl = (6 10-2 ) = 1.57 105 H-1 (3.0 10-3 )(4 10-4 ) (6 10-2 ) = 1.88 106 H-1 (200)(4 10-7 )(4 10-4 ) where s is found from Fig. 9.11, using B = 1.2, from which H = 400, and so B/H = 3.0 mH/m. The reluctance of each gap is now Rg = We now construct NI1 = R = 1.2(4 10-4 ) Rs + Rl + 2Rg = 1.74 103 0.4 10-3 = 7.96 105 H-1 (4 10-7 )(4 10-4 ) Thus I1 = (1.74 103 )/4000 = 435 mA. 158 9.30b. Find the flux density in the core if I1 = 0.3 A: We are not sure what to use for the permittivity of steel in this case, so we use the iterative approach. Since the current is down from the value obtained in part a, we can try B = 1.0 T and see what happens. From Fig. 9.11, we find H = 200 A/m. Then, in the linear material, 1.0 = 3.98 103 A/m Hl = 200(4 10-7 ) and in each gap, 1.0 = 7.96 105 A/m 4 10-7 Now Ampere's circuital law around the toroid becomes Hg = NI1 = (.06)(200 + 3.98 103 ) + 2(7.96 105 )(4 10-4 ) = 1.42 103 A-t Then I1 = (1.42 103 )/4000 = .356 A. This is still larger than the given value of .3A, so we can extrapolate down to find a better value for B: B = 1.0 - (1.2 - 1.0) .356 - .300 = 0.86 T .435 - .356 Using this value in the procedure above to evaluate Ampere's circuital law leads to a value of I1 of 0.306 A. The result of 0.86 T for B is probably good enough for this problem, considering the limited resolution of Fig. 9.11. 9.31. A toroid is constructed of a magnetic material having a cross-sectional area of 2.5 cm2 and an effective length of 8 cm. There is also a short air gap 0.25 mm length and an effective area of 2.8 cm2 . An mmf of 200 A t is applied to the magnetic circuit. Calculate the total flux in the toroid if: a) the magnetic material is assumed to have infinite permeability: In this case the core reluctance, Rc = l/(A), is zero, leaving only the gap reluctance. This is Rg = Now = d 0.25 10-3 = = 7.1 105 H 0 Ag (4 10-7 )(2.5 10-4 ) mmf 200 = = 2.8 10-4 Wb g 7.1 105 b) the magnetic material is assumed to be linear with R = 1000: Now the core reluctance is no longer zero, but Rc = The flux is then = 8 10-2 = 2.6 105 H (1000)(4 10-7 )(2.5 10-4 ) mmf 200 = = 2.1 10-4 Wb Rc + R g 9.7 105 c) the magnetic material is silicon steel: In this case we use the magnetization curve, Fig. 9.11, and employ an iterative process to arrive at the final answer. We can begin with the value of found in part a, assuming infinite permeability: (1) = 2.8 10-4 Wb. The flux density in the core (1) is then Bc = (2.8 10-4 )/(2.5 10-4 ) = 1.1 Wb/m2 . From Fig. 9.11, this corresponds to 159 magnetic field strength Hc magnetic circuit: (1) . = 270 A/m. We check this by applying Ampere's circuital law to the (1) (1) H dL = Hc Lc + Hg d where Hc Lc = (270)(810-2 ) = 22, and where Hg d = 199. But we require that H dL = 200 A t (1) (1) (1) g = (2.810-4 )(7.1105 ) = whereas the actual result in this first calculation is 199 + 22 = 221, which is too high. So, for a (2) (2) second trial, we reduce B to Bc = 1 Wb/m2 . This yields Hc = 200 A/m from Fig. 9.11, and thus (2) = 2.5 10-4 Wb. Now (2) H dL = Hc Lc + (2) Rg = 200(8 10-2 ) + (2.5 10-4 )(7.1 105 ) = 194 This is less than 200, meaning that the actual flux is slightly higher than 2.5 10-4 Wb. I will leave the answer at that, considering the lack of fine resolution in Fig. 9.11. 9.32. Determine the total energy stored in a spherical region 1cm in radius, centered at the origin in free space, in the uniform field: a) H1 = -600ay A/m: First we find the energy density: wm1 = 1 1 1 2 B1 H1 = 0 H1 = (4 10-7 )(600)2 = 0.226 J/m3 2 2 2 The energy within the sphere is then Wm1 = wm1 4 3 a 3 = 0.226 4 10-6 3 = 0.947 J b) H2 = 600ax + 1200ay A/m: In this case the energy density is wm2 = 1 5 0 (600)2 + (1200)2 = 0 (600)2 2 2 or five times the energy density that was found in part a. Therefore, the stored energy in this field is five times the amount in part a, or Wm2 = 4.74 J. c) H3 = -600ax + 1200ay . This field differs from H2 only by the negative x component, which is a non-issue since the component is squared when finding the energy density. Therefore, the stored energy will be the same as that in part b, or Wm3 = 4.74 J. d) H4 = H2 + H3 , or 2400ay A/m: The energy density is now wm4 = (1/2)0 (2400)2 = (1/2)0 (16)(600)2 J/m3 , which is sixteen times the energy density in part a. The stored energy is therefore sixteen times that result, or Wm4 = 16(0.947) = 15.2 J. e) 1000ax A/m+0.001ax T: The energy density is wm5 = (1/2)0 [1000+.001/0 ]2 = 2.03 J/m3 . Then Wm5 = 2.03[(4/3) 10-6 ] = 8.49 J. 160 9.33. A toroidal core has a square cross section, 2.5 cm < < 3.5 cm, -0.5 cm < z < 0.5 cm. The upper half of the toroid, 0 < z < 0.5 cm, is constructed of a linear material for which R = 10, while the lower half, -0.5 cm < z < 0, has R = 20. An mmf of 150 A t establishes a flux in the a direction. For z > 0, find: a) H (): Ampere's circuital law gives: 2H = NI = 150 H = 150 = 23.9/ A/m 2 b) B (): We use B = R 0 H = (10)(4 10-7 )(23.9/) = 3.0 10-4 / Wb/m2 . c) z>0 : This will be z>0 = B dS = 0 -7 .005 .035 .025 .035 3.0 10-4 ddz = (.005)(3.0 10-4 ) ln .025 = 5.0 10 Wb d) Repeat for z < 0: First, the magnetic field strength will be the same as in part a, since the calculation is material-independent. Thus H = 23.9/ A/m. Next, B is modified only by the new permeability, which is twice the value used in part a: Thus B = 6.0 10-4 / Wb/m2 . Finally, since B is twice that of part a, the flux will be increased by the same factor, since the area of integration for z < 0 is the same. Thus z<0 = 1.0 10-6 Wb. e) Find total : This will be the sum of the values found for z < 0 and z > 0, or 1.5 10-6 Wb. total = 9.34. Three planar current sheets are located in free space as follows: -100ax A/m2 at z = -1, 200ax A/m2 at z = 0, -100ax A/m2 at z = 1. Let wH = (1/2)B H J/m3 , and find wH for all z: Using the fact that the field on either side of a current sheet is given in magnitude by H = K/2, we find, in A/m: H(z > 1) = (1/2)(-200 + 100 + 100)ay = 0 H(0 < z < 1) = (1/2)(-200 - 100 + 100)ay = -100ay H(-1 < z < 0) = (1/2)(200 - 100 + 100)ay = 100ay and H(z < -1) = (1/2)(200 - 100 - 100)ay = 0 The energy densities are then wH (z > 1) = wH (z < -1) = 0 wH (0 < z < 1) = wH (-1 < z < 0) = (1/2)0 (100)2 = 6.28 mJ/m2 161 9.35. The cones = 21 and = 159 are conducting surfaces and carry total currents of 40 A, as shown in Fig. 9.18. The currents return on a spherical conducting surface of 0.25 m radius. a) Find H in the region 0 < r < 0.25, 21 < < 159 , 0 < < 2 : We can apply Ampere's circuital law and take advantage of symmetry. We expect to see H in the a direction and it would be constant at a given distance from the z axis. We thus perform the line integral of H over a circle, centered on the z axis, and parallel to the xy plane: H dL = 0 2 H a r sin a d = Iencl. = 40 A Assuming that H is constant over the integration path, we take it outside the integral and solve: H = 20 40 H= a A/m 2r sin r sin b) How much energy is stored in this region? This will be WH = 2 159 .25 1 2000 1000 2 r 2 sin dr d d = 0 H = 2 r 2 sin2 2 v 21 0 0 tan(159/2) 1000 ln = 1.35 10-4 J = tan(21/2) 159 21 d sin 9.36. A filament carrying current I in the az direction lies on the z axis, and cylindrical current sheets of 5az A/m and -2az A/m are located at = 3 and = 10, respectively. a) Find I if H = 0 for > 10. Ampere's circuital law says, for > 10: 2H = 2(3)(5) - 2(10)(2) + I = 0 from which I = 2(10)(3) - 2(3)(5) = 10 A. b) Using this value of I , calculate H for all , 3 < < 10: Again, using Ampere's circuital law, we find 1 20 H(3 < < 10) = a A/m [10 + 2(3)(5)] a = 2 c) Calculate and plot WH versus 0 , where WH is the total energy stored within the volume 0 < z < 1, 0 < < 2, 3 < < 0 : First the energy density will be wH = (1/2)0 H 2 = 2000 / 2 J/m3 . Then the energy is WH = 1 0 0 2 3 0 2000 0 0 = (1.58 10-3 ) ln d d dz = 4000 ln 2 3 3 J 162 9.36c. (continued) A plot of the energy as a function of 0 is shown below. 9.37. Find the inductance of the cone-sphere configuration described in Problem 9.35 and Fig. 9.18. The inductance is that offered at the origin between the vertices of the cone: From Problem 9.35, the magnetic flux density is B = 200 /(r sin ). We integrate this over the crossectional area defined by 0 < r < 0.25 and 21 < < 159 , to find the total flux: = 159 21 0 0.25 50 tan(159/2) 50 200 r dr d = ln = (3.37) = 6.74 10-6 Wb r sin tan(21/2) Now L = /I = 6.74 10-6 /40 = 0.17 H. Second method: Use the energy computation of Problem 9.35, and write L= 2WH 2(1.35 10-4 ) = = 0.17 H I2 (40)2 9.38. A toroidal core has a rectangular cross section defined by the surfaces = 2 cm, = 3 cm, z = 4 cm, and z = 4.5 cm. The core material has a relative permeability of 80. If the core is wound with a coil containing 8000 turns of wire, find its inductance: First we apply Ampere's circuital law to a circular loop of radius in the interior of the toroid, and in the a direction. H dL = 2H = NI H = The flux in the toroid is then the integral over the cross section of B: = B dL = .045 .04 .03 .02 NI 2 R 0 NI R 0 NI .03 d dz = (.005) ln 2 2 .02 The flux linkage is then given by N , and the inductance is L= N I = (.005)(80)(4 10-7 )(8000)2 ln(1.5) = 2.08 H 2 163 9.39. Conducting planes in air at z = 0 and z = d carry surface currents of K0 ax A/m. a) Find the energy stored in the magnetic field per unit length (0 < x < 1) in a width w (0 < y < w): First, assuming current flows in the +ax direction in the sheet at z = d, and in -ax in the sheet at z = 0, we find that both currents together yield H = K0 ay for 0 < z < d and zero elsewhere. The stored energy within the specified volume will be: WH = 1 0 H 2 dv = 2 d 0 0 w 0 1 v 1 1 2 2 0 K0 dx dy dz = wd0 K0 J/m 2 2 b) Calculate the inductance per unit length of this transmission line from WH = (1/2)LI 2 , where I is the total current in a width w in either conductor: We have I = wK0 , and so L= 2 wd 2 dw 0 d 2 2 0 K0 = 2 2 0 K0 = H/m 2 2 I w w K0 2 c) Calculate the total flux passing through the rectangle 0 < x < 1, 0 < z < d, in the plane y = 0, and from this result again find the inductance per unit length: = 0 d 0 1 0 H ay ay dx dz = d 0 0 1 0 K0 dx dy = 0 dK0 Then L= I = 0 dK0 0 d H/m = wK0 w 9.40. A coaxial cable has conductor dimensions of 1 and 5 mm. The region between conductors is air for 0 < < /2 and < < 3/2, and a non-conducting material having R = 8 for /2 < < and 3/2 < < 2. Find the inductance per meter length: The interfaces between media all occur along radial lines, normal to the direction of B and H in the coax line. B is therefore continuous (and constant at constant radius) around a circular loop centered on the z axis. Ampere's circuital law can thus be written in this form: H dL = and so B= B 0 B + 2 R 0 B + 2 0 B + 2 R 0 B = (R + 1) = I 2 R 0 R 0 I a (1 + R ) The flux in the line per meter length in z is now = 0 1 .005 .001 R 0 I R 0 I d dz = ln(5) (1 + R ) (1 + R ) And the inductance per unit length is: L= = R 0 8(4 10-7 ) ln(5) = ln(5) = 572 nH/m (1 + R ) (9) I 164 9.41. A rectangular coil is composed of 150 turns of a filamentary conductor. Find the mutual inductance in free space between this coil and an infinite straight filament on the z axis if the four corners of the coil are located at a) (0,1,0), (0,3,0), (0,3,1), and (0,1,1): In this case the coil lies in the yz plane. If we assume that the filament current is in the +az direction, then the B field from the filament penetrates the coil in the -ax direction (normal to the loop plane). The flux through the loop will thus be = 0 1 1 3 0 I -0 I ax (-ax ) dy dz = ln 3 2y 2 The mutual inductance is then M= N I = 1500 ln 3 = 33 H 2 b) (1,1,0), (1,3,0), (1,3,1), and (1,1,1): Now the coil lies in the x = 1 plane, and the field from the filament penetrates in a direction that is not normal to the plane of the coil. We write the B field from the filament at the coil location as B= The flux through the coil is now = 0 1 1 1 0 1 3 3 0 I a 2 y 2 + 1 0 I a 2 y2 +1 (-ax ) dy dz = 1 0 1 3 0 I sin 2 y 2 + 1 3 1 dy dz = 0 I 0 Iy dy dz = ln(y 2 + 1) 2 + 1) 2(y 2 = (1.6 10-7 )I The mutual inductance is then M= N I = (150)(1.6 10-7 ) = 24 H 9.42. Find the mutual inductance of this conductor system in free space: a) the solenoid of Fig. 8.11b and a square filamentary loop of side length b coaxially centered inside the solenoid, if a > b/ 2; With the given side length, the loop lies entirely inside the solenoid, and so is linked over its entire cross section by the solenoid field. The latter is given by B = 0 NI /d az T. The flux through the loop area is now = Bb2 , and the mutual inductance is M = /I = 0 Nb2 /d H. b) a cylindrical conducting shell of a radius a, axis on the z axis, and a filament at x = 0, y = d, and where d > a (omitted from problem statement); The B field from the cylinder is B = (0 I )/(2) a for > a, and so the flux per unit length between cylinder and wire is = 0 1 a d 0 I 0 I d d dz = ln 2 2 a /I = 0 /2 ln(d/a) H. 165 Wb Finally the mutual inductance is M = 9.43. a) Use energy relationships to show that the internal inductance of a nonmagnetic cylindrical wire of radius a carrying a uniformly-distributed current I is 0 /(8) H/m. We first find the magnetic field inside the conductor, then calculate the energy stored there. From Ampere's circuital law: 2H = Now WH = Now, with WH = v 2 I I H = A/m 2 a 2a 2 1 0 0 2 0 a 1 2 0 H dv = 2 0 I 2 2 0 I 2 J/m d d dz = 8 2 a 4 16 (1/2)LI 2 , we find Lint = 0 /(8) as expected. b) Find the internal inductance if the portion of the conductor for which < c < a is removed: The hollowed-out conductor still carries current I , so Ampere's circuital law now reads: 2H = and the energy is now WH = = 1 0 0 2 c a ( 2 - c2 ) I H = 2 - c2 ) (a 2 2 - c2 a 2 - c2 A/m 0 I 2 ( 2 - c2 )2 0 I 2 d d dz = 8 2 2 (a 2 - c2 )2 4(a 2 - c2 )2 1 4 a (a - c4 ) - c2 (a 2 - c2 ) + c4 ln 4 c J/m a c 3 - 2c2 + C4 d 0 I 2 4(a 2 - c2 )2 The internal inductance is then Lint = 2WH 0 = 2 I 8 a 4 - 4a 2 c2 + 3c4 + 4c4 ln(a/c) H/m (a 2 - c2 )2 166 CHAPTER 10 10.1. In Fig. 10.4, let B = 0.2 cos 120t T, and assume that the conductor joining the two ends of the resistor is perfect. It may be assumed that the magnetic field produced by I (t) is negligible. Find: a) Vab (t): Since B is constant over the loop area, the flux is = (0.15)2 B = 1.4110-2 cos 120 t Wb. Now, emf = Vba (t) = -d /dt = (120)(1.41 10-2 ) sin 120t. Then Vab (t) = -Vba (t) = -5.33 sin 120t V. b) I (t) = Vba (t)/R = 5.33 sin(120t)/250 = 21.3 sin(120t) mA 10.2. Given the time-varying magnetic field, B = (0.5ax + 0.6ay - 0.3az ) cos 5000t T, and a square filamentary loop with its corners at (2,3,0), (2,-3,0), (-2,3,0), and (-2,-3,0), find the time-varying current flowing in the general a direction if the total loop resistance is 400 k : We write emf = E dL = - d d =- dt dt B az da = d (0.3)(4)(6) cos 5000t dt loop area where the loop normal is chosen as positive az , so that the path integral for E is taken around the positive a direction. Taking the derivative, we find emf = -7.2(5000) sin 5000t so that I = emf -36000 sin 5000t = = -90 sin 5000t mA R 400 103 10.3. Given H = 300 az cos(3 108 t - y) A/m in free space, find the emf developed in the general a direction about the closed path having corners at a) (0,0,0), (1,0,0), (1,1,0), and (0,1,0): The magnetic flux will be: = 0 1 0 1 3000 cos(3 108 t - y) dx dy = 3000 sin(3 108 t - y)|1 0 = 3000 sin(3 108 t - 1) - sin(3 108 t) Wb Then emf = - d = -300(3 108 )(4 10-7 ) cos(3 108 t - 1) - cos(3 108 t) dt = -1.13 105 cos(3 108 t - 1) - cos(3 108 t) V b) corners at (0,0,0), (2,0,0), (2,2,0), (0,2,0): In this case, the flux is = 2 3000 sin(3 108 t - y)|2 = 0 0 The emf is therefore 0. 167 10.4. Conductor surfaces are located at = 1cm and = 2cm in free space. The volume 1 cm < < 2 cm contains the fields H = (2/) cos(6 108 t - 2z) A/m and E = (240/) cos(6 108 t - 2 z) V/m. a) Show that these two fields satisfy Eq. (6), Sec. 10.1: Have E= Then - 20 (6 108 ) B = sin(6 108 t - 2z) a t (8 10-7 )(6 108 ) 480 2 sin(6 108 t - 2z) = sin(6 108 t - 2z) a = E 2(240) 480 2 a = sin(6 108 t - 2z) a = sin(6 108 t - 2z)a z b) Evaluate both integrals in Eq. (4) for the planar surface defined by = 0, 1cm < < 2cm, 0 < z < 0.1m (note misprint in problem statement), and its perimeter, and show that the same results are obtained: we take the normal to the surface as positive a , so the the loop surrounding the surface (by the right hand rule) is in the negative a direction at z = 0, and is in the positive a direction at z = 0.1. Taking the left hand side first, we find E dL = 240 cos(6 108 t) a a d .02 .02 240 cos(6 108 t - 2(0.1)) a a d + .01 1 2 + 240 cos(6 108 t - 0.2) ln = 240 cos(6 108 t) ln 2 1 = 240(ln 2) cos(6 108 t - 0.2) - cos(6 108 t) Now for the right hand side. First, B dS = 0 0.1 .02 .01 0.1 0 .01 8 10-7 cos(6 108 t - 2z) a a d dz = (8 10-7 ) ln 2 cos(6 108 t - 2z) dz = -4 10-7 ln 2 sin(6 108 t - 0.2) - sin(6 108 t) Then - d dt B dS = 240(ln 2) cos(6 108 t - 0.2) - cos(6 108 t) (check) 10.5. The location of the sliding bar in Fig. 10.5 is given by x = 5t + 2t 3 , and the separation of the two rails is 20 cm. Let B = 0.8x 2 az T. Find the voltmeter reading at: a) t = 0.4 s: The flux through the loop will be = 0 0.2 0 x 0.8(x )2 dx dy = 168 0.16 3 0.16 x = (5t + 2t 3 )3 Wb 3 3 Then emf = - 0.16 d = (3)(5t + 2t 3 )2 (5 + 6t 2 ) = -(0.16)[5(.4) + 2(.4)3 ]2 [5 + 6(.4)2 ] = -4.32 V dt 3 b) x = 0.6 m: Have 0.6 = 5t + 2t 3 , from which we find t = 0.1193. Thus emf = -(0.16)[5(.1193) + 2(.1193)3 ]2 [5 + 6(.1193)2 ] = -.293 V 10.6. A perfectly conducting filament containing a small 500- resistor is formed into a square, as illustrated in Fig. 10.6. Find I (t) if a) B = 0.3 cos(120t - 30 ) az T: First the flux through the loop is evaluated, where the unit normal to the loop is az . We find = loop B dS = (0.3)(0.5)2 cos(120t - 30 ) Wb Then the current will be I (t) = 1d (120)(0.3)(0.25) emf =- = sin(120t - 30 ) = 57 sin(120t - 30 ) mA R R dt 500 b) B = 0.4 cos[(ct - y)] az T where c = 3 108 m/s: Since the field varies with y, the flux is now = loop B dS = (0.5)(0.4) 0 .5 cos(y - ct) dy = 0.2 [sin(ct - /2) - sin(ct)] Wb The current is then I (t) = emf 1d -0.2c =- = [cos(ct - /2) - cos(ct)] A R R dt 500 -0.2(3 108 ) = [sin(ct) - cos(ct)] A = 120 [cos(ct) - sin(ct)] mA 500 10.7. The rails in Fig. 10.7 each have a resistance of 2.2 /m. The bar moves to the right at a constant speed of 9 m/s in a uniform magnetic field of 0.8 T. Find I (t), 0 < t < 1 s, if the bar is at x = 2 m at t = 0 and a) a 0.3 resistor is present across the left end with the right end open-circuited: The flux in the left-hand closed loop is l = B area = (0.8)(0.2)(2 + 9t) Then, emf l = -d l /dt = -(0.16)(9) = -1.44 V. With the bar in motion, the loop resistance is increasing with time, and is given by Rl (t) = 0.3 + 2[2.2(2 + 9t)]. The current is now Il (t) = emf l -1.44 A = Rl (t) 9.1 + 39.6t Note that the sign of the current indicates that it is flowing in the direction opposite that shown in the figure. 169 b) Repeat part a, but with a resistor of 0.3 across each end: In this case, there will be a contribution to the current from the right loop, which is now closed. The flux in the right loop, whose area decreases with time, is r = (0.8)(0.2)[(16 - 2) - 9t] and emf r = -d r /dt = (0.16)(9) = 1.44 V. The resistance of the right loop is Rr (t) = 0.3 + 2[2.2(14 - 9t)], and so the contribution to the current from the right loop will be Ir (t) = -1.44 A 61.9 - 39.6t The minus sign has been inserted because again the current must flow in the opposite direction as that indicated in the figure, with the flux decreasing with time. The total current is found by adding the part a result, or IT (t) = -1.44 1 1 + 61.9 - 39.6t 9.1 + 39.6t A 10.8. Fig. 10.1 is modified to show that the rail separation is larger when y is larger. Specifically, let the separation d = 0.2 + 0.02y. Given a uniform velocity vy = 8 m/s and a uniform magnetic flux density Bz = 1.1 T, find V12 as a function of time if the bar is located at y = 0 at t = 0: The flux through the loop as a function of y can be written as = B dS = 0 y 0 .2+.02y 1.1 dx dy = 0 y 1.1(.2 + .02y ) dy = 0.22y(1 + .05y) Now, with y = vt = 8t, the above becomes V12 = - = 1.76t (1 + .40t). Finally, d = -1.76(1 + .80t) V dt 10.9. A square filamentary loop of wire is 25 cm on a side and has a resistance of 125 per meter length. The loop lies in the z = 0 plane with its corners at (0, 0, 0), (0.25, 0, 0), (0.25, 0.25, 0), and (0, 0.25, 0) at t = 0. The loop is moving with velocity vy = 50 m/s in the field Bz = 8 cos(1.5 108 t - 0.5x) T. Develop a function of time which expresses the ohmic power being delivered to the loop: First, since the field does not vary with y, the loop motion in the y direction does not produce any time-varying flux, and so this motion is immaterial. We can evaluate the flux at the original loop position to obtain: (t) = 0 .25 0 .25 8 10-6 cos(1.5 108 t - 0.5x) dx dy = -(4 10-6 ) sin(1.5 108 t - 0.13x) - sin(1.5 108 t) Wb Now, emf = V (t) = -d /dt = 6.0 102 cos(1.5 108 t - 0.13x) - cos(1.5 108 t) , The total loop resistance is R = 125(0.25 + 0.25 + 0.25 + 0.25) = 125 . Then the ohmic power is P (t) = V 2 (t) = 2.9 103 cos(1.5 108 t - 0.13x) - cos(1.5 108 t) Watts R 170 10.10a. Show that the ratio of the amplitudes of the conduction current density and the displacement current density is / for the applied field E = Em cos t. Assume = 0 . First, D = E = Em cos t. Then the displacement current density is D/t = - Em sin t. Second, Jc = E = Em cos t. Using these results we find |Jc |/|Jd | = / . b. What is the amplitude ratio if the applied field is E = Em e-t/ , where is real? As before, find D = E = Em e-t/ , and so Jd = D/t = -( / )Em e-t/ . Also, Jc = Em e-t/ . Finally, |Jc |/|Jd | = / . 10.11. Let the internal dimension of a coaxial capacitor be a = 1.2 cm, b = 4 cm, and l = 40 cm. The homogeneous material inside the capacitor has the parameters = 10-11 F/m, = 10-5 H/m, and = 10-5 S/m. If the electric field intensity is E = (106 /) cos(105 t)a V/m (note missing t in the argument of the cosine in the book), find: a) J: Use 10 J = E = cos(105 t)a A/m2 b) the total conduction current, Ic , through the capacitor: Have Ic = J dS = 2lJ = 20l cos(105 t) = 8 cos(105 t) A c) the total displacement current, Id , through the capacitor: First find Jd = Now Id = 2lJd = -2l sin(105 t) = -0.8 sin(105 t) A d) the ratio of the amplitude of Id to that of Ic , the quality factor of the capacitor: This will be 0.8 |Id | = 0.1 = |Ic | 8 10.12. Given a coaxial transmission line with b/a = e2.5 , R = R = 1, and an electric field intensity E = (200/) cos(109 t - 3.336z) a V/m, find: a) Vab , the voltage between the conductors, if it is known that electrostatic relationship E = -V is valid; We use Vab = - a b D 1 (105 )(10-11 )(106 ) = ( E) = - sin(105 t)a = - sin(105 t) A/m t t 200 b cos(109 t - 3.336z) d = 200 ln cos(109 t - 3.336z) a = 500 cos(109 t - 3.336z) V b) the displacement current density; Jd = -200 109 D = t 0 sin(109 t - 3.336z)a = - 1.77 sin(109 t - 3.336z)a A/m2 171 10.13. Consider the region defined by |x|, |y|, and |z| < 1. Let 20 cos(1.5 108 t - bx)ay A/m2 ; a) find D and E: Since Jd = D/t, we write D= R = 5, R = 4, and = 0. If Jd = 20 10-6 sin(1.5 108 - bx)ay 1.5 108 = 1.33 10-13 sin(1.5 108 t - bx)ay C/m2 Jd dt + C = where the integration constant is set to zero (assuming no dc fields are present). Then E= 1.33 10-13 sin(1.5 108 t - bx)ay (5 8.85 10-12 ) = 3.0 10-3 sin(1.5 108 t - bx)ay V/m D = b) use the point form of Faraday's law and an integration with respect to time to find B and H: In this case, E= Ey B az = -b(3.0 10-3 ) cos(1.5 108 t - bx)az = - x t Solve for B by integrating over time: B= Now H= b(3.0 10-3 ) sin(1.5 108 t - bx)az = (2.0)b 10-11 sin(1.5 108 t - bx)az T 1.5 108 B (2.0)b 10-11 = sin(1.5 108 t - bx)az -7 4 4 10 = (4.0 10-6 )b sin(1.5 108 t - bx)az A/m c) use H = Jd + J to find Jd : Since = 0, there is no conduction current, so in this case H=- Hz ay = 4.0 10-6 b2 cos(1.5 108 t - bx)ay A/m2 = Jd x d) What is the numerical value of b? We set the given expression for Jd equal to the result of part c to obtain: 20 10-6 = 4.0 10-6 b2 b = 5.0 m-1 10.14. A voltage source, V0 sin t, is connected between two concentric conducting spheres, r = a and r = b, b > a, where the region between them is a material for which = R 0 , = 0 , and = 0. Find the total displacement current through the dielectric and compare it with the source current as determined from the capacitance (Sec. 5.10) and circuit analysis methods: First, solving Laplace's equation, we find the voltage between spheres (see Eq. 20, Chapter 7): V (t) = (1/r) - (1/b) V0 sin t (1/a) - (1/b) 172 10.14 (continued) Then E = -V = Now Jd = The displacement current is then Id = 4r 2 Jd = where, from Eq. 47, Chapter 5, C= The results are consistent. 4 R 0 (1/a - 1/b) 4 R 0 V0 cos t V0 sin t a 2 (1/a - 1/b) r r D= R 0 V0 sin t a 2 (1/a - 1/b) r r D R 0 V0 cos t = 2 ar t r (1/a - 1/b) (1/a - 1/b) =C dV dt 10.15. Let = 3 10-5 H/m, = 1.2 10-10 F/m, and = 0 everywhere. If H = 2 cos(1010 t - x)az A/m, use Maxwell's equations to obtain expressions for B, D, E, and : First, B = H = 6 10-5 cos(1010 t - x)az T. Next we use H=- from which D= 2 sin(1010 t - x) dt + C = - 2 cos(1010 t - x)ay C/m2 1010 H D ay = 2 sin(1010 t - x)ay = x t where the integration constant is set to zero, since no dc fields are presumed to exist. Next, E= Now E= So B=- 1.67 2 sin(1010 t - x)az dt = (1.67 10-10 ) 2 cos(1010 t - x)az D =- 2 cos(1010 t - x)ay = -1.67 cos(1010 t - x)ay V/m -10 )(1010 ) (1.2 10 Ey B az = 1.67 2 sin(1010 t - x)az = - x t We require this result to be consistent with the expression for B originally found. So (1.67 10-10 ) 2 = 6 10-5 = 600 rad/m 173 10.16a. A certain material has = 0 and equations to find R : First find H=- So E= .04 0 R = 1. If H = 4 sin(106 t - 0.01z)ay A/m, make use of Maxwell's E t Hy ax = 0.04 cos(106 t - 0.01z)ax = z 0 cos(106 t - 0.01z)ax dt = .04 sin(106 t - 0.01z)ax 106 0 where the integration constant is zero, since we assume no dc fields present. Next E= So H= H .04(.01) Ex ay = - cos(106 t - 0.01z)ay = -R 0 6 z 10 0 t .04(.01) .04(.01) cos(106 t - 0.01z)ay dt = 12 sin(106 t - 0.01z)ay 6 10 0 0 R 10 0 0 R = 4 sin(106 t - 0.01z)ay where the last equality is required for consistency. Therefore .04(.01) (.01)2 (9 1016 ) = 4 R = =9 1012 0 0 R 1012 b) Find E(z, t): This we already found during the development in part a: We have E(z, t) = .04 sin(106 t - 0.01z)ax V/m = 4.5 sin(106 t - 0.01z)ax kV/m 106 0 10.17. The electric field intensity in the region 0 < x < 5, 0 < y < /12, 0 < z < 0.06 m in free space is given by E = C sin(12y) sin(az) cos(2 1010 t) ax V/m. Beginning with the E relationship, use Maxwell's equations to find a numerical value for a, if it is known that a is greater than zero: In this case we find E= Ez Ex ay - az z y = C a sin(12y) cos(az)ay - 12 cos(12y) sin(az)az cos(2 1010 t) = - Then H=- =- 1 0 E dt + C1 B t C a sin(12y) cos(az)ay - 12 cos(12y) sin(az)az sin(2 1010 t) A/m 0 (2 1010 where the integration constant, C1 = 0, since there are no initial conditions. Using this result, we now find H= Hy Hz - y z ax = - C(144 + a 2 ) D sin(12y) sin(az) sin(2 1010 t) ax = 10 ) 0 (2 10 t 174 10.17. (continued) Now E= D 0 = 1 0 H dt + C2 = C(144 + a 2 ) sin(12y) sin(az) cos(2 1010 t) ax 10 )2 0 0 (2 10 where C2 = 0. This field must be the same as the original field as stated, and so we require that C(144 + a 2 ) =1 0 0 (2 1010 )2 Using 0 0 = (3 108 )-2 , we find a= (2 1010 )2 - 144 (3 108 )2 1/2 = 66 m-1 10.18. The parallel plate transmission line shown in Fig. 10.8 has dimensions b = 4 cm and d = 8 mm, while the medium between plates is characterized by R = 1, R = 20, and = 0. Neglect fields outside the dielectric. Given the field H = 5 cos(109 t - z)ay A/m, use Maxwell's equations to help find: a) , if > 0: Take H=- So E= Then E= So that H= - 2 (4 109 )0 sin(109 t - z)ax dt = 2 (4 1018 )0 cos(109 t - z) Hy E ax = -5 sin(109 t - z)ax = 20 0 z t cos(109 t - z)ax H t -5 sin(109 t - z)ax dt = 20 0 (4 109 ) Ex 2 ay = z (4 109 ) 0 0 sin(109 t - z)ay = -0 0 0 = 5 cos(109 t - z)ay where the last equality is required to maintain consistency. Therefore 2 (4 1018 )0 = 5 = 14.9 m-1 0 b) the displacement current density at z = 0: Since = 0, we have H = Jd = -5 sin(109 t - z) = -74.5 sin(109 t - 14.9z)ax = -74.5 sin(109 t)ax A/m at z = 0 c) the total displacement current crossing the surface x = 0.5d, 0 < y < b, and 0 < z < 0.1 m in the ax direction. We evaluate the flux integral of Jd over the given cross section: Id = -74.5b 0.1 0 sin(109 t - 14.9z) ax ax dz = 0.20 cos(109 t - 1.49) - cos(109 t) A 175 1 10.19. In the first section of this chapter, Faraday's law was used to show that the field E = - 2 kB0 ekt a results from the changing magnetic field B = B0 ekt az (note that the factor of appearing in E was omitted from the original problem statement). a) Show that these fields do not satisfy Maxwell's other curl equation: Note that B as stated is constant with position, and so will have zero curl. The electric field, however, varies with time, and so H = D would have a zero left-hand side and a non-zero right-hand side. The equation is t thus not valid with these fields. b) If we let B0 = 1 T and k = 106 s -1 , we are establishing a fairly large magnetic flux density in 1 s. Use the H equation to show that the rate at which Bz should (but does not) change with is only about 5 10-6 T/m in free space at t = 0: Assuming that B varies with , we write H=- Thus Hz 1 dB0 kt a = - e = 0 d 0 1 E = - 0 k 2 B0 ekt t 2 dB0 1012 (1) 1 = 5.6 10-6 = 0 0 k 2 B0 = d 2 2(3 108 )2 which is near the stated value if is on the order of 1m. 10.20. Point C(-0.1, -0.2, 0.3) lies on the surface of a perfect conductor. The electric field intensity at C is (500ax - 300ay + 600az ) cos 107 t V/m, and the medium surrounding the conductor is characterized by R = 5, R = 10, and = 0. a) Find a unit vector normal to the conductor surface at C, if the origin lies within the conductor: At t = 0, the field must be directed out of the surface, and will be normal to it, since we have a perfect conductor. Therefore n= 5ax - 3ay + 6az +E(t = 0) = = 0.60ax - 0.36ay + 0.72az |E(t = 0)| 25 + 9 + 36 b) Find the surface charge density at C: Use s = D n|surf ace = 10 = 10 0 [300 0 500ax - 300ay + 600az cos(107 t) .60ax - .36ay + .72az + 108 + 432] cos(107 t) = 7.4 10-8 cos(107 t) C/m2 = 74 cos(107 t) nC/m2 10.21. The surfaces = 3 and 10 mm, and z = 0 and 25 cm are perfect conductors. The region enclosed by these surfaces has = 2.5 10-6 H/m, = 4 10-11 F/m, and = 0. Let H = (2/) cos(10z) cos(t) a A/m. Make use of Maxwell's equations to find a) : We start with H=- We then find E= 20 20 sin(10z) cos(t) dt a = sin(10z) sin(t) a H 20 E a = sin(10z) cos(t) a = z t 176 10.21a. (continued) At this point, a flaw in the problem statement becomes apparent, since this field should vanish on the surface of the perfect conductor located at z = 0.25m. This does not happen with the sin(10z) function. Nevertheless, we press on: E= So H= E (20)(10) H a = cos(10z) sin(t) a = - z t 200 2 -200 2 cos(10z) sin(t) a dt = 2 cos(10z) cos(t) a This result must equal the given H field, so we require that 2 10 200 2 = = = 2 b) E: We use the result of part a: E= 500 20 sin(10z) sin(t) a = sin(10z) sin(t) a V/m 10 (2.5 10-6 )(4 10-11 ) = 109 sec-1 10.22. In free space, where = 0 , = 0 , = 0, J = 0, and v = 0, assume a cartesian coordinate system in which E and H are both functions only of z and t. a) If E = Ey ay and H = Hx ax , begin with Maxwell's equations and determine the second order partial differential equation that Ey must satisfy: The procedure here is similar to the development that leads to Eq. 53. Begin by taking the curl of both sides of the Faraday law equation: E = -0 where H = 0 E/t. H t = -0 ( H) t Therefore 0 E = ( E) - 2 E = -0 2E t 2 0 where the first equality is found from Eq. 52. Noting that in free space, D = obtain, 2 Ey 2 Ey 2E 2 E = 0 0 2 = 0 0 2 t z2 t since E varies only with z and t, and is y-directed. E = 0, we b) Show that Ey = 5(300t + bz)2 is a solution of that equation for a particular value of b, and find that value: Substituting, we find 2 Ey = 10b2 = 0 z2 Therefore 10b2 = 9 105 0 0 0 Ey = 9 105 0 t 2 0 b = 1.0 10-6 m-1 177 10.23. In region 1, z < 0, 1 = 2 10-11 F/m, 1 = 2 10-6 H/m, and 1 = 4 10-3 S/m; in region 2, z > 0, 2 = 1 /2, 2 = 21 , and 2 = 1 /4. It is known that E1 = (30ax + 20ay + 10az ) cos(109 t) V/m at P1 (0, 0, 0- ). a) Find EN 1 , Et1 , DN 1 , and Dt1 : These will be EN 1 = 10 cos(109 t)az V/m Et1 = (30ax + 20ay ) cos(109 t) V/m DN 1 = Dt1 = 1 Et1 1 EN 1 = (2 10-11 )(10) cos(109 t)az C/m2 = 200 cos(109 t)az pC/m2 = (2 10-11 )(30ax + 20ay ) cos(109 t) = (600ax + 400ay ) cos(109 t) pC/m2 b) Find JN 1 and Jt1 at P1 : JN 1 = 1 EN 1 = (4 10-3 )(10 cos(109 t))az = 40 cos(109 t)az mA/m2 Jt1 = 1 Et1 = (4 10-3 )(30ax + 20ay ) cos(109 t) = (120ax + 80ay ) cos(109 t) mA/m2 c) Find Et2 , Dt2 , and Jt2 at P1 : By continuity of tangential E, Et2 = Et1 = (30ax + 20ay ) cos(109 t) V/m Then Dt2 = 2 Et2 = (10-11 )(30ax + 20ay ) cos(109 t) = (300ax + 200ay ) cos(109 t) pC/m2 Jt2 = 2 Et2 = (10-3 )(30ax + 20ay ) cos(109 t) = (30ax + 20ay ) cos(109 t) mA/m2 d) (Harder) Use the continuity equation to help show that JN 1 - JN 2 = DN 2 /t - DN 1 /t (note misprint in problem statement) and then determine EN 2 , DN 2 , and JN 2 : We assume the existence of a surface charge layer at the boundary having density s C/m2 . If we draw a cylindrical "pillbox" whose top and bottom surfaces (each of area a) are on either side of the interface, we may use the continuity condition to write s a (JN 2 - JN 1 ) a = - t where s = DN 2 - DN 1 . Therefore, J N 1 - JN 2 = (DN 2 - DN 1 ) t In terms of the normal electric field components, this becomes 1 EN 1 - 2 EN 2 = ( 2 EN 2 - t 1 EN 1 ) Now let EN 2 = A cos(109 t) + B sin(109 t), while from before, EN 1 = 10 cos(109 t). 178 10.23. (continued) These, along with the permittivities and conductivities, are substituted to obtain (4 10-3 )(10) cos(109 t) - 10-3 [A cos(109 t) + B sin(109 t)] 10-11 [A cos(109 t) + B sin(109 t)] - (2 10-11 )(10) cos(109 t) = t = -(10-2 A sin(109 t) + 10-2 B cos(109 t) + (2 10-1 ) sin(109 t) We now equate coefficients of the sin and cos terms to obtain two equations: 4 10-2 - 10-3 A = 10-2 B -10-3 B = -10-2 A + 2 10-1 These are solved together to find A = 20.2 and B = 2.0. Thus EN 2 = 20.2 cos(109 t) + 2.0 sin(109 t) az = 20.3 cos(109 t + 5.6 )az V/m Then DN 2 = and 2 EN 2 = 203 cos(109 t + 5.6 )az pC/m2 JN 2 = 2 EN 2 = 20.3 cos(109 t + 5.6 )az mA/m2 10.24. Given the fields V = 80z cos x cos 3 108 t. kV and A = 26.7z sin x sin 3 108 t ax mWb/m in free space, find E and H: First, find E through E = -V - where -V = 80 cos(3 108 t)[z sin xax - cos xaz ] kV/m and -A/t = -(3 108 )(26.7)z sin x cos(3 108 t)ax mV/m Finally, E = - 7.9 106 z sin x ax + 8.0 104 cos x az cos(3 108 t) V/m Now B= A= Then H= Ax ay = 26.7 sin x sin(3 108 t)ay mWb/m2 z A t B = 2.12 104 sin x sin(3 108 t) ay A/m 0 179 10.25. In a region where R = R = 1 and = 0, the retarded potentials are given by V = x(z - ct) V and A = x[(z/c) - t]az Wb/m, where c = 1/ 0 0 . a) Show that A = - (V /t): First, A= Second, Az x = = x 0 z c 0 V x = -cx = - t 0 0 so we observe that A = -0 0 (V /t) in free space, implying that the given statement would hold true in general media. b) Find B, H, E, and D: Use B= A=- Then H= Now, E = -V - Then D= 0E Ax z ay = t - ay T x c B 1 z ay A/m = t- 0 0 c A = -(z - ct)ax - xaz + xaz = (ct - z)ax V/m t = 0 (ct - z)ax C/m2 c) Show that these results satisfy Maxwell's equations if J and v are zero: i. D = 0 (ct - z)ax = 0 ii. B = (t - z/c)ay = 0 iii. H=- which we require to equal D/t: D = t iv. E= which we require to equal -B/t: 0 cax Hy 1 ax = ax = z 0 c 0 0 ax = 0 0 ax Ex ay = -ay z B = ay t So all four Maxwell equations are satisfied. 180 10.26. Let the current I = 80t A be present in the az direction on the z axis in free space within the interval -0.1 < z < 0.1 m. a) Find Az at P (0, 2, 0): The integral for the retarded vector potential will in this case assume the form A= where R = Az = .1 -.1 0 80(t - R/c) az dz 4R z2 + 4 and c = 3 108 m/s. We obtain .1 .1 -.1 t dz - z2 + 4 -.1 .1 + 4.01 -6 = 8 10 ln -.1 + 4.01 800 4 1 dz = 8 10-6 t ln(z + c z2 + 4) .1 -.1 - 8 10-6 z 3 108 .1 -.1 - 0.53 10-14 = 8.0 10-7 t - 0.53 10-14 So finally, A = 8.0 10-7 t - 5.3 10-15 az Wb/m. b) Sketch Az versus t over the time interval -0.1 < t < 0.1 s: The sketch is linearly increasing with time, beginning with Az = -8.53 10-14 Wb/m at t = -0.1 s, crossing the time axis and going positive at t = 6.6 ns, and reaching a maximum value of 7.46 10-14 Wb/m at t = 0.1 s. 181 CHAPTER 11 11.1. Show that Exs = Aej k0 z+ is a solution to the vector Helmholtz equation, Sec. 11.1, Eq. (16), for k0 = 0 0 and any and A: We take d2 2 Aej k0 z+ = (j k0 )2 Aej k0 z+ = -k0 Exs dz2 11.2. Let E(z, t) = 200 sin 0.2z cos 108 tax + 500 cos(0.2z + 50 ) sin 108 tay V/m. Find: a) E at P (0, 2, 0.6) at t = 25 ns: Obtain EP (t = 25) = 200 sin [(0.2)(0.6)] cos(2.5)ax + 500 cos [(0.2)(0.6) + 50(2)/360] sin(2.5)ay = -19.2ax + 164ay V/m b) |E| at P at t = 20 ns: EP (t = 20) = 200 sin [(0.2)(0.6)] cos(2.0)ax + 500 cos [(0.2)(0.6) + 50(2)/360] sin(2.0)ay = -9.96ax + 248ay V/m Thus |EP | = (9.96)2 + (248)2 = 249 V/m. c) Es at P : Es = 200 sin 0.2zax - j 500 cos(0.2z + 50 )ay . Thus EsP = 200 sin [(0.2)(0.6)] ax - j 500 cos [(0.2)(0.6) + 2(50)/360] ay = 23.9ax - j 273ay V/m 11.3. An H field in free space is given as H(x, t) = 10 cos(108 t - x)ay A/m. Find a) : Since we have a uniform plane wave, = /c, where we identify = 108 sec-1 . Thus = 108 /(3 108 ) = 0.33 rad/m. b) : We know = 2/ = 18.9 m. c) E(x, t) at P (0.1, 0.2, 0.3) at t = 1 ns: Use E(x, t) = -0 H (x, t) = -(377)(10) cos(108 t - x) = -3.77 103 cos(108 t - x). The vector direction of E will be -az , since we require that S = E H, where S is x-directed. At the given point, the relevant coordinate is x = 0.1. Using this, along with t = 10-9 sec, we finally obtain E(x, t) = -3.77 103 cos[(108 )(10-9 ) - (0.33)(0.1)]az = -3.77 103 cos(6.7 10-2 )az = -3.76 103 az V/m 11.4. In phasor form, the electric field intensity of a uniform plane wave in free space is expressed as Es = (40 - j 30)e-j 20z ax V/m. Find: a) : From the given expression, we identify = 20 rad/m. Then = c = (3 108 )(20) = 6.0 109 rad/s. b) = 20 rad/m from part a. 182 11.4. (continued) c) f = /2 = 956 MHz. d) = 2/ = 2/20 = 0.314 m. e) Hs : In free space, we find Hs by dividing Es by 0 , and assigning vector components such that Es Hs gives the required direction of wave travel: We find Hs = 40 - j 30 -j 20z ay = (0.11 - j 0.08)e-j 20z ay A/m e 377 f) H(z, t) at P (6, -1, 0.07), t = 71 ps: H(z, t) = Re Hs ej t = 0.11 cos(6.0 109 t - 20z) + 0.08 sin(6.0 109 t - 20z) ay Then H(.07, t = 71ps) = 0.11 cos (6.0 109 )(7.1 10-11 ) - 20(.07) + .08 sin (6.0 109 )(7.1 10-11 ) - 20(.07) ay = [0.11(0.562) - 0.08(0.827)]ay = -6.2 10-3 ay A/m 11.5. A 150-MHz uniform plane wave in free space is described by Hs = (4 + j 10)(2ax + j ay )e-jz A/m. a) Find numerical values for , , and : First, = 2 150 106 = 3 108 sec-1 . Second, for a uniform plane wave in free space, = 2c/ = c/f = (3 108 )/(1.5 108 ) = 2 m. Third, = 2/ = rad/m. b) Find H(z, t) at t = 1.5 ns, z = 20 cm: Use H(z, t) = Re{Hs ej t } = Re{(4 + j 10)(2ax + j ay )(cos(t - z) + j sin(t - z)} = [8 cos(t - z) - 20 sin(t - z)] ax - [10 cos(t - z) + 4 sin(t - z)] ay . Now at the given position time, t - z = (3 108 )(1.5 10-9 ) - (0.20) = /4. And and cos(/4) = sin(/4) = 1/ 2. So finally, 1 H(z = 20cm, t = 1.5ns) = - 12ax + 14ay = -8.5ax - 9.9ay A/m 2 c) What is |E|max ? Have |E|max = 0 |H |max , where |H |max = Hs Hs = [4(4 + j 10)(4 - j 10) + (j )(-j )(4 + j 10)(4 - j 10)]1/2 = 24.1 A/m Then |E|max = 377(24.1) = 9.08 kV/m. 183 11.6. Let R = R = 1 for the field E(z, t) = (25ax - 30ay ) cos(t - 50z) V/m. a) Find : = c = (3 108 )(50) = 15.0 109 s-1 . b) Determine the displacement current density, Jd (z, t): Jd (z, t) = D = - 0 (25ax - 30ay ) sin(t - 50z) t = (-3.32ax + 3.98ay ) sin(1.5 1010 t - 50z) A/m2 c) Find the total magnetic flux passing through the rectangle defined by 0 < x < 1, y = 0, 0 < z < 1, at t = 0: In free space, the magnetic field of the uniform plane wave can be easily found using the intrinsic impedance: H(z, t) = 30 25 ay + ax cos(t - 50z) A/m 0 0 Then B(z, t) = 0 H(z, t) = (1/c)(25ay + 30ax ) cos(t - 50z) Wb/m2 , where 0 /0 = 0 0 = 1/c. The flux at t = 0 is now = 0 1 0 1 B ay dx dz = 1 0 25 25 cos(50z) dz = sin(50) = -0.44 nWb c 50(3 108 ) 11.7. The phasor magnetic field intensity for a 400-MHz uniform plane wave propagating in a certain lossless material is (2ay - j 5az )e-j 25x A/m. Knowing that the maximum amplitude of E is 1500 V/m, find , , , vp , R , R , and H(x, y, z, t): First, from the phasor expression, we identify = 25 m-1 from the argument of the exponential function. Next, we evaluate H0 = |H| = H H = 22 + 52 = 29. Then = E0 /H0 = 1500/ 29 = 278.5 . Then = 2/ = 2/25 = .25 m = 25 cm. Next, vp = Now we note that = 278.5 = 377 And 2 400 106 = = 1.01 108 m/s 25 R R R R = 0.546 = 8.79 c vp = 1.01 108 = R R R R R We solve the above two equations simultaneously to find H(x, y, z, t) = Re (2ay - j 5az )e-j 25x ej t = 4.01 and R = 2.19. Finally, = 2 cos(2 400 106 t - 25x)ay + 5 sin(2 400 106 t - 25x)az = 2 cos(8 108 t - 25x)ay + 5 sin(8 108 t - 25x)az A/m 184 11.8. Let the fields, E(z, t) = 1800 cos(107 t - z)ax V/m and H(z, t) = 3.8 cos(107 t - z)ay A/m, represent a uniform plane wave propagating at a velocity of 1.4 108 m/s in a perfect dielectric. Find: a) = /v = (107 )/(1.4 108 ) = 0.224 m-1 . b) = 2/ = 2/.224 = 28.0 m. c) = |E|/|H| = 1800/3.8 = 474 . d) R : Have two equations in the two unknowns, R and Eliminate R to find R = e) R R: = 0 R / 2 R and = R R /c. c 0 2 = (.224)(3 108 )(474) (107 )(377) = 2.69 = R (0 /)2 = (2.69)(377/474)2 = 1.70. 11.9. A certain lossless material has R = 4 and R = 9. A 10-MHz uniform plane wave is propagating in the ay direction with Ex0 = 400 V/m and Ey0 = Ez0 = 0 at P (0.6, 0.6, 0.6) at t = 60 ns. a) Find , , vp , and : For a uniform plane wave, = = R c R = 2 107 (4)(9) = 0.4 rad/m 3 108 Then = (2)/ = (2)/(0.4) = 5 m. Next, vp = Finally, = = 0 R R 2 107 = = 5 107 m/s 4 10-1 4 = 251 9 = 377 b) Find E(t) (at P ): We are given the amplitude at t = 60 ns and at y = 0.6 m. Let the maximum amplitude be Emax , so that in general, Ex = Emax cos(t - y). At the given position and time, Ex = 400 = Emax cos[(2 107 )(60 10-9 ) - (4 10-1 )(0.6)] = Emax cos(0.96 ) = -0.99Emax So Emax = (400)/(-0.99) = -403 V/m. Thus at P, E(t) = -403 cos(2 107 t) V/m. c) Find H (t): First, we note that if E at a given instant points in the negative x direction, while the wave propagates in the forward y direction, then H at that same position and time must point in the positive z direction. Since we have a lossless homogeneous medium, is real, and we are allowed to write H (t) = E(t)/, where is treated as negative and real. Thus H (t) = Hz (t) = Ex (t) -403 = cos(2 10-7 t) = 1.61 cos(2 10-7 t) A/m -251 185 11.10. Given a 20MHz uniform plane wave with Hs = (6ax - j 2ay )e-j z A/m, assume propagation in a lossless medium characterized by R = 5 and an unknown R . a) Find , vp , R , and : First, = 1, so = 2/ = 2 m. Next, vp = / = 2 20 106 = 4 107 m/s. Then, R = ( 2 c2 )/(2 R ) = (3 108 )2 /(4 107 )2 (5) = 1.14. Finally, = 0 R / R = 377 1.14/5 = 180. b) Determine E at the origin at t = 20ns: We use the relation |E| = |H| and note that for positive z propagation, a positive x component of H is coupled to a negative y component of E, and a negative y component of H is coupled to a negative x component of E. We obtain Es = -(6ay +j 2ax )e-j z . Then E(z, t) = Re Es ej t = -6 cos(t - z)ay + 2 sin(t - z)ax = 360 sin(t - z)ax - 1080 cos(t - z)ay . With = 4 107 sec-1 , t = 2 10-8 s, and z = 0, E evaluates as E(0, 20ns) = 360(0.588)ax - 1080(-0.809)ay = 212ax + 874ay V/m. 11.11. A 2-GHz uniform plane wave has an amplitude of Ey0 = 1.4 kV/m at (0, 0, 0, t = 0) and is propagating in the az direction in a medium where = 1.610-11 F/m, = 3.010-11 F/m, and = 2.5 H/m. Find: a) Ey at P (0, 0, 1.8cm) at 0.2 ns: To begin, we have the ratio, 1/2 2 = 1+ - 1 2 = (2 2 109 ) Then = Thus in general, (2.5 10-6 )(3.0 10-11 ) 2 1+ 2 2 / = 1.6/3.0 = 0.533. So 1 + (.533)2 - 1 1/2 1/2 = 28.1 Np/m + 1 = 112 rad/m Ey (z, t) = 1.4e-28.1z cos(4 109 t - 112z) kV/m Ey (1.8 cm, 0.2 ns) = 0.74 kV/m Evaluating this at t = 0.2 ns and z = 1.8 cm, find b) Hx at P at 0.2 ns: We use the phasor relation, Hxs = -Eys / where = So now Hxs = - Then Eys (1.4 103 )e-28.1z e-j 112z =- = -5.16e-28.1z e-j 112z e-j 14 A/m j 14 271e Hx (z, t) = -5.16e-28.1z cos(4 10-9 t - 112z - 14 ) Hx (1.8 cm, 0.2 ns) = -3.0 A/m 186 1 1 - j( / ) = 2.5 10-6 1 = 263 + j 65.7 = 271 14 -11 1 - j (.533) 3.0 10 This, when evaluated at t = 0.2 ns and z = 1.8 cm, yields 11.12. The plane wave Es = 300e-j kx ay V/m is propagating in a material for which = 2.25 H/m, pF/m, and = 7.8 pF/m. If = 64 Mrad/s, find: a) : We use the general formula, Eq. (35): = 1+ 2 2 =9 1/2 - 1 1/2 = (64 106 ) b) : Using (36), we write (2.25 10-6 )(9 10-12 ) 2 = 1+ 2 1 + (.867)2 - 1 = 0.116 Np/m 2 1/2 + 1 = .311 rad/m c) vp = / = (64 106 )/(.311) = 2.06 108 m/s. d) = 2/ = 2/(.311) = 20.2 m. e) : Using (39): = 1 1 - j( / ) = 2.25 10-6 1 = 407 + j 152 = 434.5ej.36 -12 9 10 1 - j (.867) f) Hs : With Es in the positive y direction (at a given time) and propagating in the positive x direction, we would have a positive z component of Hs , at the same time. We write (with j k = + j): Hs = Es 300 az = e-j kx az = 0.69e-x e-jx e-j.36 az 434.5ej.36 = 0.69e-.116x e-j.311x e-j.36 az A/m g) E(3, 2, 4, 10ns): The real instantaneous form of E will be E(x, y, z, t) = Re Es ej t = 300e-x cos(t - x)ay Therefore E(3, 2, 4, 10ns) = 300e-.116(3) cos[(64 106 )(10-8 ) - .311(3)]ay = 203 V/m 11.13. Let j k = 0.2 + j 1.5 m-1 and = 450 + j 60 for a uniform plane wave propagating in the az direction. If = 300 Mrad/s, find , , and : We begin with = and jk = j 1 - j ( / ) = 0.2 + j 1.5 187 1 1 - j( / ) = 450 + j 60 11.13. (continued) Then = and (j k)(j k) = 2 Taking the ratio of (2) to (1), (j k)(j k) 2.29 = 2 ( )2 1 + ( / )2 = = 1.11 10-5 5 2.06 10 Then with = 3 108 , ( )2 = 1.11 10-5 1.23 10-22 = 1 + ( / )2 (3 108 )2 1 + ( / )2 (3) 1 + ( / )2 = (0.2 + j 1.5)(0.2 - j 1.5) = 2.29 (2) 1 1 + ( / )2 = (450 + j 60)(450 - j 60) = 2.06 105 (1) Now, we use Eqs. (35) and (36). Squaring these and taking their ratio gives 2 = 2 1 + ( / )2 1 + ( / )2 = (0.2)2 (1.5)2 We solve this to find / = 0.271. Substituting this result into (3) gives = 1.07 10-11 F/m. Since / = 0.271, we then find = 2.90 10-12 F/m. Finally, using these results in either (1) or (2) we find = 2.28 10-6 H/m. Summary: = 2.28 10-6 H/m, = 1.07 10-11 F/m, and = 2.90 10-12 F/m. 11.14. A certain nonmagnetic material has the material constants R = 2 and / = 4 10-4 at = 1.5 Grad/s. Find the distance a uniform plane wave can propagate through the material before: a) it is attenuated by 1 Np: First, = (4 104 )(2)(8.854 10-12 ) = 7.1 10-15 F/m. Then, since / << 1, we use the approximate form for , given by Eq. (51) (written in terms of ): . = 2 = (1.5 109 )(7.1 10-15 ) 377 = 1.42 10-3 Np/m 2 2 The required distance is now z1 = (1.42 10-3 )-1 = 706 m b) the power level is reduced by one-half: The governing relation is e-2z1/2 = 1/2, or z1/2 = ln 2/2 = ln 2/2(1.42 10- 3) = 244 m. c) the phase shifts 360 : This distance is defined as one wavelength, where = 2/ = (2c)/( R ) = [2(3 108 )]/[(1.5 109 ) 2] = 0.89 m. 11.15. A 10 GHz radar signal may be represented as a uniform plane wave in a sufficiently small region. Calculate the wavelength in centimeters and the attenuation in nepers per meter if the wave is propagating in a non-magnetic material for which a) R = 1 and R = 0: In a non-magnetic material, we would have: 1/2 2 0 0 R 1+ R = - 1 2 R 188 11.15. (continued) and = With the given values of and 0 0 R 2 R, 1+ 2 R R 1/2 + 1 = 0 0 R it is clear that = /c, and so = 2/ = 2c/ = 3 1010 /1010 = 3 cm. It is also clear that = 0. b) = 1.04 and R = 9.00 10-4 : In this case Thus = 2/ = 2.95 cm. Then R R/ R . << 1, and so = R /c = 2.13 cm-1 . . = 2 R = 2 -2 0 R 0 = 2c R R = 2 1010 (9.00 10-4 ) 2 3 108 1.04 = 9.24 10 c) = 2.5 and Np/m R R = 7.2: Using the above formulas, we obtain 2.5 2 = 1+ 10 ) 2 (3 10 1010 7.2 2.5 2 1/2 + 1 = 4.71 cm-1 and so = 2/ = 1.33 cm. Then 2.5 2 = 1+ 8) 2 (3 10 1010 7.2 2.5 2 1/2 - 1 = 335 Np/m 11.16. The power factor of a capacitor is defined as the cosine of the impedance phase angle, and its Q is CR, where R is the parallel resistance. Assume an idealized parallel plate capacitor having a dielecric characterized by , , and R . Find both the power factor and Q in terms of the loss tangent: First, the impedance will be: R R+ 1 j C 1 j C Z= Now R = d/( A) and C = =R 1 - j RC 1 - jQ =R 2 1 + (RC) 1 + Q2 A/d, and so Q = / = 1/ l.t. Then the power factor is P.F = cos[tan-1 (-Q)] = 1/ 1 + Q2 . 189 11.17. Let = 250 + j 30 and j k = 0.2 + j 2 m-1 for a uniform plane wave propagating in the az direction in a dielectric having some finite conductivity. If |Es | = 400 V/m at z = 0, find: a) Pz,av at z = 0 and z = 60 cm: Assume x-polarization for the electric field. Then Pz,av = 1 1 400 Re Es Hs = Re 400e-z e-jz ax e-z ejz ay 2 2 1 1 1 = (400)2 e-2z Re az = 8.0 104 e-2(0.2)z Re az 2 250 - j 30 = 315 e-2(0.2)z az W/m2 Evaluating at z = 0, obtain Pz,av (z = 0) = 315 az W/m2 , and at z = 60 cm, Pz,av (z = 0.6) = 315e-2(0.2)(0.6) az = 248 az W/m2 . b) the average ohmic power dissipation in watts per cubic meter at z = 60 cm: At this point a flaw becomes evident in the problem statement, since solving this part in two different ways gives results that are not the same. I will demonstrate: In the first method, we use Poynting's theorem in point form (first equation at the top of p. 366), which we modify for the case of time-average fields to read: - Pz,av =< J E > where the right hand side is the average power dissipation per volume. Note that the additional right-hand-side terms in Poynting's theorem that describe changes in energy stored in the fields will both be zero in steady state. We apply our equation to the result of part a: < J E >= - Pz,av = - d 315 e-2(0.2)z = (0.4)(315)e-2(0.2)z = 126e-0.4z W/m3 dz At z = 60 cm, this becomes < J E >= 99.1 W/m3 . In the second method, we solve for the conductivity and evaluate < J E >= < E 2 >. We use jk = j and = We take the ratio, jk = j Identifying = , we find = Re jk = Re 0.2 + j 2 250 + j 30 = 1.74 10-3 S/m 1-j = j + 1 1 - j( / ) 1 - j( / ) Now we find the dissipated power per volume: < E 2 >= 1.74 10-3 1 2 400e-0.2z 2 190 11.17b. (continued) At z = 60 cm, this evaluates as 109 W/m3 . One can show that consistency between the two methods requires that 1 Re = 2 This relation does not hold using the numbers as given in the problem statement and the value of found above. Note that in Problem 11.13, where all values are worked out, the relation does hold and consistent results are obtained using both methods. 11.18a. Find P (r, t) if Es = 400e-j 2x ay V/m in free space: A positive y component of E requires a positive z component of H for propagation in the forward x direction. Thus Hs = (400/0 )e-j 2x az = 1.06e-j 2x az A/m. In real form, the field are E(x, t) = 400 cos(t -2x)ay and H(x, t) = 1.06 cos(t - 2x)az . Now P (r, t) = P (x, t) = E(x, t) H(x, t) = 424.4 cos2 (t - 2x)ax W/m2 . b) Find P at t = 0 for r = (a, 5, 10), where a = 0,1,2, and 3: At t = 0, we find from part a, P (a, 0) = 424.4 cos2 (2a), which leads to the values (in W/m2 ): 424.4 at a = 0, 73.5 at a = 1, 181.3 at a = 2, and 391.3 at a = 3. c) Find P at the origin for T = 0, 0.2T , 0.4T , and 0.6T , where T is the oscillation period. At the origin, we have P (0, t) = 424.4 cos2 (t) = 424.4 cos2 (2t/T ). Using this, we obtain the following values (in W/m2 ): 424.4 at t = 0, 42.4 at t = 0.2T , 277.8 at t = 0.4T , and 277.8 at t = 0.6T . 11.19. Perfectly-conducting cylinders with radii of 8 mm and 20 mm are coaxial. The region between the cylinders is filled with a perfect dielectric for which = 10-9 /4 F/m and R = 1. If E in this region is (500/) cos(t - 4z)a V/m, find: a) , with the help of Maxwell's equations in cylindrical coordinates: We use the two curl equations, beginning with E = -B/t, where in this case, E= So B = Then H = B E 2000 a = sin(t - 4z)a = - a z t 2000 2000 sin(t - 4z)dt = cos(t - 4z) T B 2000 cos(t - 4z) A/m = 0 (4 10-7 ) H 1 (H ) a + az z We next use H = D/t, where in this case H=- where the second term on the right hand side becomes zero when substituting our H . So H=- And D = - 8000 8000 sin(t - 4z)dt = cos(t - 4z) C/m2 -7 ) (4 10 (4 10-7 )2 191 H D 8000 a = - sin(t - 4z)a = a z (4 10-7 ) t 11.19a. (continued) Finally, using the given , E = D = 8000 (10-16 )2 cos(t - 4z) V/m This must be the same as the given field, so we require 8000 (10-16 )2 b) H(, z, t): From part a, we have H(, z, t) = 2000 4.0 cos(t - 4z)a = cos(4 108 t - 4z)a A/m (4 10-7 ) = 500 = 4 108 rad/s c) P(, , z): This will be P(, , z) = E H = = 500 4.0 cos(4 108 t - 4z)a cos(4 108 t - 4z)a 2.0 10-3 cos2 (4 108 t - 4z)az W/m2 2 d) the average power passing through every cross-section 8 < < 20 mm, 0 < < 2 . Using the result of part c, we find Pavg = (1.0 103 )/ 2 az W/m2 . The power through the given cross-section is now P= 0 2 .020 .008 1.0 103 20 d d = 2 103 ln 2 8 = 5.7 kW 11.20. If Es = (60/r) sin e-j 2r a V/m, and Hs = (1/4r) sin e-j 2r a A/m in free space, find the average power passing outward through the surface r = 106 , 0 < < /3, and 0 < < 2. Pavg = Then, the requested power will be /3 15 sin2 ar ar r 2 sin dd = 15 sin3 d 2r 2 0 0 0 /3 25 1 2 = 3.13 W = = 15 - cos (sin + 2) 0 3 8 1 15 sin2 Re Es Hs = ar W/m2 2 2r 2 = 2 /3 Note that the radial distance at the surface, r = 106 m, makes no difference, since the power density dimishes as 1/r 2 . 192 11.21. The cylindrical shell, 1 cm < < 1.2 cm, is composed of a conducting material for which = 106 S/m. The external and internal regions are non-conducting. Let H = 2000 A/m at = 1.2 cm. a) Find H everywhere: Use Ampere's circuital law, which states: H dL = 2(2000) = 2(1.2 10-2 )(2000) = 48 A = Iencl Then in this case J= I 48 az = az = 1.09 106 az A/m2 Area (1.44 - 1.00) 10-4 With this result we again use Ampere's circuital law to find H everywhere within the shell as a function of (in meters): H1 () = 1 2 2 0 .01 1.09 106 d d = 54.5 4 2 (10 - 1) A/m (.01 < < .012) Outside the shell, we would have H2 () = 48 = 24/ A/m ( > .012) 2 Inside the shell ( < .01 m), H = 0 since there is no enclosed current. b) Find E everywhere: We use E= J 1.09 106 = az = 1.09 az V/m 106 which is valid, presumeably, outside as well as inside the shell. c) Find P everywhere: Use P = E H = 1.09 az =- Outside the shell, P = 1.09 az 24 26 a = - a W/m2 ( > .012 m) 54.5 4 2 (10 - 1) a 59.4 (104 2 - 1) a W/m2 (.01 < < .012 m) 193 11.22. The inner and outer dimensions of a copper coaxial transmission line are 2 and 7 mm, respectively. Both conductors have thicknesses much greater than . The dielectric is lossless and the operating frequency is 400 MHz. Calculate the resistance per meter length of the: a) inner conductor: First 1 = = f 1 (4 108 )(4 10-7 )(5.8 107 ) = 3.3 10-6 m = 3.3m Now, using (70) with a unit length, we find Rin = 1 1 = 0.42 ohms/m = -3 )(5.8 107 )(3.3 10-6 ) 2(2 10 2a b) outer conductor: Again, (70) applies but with a different conductor radius. Thus Rout = a 2 Rin = (0.42) = 0.12 ohms/m b 7 c) transmission line: Since the two resistances found above are in series, the line resistance is their sum, or R = Rin + Rout = 0.54 ohms/m. 11.23. A hollow tubular conductor is constructed from a type of brass having a conductivity of 1.2 107 S/m. The inner and outer radii are 9 mm and 10 mm respectively. Calculate the resistance per meter length at a frequency of a) dc: In this case the current density is uniform over the entire tube cross-section. We write: R(dc) = L 1 = = 1.4 10-3 7 )(.012 - .0092 ) A (1.2 10 /m b) 20 MHz: Now the skin effect will limit the effective cross-section. At 20 MHz, the skin depth is (20MHz) = [f 0 ]-1/2 = [(20 106 )(4 10-7 )(1.2 107 )]-1/2 = 3.25 10-5 m This is much less than the outer radius of the tube. Therefore we can approximate the resistance using the formula: R(20MHz) = L 1 1 = 4.1 10-2 = = 7 )(2(.01))(3.25 10-5 ) A 2b (1.2 10 /m c) 2 GHz: Using the same formula as in part b, we find the skin depth at 2 GHz to be = 3.25 10-6 m. The resistance (using the other formula) is R(2GHz) = 4.1 10-1 /m. 194 11.24a. Most microwave ovens operate at 2.45 GHz. Assume that = 1.2 106 S/m and R = 500 for the stainless steel interior, and find the depth of penetration: 1 = = f 1 (2.45 109 )(4 10-7 )(1.2 106 ) = 9.28 10-6 m = 9.28m b) Let Es = 50 0 V/m at the surface of the conductor, and plot a curve of the amplitude of Es vs. propagates into the stainless steel: Since the conductivity is high, we the angle of Es as the field . . use (62) to write = = f = 1/. So, assuming that the direction into the conductor is z, the depth-dependent field is written as Es (z) = 50e-z e-jz = 50e-z/ e-j z/ = 50 exp(-z/9.28) exp(-j z/9.28) amplitude angle where z is in microns. Therefore, the plot of amplitude versus angle is simply a plot of e-x versus x, where x = z/9.28; the starting amplitude is 50 and the 1/e amplitude (at z = 9.28 m) is 18.4. 11.25. A good conductor is planar in form and carries a uniform plane wave that has a wavelength of 0.3 mm and a velocity of 3 105 m/s. Assuming the conductor is non-magnetic, determine the frequency and the conductivity: First, we use f = Next, for a good conductor, = 1 = 2 f = 4 2 f = 4 (9 10-8 )(109 )(4 10-7 ) = 1.1 105 S/m 3 105 v = = 109 Hz = 1 GHz 3 10-4 11.26. The dimensions of a certain coaxial transmission line are a = 0.8mm and b = 4mm. The outer conductor thickness is 0.6mm, and all conductors have = 1.6 107 S/m. a) Find R, the resistance per unit length, at an operating frequency of 2.4 GHz: First 1 = = f 1 (2.4 108 )(4 10-7 )(1.6 107 ) = 2.57 10-6 m = 2.57m Then, using (70) with a unit length, we find Rin = 1 1 = = 4.84 ohms/m 2a 2(0.8 10-3 )(1.6 107 )(2.57 10-6 ) The outer conductor resistance is then found from the inner through Rout = a 0.8 Rin = (4.84) = 0.97 ohms/m b 4 The net resistance per length is then the sum, R = Rin + Rout = 5.81 ohms/m. 195 11.26b. Use information from Secs. 5.10 and 9.10 to find C and L, the capacitance and inductance per unit length, respectively. The coax is air-filled. From those sections, we find (in free space) C= L= 2(8.854 10-12 ) 2 0 = = 3.46 10-11 F/m ln(b/a) ln(4/.8) 4 10-7 0 ln(b/a) = ln(4/.8) = 3.22 10-7 H/m 2 2 c) Find and if + j = j C(R + j L): Taking real and imaginary parts of the given expression, we find 1/2 LC R 2 = Re j C(R + j L) = 1+ - 1 L 2 1/2 2 LC R j C(R + j L) = 1+ + 1 = Im L 2 These can be found by writing out = Re j C(R + j L) = (1/2) j C(R + j L)+c.c., where c.c denotes the complex conjugate. The result is squared, terms collected, and the square root taken. Now, using the values of R, C, and L found in parts a and b, we find = 3.0 10-2 Np/m and = 50.3 rad/m. 11.27. The planar surface at z = 0 is a brass-Teflon interface. Use data available in Appendix C to evaluate the following ratios for a uniform plane wave having = 4 1010 rad/s: a) Tef /brass : From the appendix we find / = .0003 for Teflon, making the material a good dielectric. Also, for Teflon, R = 2.1. For brass, we find = 1.5 107 S/m, making brass a good conductor at the stated frequency. For a good dielectric (Teflon) we use the approximations: . = 2 = 1+ 1 8 1 2 = 1 2 = c c R R and . = For brass (good conductor) we have . . == Now / (/c) 1/2 Tef = brass f brass b) R . = f brass = 1 2 (4 1010 )(4 10-7 )(1.5 107 ) = 6.14 105 m-1 (1/2)(.0003)(4 1010 /3 108 ) 2.1 = = 4.7 10-8 6.14 105 Tef brass (2/Tef ) c f brass (3 108 )(6.14 105 ) = = = = = 3.2 103 brass (2/brass ) Tef (4 1010 ) 2.1 R Tef 196 11.27. (continued) c) vTef brass (/Tef ) = = = 3.2 103 as before vbrass (/brass ) Tef 11.28. A uniform plane wave in free space has electric field given by Es = 10e-jx az + 15e-jx ay V/m. a) Describe the wave polarization: Since the two components have a fixed phase difference (in this case zero) with respect to time and position, the wave has linear polarization, with the field vector in the yz plane at angle = tan-1 (10/15) = 33.7 to the y axis. b) Find Hs : With propagation in forward x, we would have Hs = -10 -jx 15 -jx e e ay + az A/m = -26.5e-jx ay + 39.8e-jx az mA/m 377 377 c) determine the average power density in the wave in W/m2 : Use Pavg = 1 (10)2 1 (15)2 Re Es Hs = ax + ax = 0.43ax W/m2 or Pavg = 0.43 W/m2 2 2 377 377 11.29. Consider a left-circularly polarized wave in free space that propagates in the forward z direction. The electric field is given by the appropriate form of Eq. (80). a) Determine the magnetic field phasor, Hs : We begin, using (80), with Es = E0 (ax + j ay )e-jz . We find the two components of Hs separately, using the two components of Es . Specifically, the x component of Es is associated with a y component of Hs , and the y component of Es is associated with a negative x component of Hs . The result is E0 Hs = ay - j ax e-jz 0 b) Determine an expression for the average power density in the wave in W/m2 by direct application of Eq. (57): We have Pz,avg = = 1 1 E0 Re(Es Hs ) = Re E0 (ax + j ay )e-jz (ay - j ax )e+jz 2 2 0 2 E0 az W/m2 (assuming E0 is real) 0 197 11.30. The electric field of a uniform plane wave in free space is given by Es = 10(ay + j az )e-j 25x . a) Determine the frequency, f : Use f = c (25)(3 108 ) = = 1.2 GHz 2 2 b) Find the magnetic field phasor, Hs : With the Poynting vector in the positive x direction, a positive y component for E requires a positive z component for H. Similarly, a positive z component for E requires a negative y component for H. Therefore, Hs = 10 az - j ay e-j 25x 0 c) Describe the polarization of the wave: This is most clearly seen by first converting the given field to real instantaneous form: E(x, t) = Re Es ej t = 10 cos(t - 25x)ay - sin(t - 25x)az At x = 0, this becomes, E(0, t) = 10 cos(t)ay - sin(t)az With the wave traveling in the forward x direction, we recognize the polarization as left circular. 11.31. A linearly-polarized uniform plane wave, propagating in the forward z direction, is input to a lossless anisotropic material, in which the dielectric constant encountered by waves polarized along y ( Ry ) differs from that seen by waves polarized along x ( Rx ). Suppose Rx = 2.15, Ry = 2.10, and the wave electric field at input is polarized at 45 to the positive x and y axes. Assume free space wavelength . a) Determine the shortest length of the material such that the wave as it emerges from the output end is circularly polarized: With the input field at 45 , the x and y components are of equal magnitude, and circular polarization will result if the phase difference between the components is /2. Our requirement over length L is thus x L - y L = /2, or L= With the given values, we find, L= (58.3)c = 58.3 = 14.6 2 4 c = 2(x - y ) 2( Rx - Ry ) b) Will the output wave be right- or left-circularly-polarized? With the dielectric constant greater for x-polarized waves, the x component will lag the y component in time at the output. The field can thus be written as E = E0 (ay - j ax ), which is left circular polarization. 198 11.32. Suppose that the length of the medium of Problem 11.31 is made to be twice that as determined in the problem. Describe the polarization of the output wave in this case: With the length doubled, a phase shift of radians develops between the two components. At the input, we can write the field as Es (0) = E0 (ax + ay ). After propagating through length L, we would have, Es (L) = E0 [e-jx L ax + e-jy L ay ] = E0 e-jx L [ax + e-j (y -x )L ay ] where (y - x )L = - (since x > y ), and so Es (L) = E0 e-jx L [ax - ay ]. With the reversal of the y component, the wave polarization is rotated by 90 , but is still linear polarization. 11.33. Given a wave for which Es = 15e-jz ax + 18e-jz ej ay V/m, propagating in a medium characterized by complex intrinsic impedance, . a) Find Hs : With the wave propagating in the forward z direction, we find: Hs = 1 -18ej ax + 15ay e-jz A/m b) Determine the average power density in W/m2 : We find Pz,avg = (15)2 1 (18)2 1 Re Es Hs = Re + 2 2 = 275 Re 1 W/m2 11.34. Given the general elliptically-polarized wave as per Eq. (73): Es = [Ex0 ax + Ey0 ej ay ]e-jz a) Show, using methods similar to those of Example 11.7, that a linearly polarized wave results when superimposing the given field and a phase-shifted field of the form: Es = [Ex0 ax + Ey0 e-j ay ]e-jz ej where is a constant: Adding the two fields gives Es,tot = Ex0 1 + ej ax + Ey0 ej + e-j ej ay e-jz = Ex0 ej /2 e-j /2 + ej /2 ax + Ey0 ej /2 e-j /2 ej + e-j ej /2 ay e-jz 2 cos(/2) 2 cos(-/2) This simplifies to Es,tot = 2 Ex0 cos(/2)ax + Ey0 cos( - /2)ay ej /2 e-jz , which is linearly polarized. b) Find in terms of such that the resultant wave is polarized along x: By inspecting the part a result, we achieve a zero y component when 2 - = (or odd multiples of ). 199 CHAPTER 12 + + 12.1. A uniform plane wave in air, Ex1 = Ex10 cos(1010 t -z) V/m, is normally-incident on a copper surface at z = 0. What percentage of the incident power density is transmitted into the copper? We need to find the reflection coefficient. The intrinsic impedance of copper (a good conductor) is c = j 1010 (4 107 ) = (1 + j ) = (1 + j ) = (1 + j )(.0104) 2 2(5.8 107 ) Note that the accuracy here is questionable, since we know the conductivity to only two significant figures. We nevertheless proceed: Using 0 = 376.7288 ohms, we write = .0104 - 376.7288 + j.0104 c - 0 = -.9999 + j.0001 = c + 0 .0104 + 376.7288 + j.0104 Now | |2 = .9999, and so the transmitted power fraction is 1 - | |2 = .0001, or about 0.01% is transmitted. 12.2. The plane y = 0 defines the boundary between two different dielectrics. For y < 0, R1 = 1, 1 = 0 , + and R1 = 0; and for y > 0, R2 = 5, 2 = 0 , and R2 = 0. Let Ez1 = 150 cos(t - 8y) V/m, and find a) : Have = 8 = /c = 8c = 2.4 109 sec-1 . + b) H1 : With E in the z direction, and propagation in the forward y direction, H will lie in the positive x direction, and its amplitude will be Hx = Ey /0 in region 1. + Thus H1 = (150/0 ) cos(t - 8y)ax = 0.40 cos(2.4 109 t - 8y)ax A/m. - c) H1 : First, - Ez1 = + Ez1 0 / 5 - 0 /1 1- 5 + + = = Ez1 = -0.38Ez1 0 / 5 + 0 /1 1+ 5 0.38(150) cos(t + 8y) 377 Then - + Hx1 = +(0.38/0 )Ez1 = - So finally, Hx1 = 0.15 cos(2.4 109 t + 8y)ax A/m. 12.3. A uniform plane wave in region 1 is normally-incident on the planar boundary separating regions 1 and 2. If 1 = 2 = 0, while R1 = 3 and R2 = 3 , find the ratio R2 / R1 if 20% of the energy in R1 R2 the incident wave is reflected at the boundary. There are two possible answers. First, since | |2 = .20, and since both permittivities and permeabilities are real, = 0.447. we then set up = 0.447 = 0 (R2 / 2 - 1 = 2 + 1 0 (R2 / (R1 /3 ) R1 (R1 /3 ) R1 = R2 ) - 0 R2 ) + 0 (R1 / (R1 / R1 ) R1 ) = (R2 /3 ) - R2 (R2 /3 ) + R2 R1 - R2 R1 + R2 200 12.3. (continued) Therefore 1 0.447 R2 = = (0.382, 2.62) R1 1 0.447 R2 R1 = R2 R1 3 = (0.056, 17.9) 12.4. The magnetic field intensity in a region where = 0 is given as H = 5 cos t cos z ay A/m, where = 5 Grad/s and = 30 rad/m. If the amplitude of the associated electric field intensity is 2kV/m, find a) and for the medium: In phasor form, the magnetic field is Hys = H0 e-jz + H0 e+z = 5 cos z H0 = 2.5. The electric field will be x directed, and is Exs = (2.5)e-jz - (2.5)e+jz = (2j )(2.5) sin z. Given the electric field amplitude of 2 kV/m, we write 2103 = 5, or = 400 . Now = 400 = 0 r / R and we also have = 30 = (/c) R R . We solve these two equations simultaneously for R and R to find R = 1.91 and R = 1.70. Therefore = 1.91 4 10-7 = 2.40 H/m and = 1.70 8.854 10-12 = 15.1 pF/m. b) E: From part a, electric field in phasor form is Exs = j 2 sin z kV/m, and so, in real form: E(z, t) = Re(Exs ej t )ax = 2 sin z sin t ax kV/m with and as given. 12.5. The region z < 0 is characterized by R = R = 1 and R = 0. The total E field here is given as the sum of the two uniform plane waves, Es = 150e-j 10z ax + (50 20 )ej 10z ax V/m. a) What is the operating frequency? In free space, = k0 = 10 = /c = /3 108 . Thus, = 3 109 s-1 , or f = /2 = 4.7 108 Hz. b) Specify the intrinsic impedance of the region z > 0 that would provide the appropriate reflected wave: Use 50ej 20 - 0 Er 1 = = = ej 20 = 0.31 + j 0.11 = Einc 150 3 + 0 Now = 0 1+ 1- = 377 1 + 0.31 + j 0.11 1 - 0.31 - j 0.31 = 691 + j 177 c) At what value of z (-10 cm < z < 0) is the total electric field intensity a maximum amplitude? We found the phase of the reflection coefficient to be = 20 = .349rad, and we use zmax = - -.349 = = -0.017 m = -1.7 cm 2 20 12.6. Region 1, z < 0, and region 2, z > 0, are described by the following parameters: 1 = 100 pF/m, 1 = 25 H/m, 1 = 0, 2 = 200 pF/m, 2 = 50 H/m, and 2 / 2 = 0.5. + If E1 = 600e-1 z cos(5 1010 t - 1 z)ax V/m, find: a) 1 : From Eq. (35), Chapter 11, we note that since 1 = 0, it follows that 1 = 0. b) 1 : 1 = 1 1 = (5 1010 ) (25 10-6 )(100 10-12 ) = 2.50 103 rad/m. + c) Es1 = 600e-j 2.5010 z ax V/m. 3 - d) Es1 : To find this, we need to evaluate the reflection coefficient, which means that we first need the two intrinsic impedances. First, 1 = 1 / 1 = (25 10-6 )/(100 10-12 ) = 500. 201 12.6d) (continued) Next, using Eq. (39), Chapter 11, 2 = Then = 460 + j 109 - 500 2 - 1 = -2.83 10-2 + j 1.16 10-1 = 0.120ej 104 = 2 + 1 460 + j 109 + 500 and reverse the propagation direction to obtain - Es1 = 71.8ej 104 ej 2.510 3z 2 2 1 1 - j ( 2 / 2) = 50 10-6 1 = 460 + j 109 -10 2 10 1 - j 0.5 + Now we multiply Es1 by V/m + e) Es2 : This wave will experience loss in region 2, along with a different phase constant. We need to evaluate 2 and 2 . First, using Eq. (35), Chapter 11, 2 = 2 2 2 1+ 2 2 2 1/2 - 1 1/2 = (5 1010 ) (50 106 )(200 10-12 ) 2 1 + (0.5)2 - 1 = 1.21 103 Np/m Then, using Eq. (36), Chapter 11, 2 = 2 2 2 1+ 2 2 2 1/2 + 1 = 5.15 103 rad/m Then, the transmission coefficient will be =1+ = 1 - 2.83 10-2 + j 1.16 10-1 = 0.972ej 7 + + The complex amplitude of Es2 is then found by multiplying the amplitude of Es1 by . The field in region 2 is then constructed by using the resulting amplitude, along with the attenuation and phase constants that are appropriate for region 2. The result is + Es2 = 587e-1.2110 z ej 7 e-j 5.1510 3 3z V/m 12.7. The semi-infinite regions z < 0 and z > 1 m are free space. For 0 < z < 1 m, R = 4, R = 1, and R = 0. A uniform plane wave with = 4 108 rad/s is travelling in the az direction toward the interface at z = 0. a) Find the standing wave ratio in each of the three regions: First we find the phase constant in the middle region, 2 = c R = 2(4 108 ) = 2.67 rad/m 3 108 202 12.7a. (continued) Then, with the middle layer thickness of 1 m, 2 d = 2.67 rad. Also, the intrinsic impedance of the middle layer is 2 = 0 / in = 2 R = 0 /2. We now find the input impedance: 377 2 cos(2.67) + j sin(2.67) 0 cos(2 d) + j 2 sin(2 d) = = 231 + j 141 2 cos(2 d) + j 0 sin(2 d) 2 cos(2.67) + j 2 sin(2.67) Now, at the first interface, 12 = in - 0 231 + j 141 - 377 = = -.176 + j.273 = .325 123 in + 0 231 + j 141 + 377 The standing wave ratio measured in region 1 is thus s1 = 1+| 1-| 12 | 12 | = 1 + 0.325 = 1.96 1 - 0.325 In region 2 the standing wave ratio is found by considering the reflection coefficient for waves incident from region 2 on the second interface: 23 = 1 - 1/2 1 0 - 0 /2 = = 0 + 0 /2 1 + 1/2 3 s2 = 1 + 1/3 =2 1 - 1/3 Then Finally, s3 = 1, since no reflected waves exist in region 3. b) Find the location of the maximum |E| for z < 0 that is nearest to z = 0. We note that the phase of 12 is = 123 = 2.15 rad. Thus zmax = -2.15 - = = -.81 m 2 2(4/3) 12.8. A wave starts at point a, propagates 100m through a lossy dielectric for which = 0.5 Np/m, reflects at normal incidence at a boundary at which = 0.3 + j 0.4, and then returns to point a. Calculate the ratio of the final power to the incident power after this round trip: Final power, Pf , and incident power, Pi , are related through Pf = Pi e-2L | |2 e-2L Try measuring that. 12.9. Region 1, z < 0, and region 2, z > 0, are both perfect dielectrics ( = 0 , = 0). A uniform plane 10 rad/s. Its wavelengths in the two wave traveling in the az direction has a radian frequency of 3 10 regions are 1 = 5 cm and 2 = 3 cm. What percentage of the energy incident on the boundary is a) reflected; We first note that R1 Pf = |0.3 + j 0.4|2 e-2(0.5)100 = 3.5 10-88 (!) Pi = 2c 1 2 and R2 = 2c 2 2 203 12.9a. (continued) Therefore = = R1 / R2 = (2 /1 )2 . Then with = 0 in both regions, we find R2 R2 0 1/ 2 - 1 = 2 + 1 0 1/ - 0 1/ + 0 1/ R1 R1 = R1 / R2 R1 / R2 -1 +1 = (2 /1 ) - 1 (2 /1 ) + 1 1 3-5 2 - 1 =- = 2 + 1 3+5 4 The fraction of the incident energy that is reflected is then | |2 = 1/16 = 6.25 10-2 . b) transmitted? We use part a and find the transmitted fraction to be 1 - | |2 = 15/16 = 0.938. c) What is the standing wave ratio in region 1? Use s= 1+| | 1 + 1/4 5 = = = 1.67 1-| | 1 - 1/4 3 12.10. In Fig. 12.1, let region 2 be free space, while R1 = 1, R1 = 0, and R1 is unknown. Find R! if - + a) the amplitude of E1 is one-half that of E1 : Since region 2 is free space, the reflection coefficient is 0 - 0 / R1 |E- | R1 - 1 1 0 - 1 = = = 1 = R1 = 9 = + 0 + 1 2 |E1 | 0 + 0 / R1 +1 R1 . - + b) P1,avg is one-half of P1,avg : This time R1 - 1 R1 + 1 2 | |2 = = 1 2 R1 = 34 c) |E1 |min is one-half |E1 |max : Use |E1 |max 1+| | =2 | |= =s= |E1 |min 1-| | = 1 = 3 R1 -1 R1 + 1 R1 =4 12.11. A 150 MHz uniform plane wave in normally-incident from air onto a material whose intrinsic impedance is unknown. Measurements yield a standing wave ratio of 3 and the appearance of an electric field minimum at 0.3 wavelengths in front of the interface. Determine the impedance of the unknown material: First, the field minimum is used to find the phase of the reflection coefficient, where zmin = - 1 ( + ) = -0.3 = 0.2 2 s-1 3-1 1 = = s+1 3+1 2 204 where = 2/ has been used. Next, | |= 12.11. (continued) So we now have = 0.5ej 0.2 = We solve for u to find u - 0 u + 0 u = 0 (1.70 + j 1.33) = 641 + j 501 12.12. A 50MHz uniform plane wave is normally incident from air onto the surface of a calm ocean. For seawater, = 4 S/m, and R = 78. a) Determine the fractions of the incident power that are reflected and transmitted: First we find the loss tangent: 4 = 18.4 = 2(50 106 )(78)(8.854 10-12 ) This value is sufficiently greater than 1 to enable seawater to be considered a good conductor at 50MHz. Then, using the approximation (Eq. 65, Chapter 11), the intrinsic impedance is s = f / (1 + j ), and the reflection coefficient becomes f / (1 + j ) - 0 = f / (1 + j ) + 0 where f / = (50 106 )(4 10-7 )/4 = 7.0. The fraction of the power reflected is Pr [ f / - 0 ]2 + f / [7.0 - 377]2 + 49.0 2 =| | = = 0.93 = Pi [7.0 + 377]2 + 49.0 [ f / + 0 ]2 + f / The transmitted fraction is then Pt = 1 - | |2 = 1 - 0.93 = 0.07 Pi b) Qualitatively, how will these answers change (if at all) as the frequency is increased? Within the limits of our good conductor approximation (loss tangent greater than about ten), the reflected power fraction, using the formula derived in part a, is found to decrease with increasing frequency. The transmitted power fraction thus increases. 12.13. A right-circularly-polarized plane wave is normally incident from air onto a semi-infinite slab of plexiglas ( R = 3.45, R = 0). Calculate the fractions of the incident power that are reflected and transmitted. Also, describe the polarizations of the reflected and transmitted waves. First, the impedance of the plexiglas will be = 0 / 3.45 = 203 . Then = 203 - 377 = -0.30 203 + 377 The reflected power fraction is thus | |2 = 0.09. The total electric field in the plane of the interface must rotate in the same direction as the incident field, in order to continually satisfy the boundary condition of tangential electric field continuity across the interface. Therefore, the reflected wave will have to be left circularly polarized in order to make this happen. The transmitted power fraction is now 1 - | |2 = 0.91. The transmitted field will be right circularly polarized (as the incident field) for the same reasons. 205 12.14. A left-circularly-polarized plane wave is normally-incident onto the surface of a perfect conductor. a) Construct the superposition of the incident and reflected waves in phasor form: Assume positive z travel for the incident electric field. Then, with reflection coefficient, = -1, the incident and reflected fields will add to give the total field: Etot = Ei + Er = E0 (ax + j ay )e-jz - E0 (ax + j ay )e+jz = E0 e-jz - ejz ax + j e-jz - ejz ay = 2E0 sin(z) ay - j ax -2j sin(z) -2j sin(z) b) Determine the real instantaneous form of the result of part a: E(z, t) = Re Etot ej t = 2E0 sin(z) cos(t)ay + sin(t)ax c) Describe the wave that is formed: This is a standing wave exhibiting circular polarization in time. At each location along the z axis, the field vector rotates clockwise in the xy plane, and has amplitude (constant with time) given by 2E0 sin(z). 12.15. Consider these regions in which = 0: region 1, z < 0, 1 = 4 H/m and 1 = 10 pF/m; region 2, 0 < z < 6 cm, 2 = 2 H /m, 2 = 25 pF/m; region 3, z > 6 cm, 3 = 1 and 3 = 1 . a) What is the lowest frequency at which a uniform plane wave incident from region 1 onto the boundary at z = 0 will have no reflection? This frequency gives the condition 2 d = , where d = 6 cm, and 2 = 2 2 Therefore 2 d = = (.06) 2 2 f = 1 0.12 (2 10-6 )(25 10-12 ) = 1.2 GHz b) If f = 50 MHz, what will the standing wave ratio be in region 1? At the given frequency, 2 = (2 5 107 ) (2 10-6 )(25 10-12 ) = 2.22 rad/m. Thus 2 d = 2.22(.06) = 0.133. The intrinsic impedance of regions 1 and 3 is 1 = 3 = (4 10-6 )/(10-11 ) = 632 . The input impedance at the first interface is now in = 283 632 cos(.133) + j 283 sin(.133) = 589 - j 138 = 605 - .23 283 cos(.133) + j 632 sin(.133) The reflection coefficient is now in - 1 589 - j 138 - 632 = = = .12 - 1.7 in + 1 589 - j 138 + 632 The standing wave ratio is now 1+| | 1 + .12 s= = = 1.27 1-| | 1 - .12 12.16. A uniform plane wave in air is normally-incident onto a lossless dielectric plate of thickness /8, and of intrinsic impedance = 260 . Determine the standing wave ratio in front of the plate. Also find the fraction of the incident power that is transmitted to the other side of the plate: With the a thickness of /8, we have d = /4, and so cos(d) = sin(d) = 1 2. The input impedance thus becomes in = 260 377 + j 260 = 243 - j 92 260 + j 377 206 12.16. (continued) The reflection coefficient is then = Therefore s= (243 - j 92) - 377 = -0.19 - j 0.18 = 0.26 - 2.4rad (243 - j 92) + 377 1 + .26 = 1.7 and 1 - | |2 = 1 - (.26)2 = 0.93 1 - .26 12.17. Repeat Problem 12.16 for the cases in which the frequency is a) doubled: If this is true, then d = /4, and thus in = (260)2 /377 = 179. The reflection coefficient becomes 179 - 377 1 + .36 = = -0.36 s = = 2.13 179 + 377 1 - .36 Then 1 - | |2 = 1 - (.36)2 = 0.87. b) quadrupled: Now, d = /2, and so we have a half-wave section surrounded by air. Transmission will be total, and so s = 1 and 1 - | |2 = 1. 12.18. In Fig. 12.6, let 1 = 3 = 377 , and 2 = 0.41 . A uniform plane wave is normally incident from the left, as shown. Plot a curve of the standing wave ratio, s, in the region to the left: a) as a function of l if f = 2.5GHz: With 1 = 3 = 0 and with 2 = 0.40 , Eq. (41) becomes in = 0.40 = 0 Then 0.4 cos(l) - j sin(l) cos(l) + j 0.4 sin(l) 0.4 cos(l) + j sin(l) 0.4 cos(l) - j sin(l) 1 - j 1.05 sin(2l) cos2 (l) + 6.25 sin2 (l) = (in - 0 )/(in + 0 ), from which we find | |= = 1 - cos2 (l) - 6.25 sin2 (l) 1 + cos2 (l) + 6.25 sin2 (l) 2 2 + (1.05)2 sin2 (2l) + (1.05)2 sin2 (2l) 1/2 Then s = (1 + | |)/(1 - | |). Now for a uniform plane wave, = = n/c. Given that 2 = 0.40 = 0 /n, we find n = 2.5 (assuming = 0 ). Thus, at 2.5 GHz, l = (2.5)(2)(2.5 109 ) n l= l = 12.95 l (l in m) = 0.1295 l (l in cm) c 3 108 Using this in the expression for | |, and calculating s as a function of l in cm leads to the first plot shown on the next page. b) as a function of frequency if l = 2cm. In this case we use l = (2.5)(2)(0.02) f = 1.04 10-10 f (f in Hz) = 0.104 f (f in GHz) 3 108 Using this in the expression for | |, and calculating s as a function of f in GHz leads to the second plot shown on the next page. MathCad was used in both cases. 207 12.18 (continued) Plots for parts a and b 12.19. You are given four slabs of lossless dielectric, all with the same intrinsic impedance, , known to be different from that of free space. The thickness of each slab is /4, where is the wavelength as measured in the slab material. The slabs are to be positioned parallel to one another, and the combination lies in the path of a uniform plane wave, normally-incident. The slabs are to be arranged such that the air spaces between them are either zero, one-quarter wavelength, or one-half wavelength in thickness. Specify an arrangement of slabs and air spaces such that a) the wave is totally transmitted through the stack: In this case, we look for a combination of halfwave sections. Let the inter-slab distances be d1 , d2 , and d3 (from left to right). Two possibilities are i.) d1 = d2 = d3 = 0, thus creating a single section of thickness , or ii.) d1 = d3 = 0, d2 = /2, thus yielding two half-wave sections separated by a half-wavelength. b) the stack presents the highest reflectivity to the incident wave: The best choice here is to make d1 = d2 = d3 = /4. Thus every thickness is one-quarter wavelength. The impedances transform as follows: First, the input impedance at the front surface of the last slab (slab 4) is in,1 = 2 /0 . We transform this back to the back surface of slab 3, moving through a distance of /4 in free 2 3 space: in,2 = 0 /in,1 = 0 /2 . We next transform this impedance to the front surface of slab 3, 3 2 / producing in,3 = in,2 = 4 /0 . We continue in this manner until reaching the front surface n-1 7 of slab 1, where we find in,7 = 8 /0 . Assuming < 0 , the ratio n /0 becomes smaller as n increases (as the number of slabs increases). The reflection coefficient for waves incident on the front slab thus gets close to unity, and approaches 1 as the number of slabs approaches infinity. 12.20. The 50MHz plane wave of Problem 12.12 is incident onto the ocean surface at an angle to the normal of 60 . Determine the fractions of the incident power that are reflected and transmitted for a) s polarization: To review Problem 12, we first we find the loss tangent: 4 = = 18.4 6 )(78)(8.854 10-12 ) 2(50 10 This value is sufficiently greater than 1 to enable seawater to be considered a good conductor at 50MHz. Then, using the approximation (Eq. 65, Chapter 11), and with = 0 , the intrinsic impedance is s = f / (1 + j ) = 7.0(1 + j ). 208 12.20a. (continued) Next we need the angle of refraction, which means that we need to know the refractive index of seawater at 50MHz. For a uniform plane wave in a good conductor, the phase constant is = nsea . = c . f nsea = c = 26.8 4f Then, using Snell's law, the angle of refraction is found: sin 2 = nsea sin 1 = 26.8 sin(60 ) 2 = 1.9 n1 . This angle is small enough so that cos 2 = 1. Therefore, for s polarization, s 7.0(1 + j ) - 377/ cos 60 . s2 - s1 = = = -0.98 + j 0.018 = 0.98 179 s2 + s1 7.0(1 + j ) + 377/ cos 60 s| 2 The fraction of the power reflected is now | = 0.96. The fraction transmitted is then 0.04. b) p polarization: Again, with the refracted angle close to zero, the relection coefficient for p polarization is p 7.0(1 + j ) - 377 cos 60 . p2 - p1 = = = -0.93 + j 0.069 = 0.93 176 p2 + p1 7.0(1 + j ) + 377 cos 60 2 p| The fraction of the power reflected is now | = 0.86. The fraction transmitted is then 0.14. 12.21. A right-circularly polarized plane wave in air is incident at Brewster's angle onto a semi-infinite slab of plexiglas ( R = 3.45, R = 0, = 0 ). a) Determine the fractions of the incident power that are reflected and transmitted: In plexiglas, Brewster's angle is B = 1 = tan-1 ( R2 / R1 ) = tan-1 ( 3.45) = 61.7 . Then the angle of refraction is 2 = 90 - B (see Example 12.9), or 2 = 28.3 . With incidence at Brewster's angle, all p-polarized power will be transmitted -- only s-polarized power will be reflected. This is found through 2s - 1s .6140 - 2.110 = = -0.549 s = 2s + 1s .6140 + 2.110 where 1s = 1 sec 1 = 0 sec(61.7 ) = 2.110 , and 2s = 2 sec 2 = (0 / 3.45) sec(28.3 ) = 0.6140 . Now, the reflected power fraction is | |2 = (-.549)2 = .302. Since the wave is circularly-polarized, the s-polarized component represents one-half the total incident wave power, and so the fraction of the total power that is reflected is .302/2 = 0.15, or 15%. The fraction of the incident power that is transmitted is then the remainder, or 85%. b) Describe the polarizations of the reflected and transmitted waves: Since all the p-polarized component is transmitted, the reflected wave will be entirely s-polarized (linear). The transmitted wave, while having all the incident p-polarized power, will have a reduced s-component, and so this wave will be right-elliptically polarized. 209 12.22. A dielectric waveguide is shown in Fig. 12.18 with refractive indices as labeled. Incident light enters the guide at angle from the front surface normal as shown. Once inside, the light totally reflects at the upper n1 - n2 interface, where n1 > n2 . All subsequent reflections from the upper an lower boundaries will be total as well, and so the light is confined to the guide. Express, in terms of n1 and n2 , the maximum value of such that total confinement will occur, with n0 = 1. The quantity sin is known as the numerical aperture of the guide. From the illustration we see that 1 maximizes when 1 is at its minimum value. This minimum will be the critical angle for the n1 - n2 interface, where sin c = sin 1 = n2 /n1 . Let the refracted angle to the right of the vertical interface (not shown) be 2 , where n0 sin 1 = n1 sin 2 . Then we see that 2 + 1 = 90 , and so sin 1 = cos 2 . Now, the numerical aperture becomes sin 1max = n1 sin 2 = n1 cos 1 = n1 1 - sin2 1 = n1 1 - (n2 /n1 )2 = n0 n2 - n2 1 2 is the numerical aperture angle. n2 - n2 1 2 Finally, 1max = sin-1 12.23. Suppose that 1 in Fig. 12.18 is Brewster's angle, and that 1 is the critical angle. Find n0 in terms of n1 and n2 : With the incoming ray at Brewster's angle, the refracted angle of this ray (measured from the inside normal to the front surface) will be 90 - 1 . Therefore, 1 = 1 , and thus sin 1 = sin 1 . Thus n1 n2 sin 1 = = sin 1 = n0 = (n1 /n2 ) n2 - n2 1 2 n1 2 + n2 n0 1 Alternatively, we could have used the result of Problem 12.22, in which it was found that sin 1 = (1/n0 ) n2 - n2 , which we then set equal to sin 1 = n2 /n1 to get the same result. 1 2 12.24. A Brewster prism is designed to pass p-polarized light without any reflective loss. The prism of Fig. 12.19 is made of glass (n = 1.45), and is in air. Considering the light path shown, determine the apex angle, : With entrance and exit rays at Brewster's angle (to eliminate reflective loss), the interior ray must be horizontal, or parallel to the bottom surface of the prism. From the geometry, the angle between the interior ray and the normal to the prism surfaces that it intersects is /2. Since this angle is also Brewster's angle, we may write: = 2 sin-1 1 1 + n2 = 2 sin-1 1 1 + (1.45)2 = 1.21 rad = 69.2 12.25. In the Brewster prism of Fig. 12.19, determine for s-polarized light the fraction of the incident power that is transmitted through the prism: We use s = (s2 - s1 )/(s2 + s1 ), where s2 = and s1 = 2 0 2 = 2 = cos(B2 ) n n/ 1 + n2 1 + n2 1 1 = = 0 1 + n2 cos(B1 ) 1/ 1 + n2 210 12.25. (continued) Thus, at the first interface, = (1 - n2 )/(1 + n2 ). At the second interface, will be equal but of opposite sign to the above value. The power transmission coefficient through each interface is 1 - | |2 , so that for both interfaces, we have, with n = 1.45: Ptr = 1 - | |2 Pinc 2 = 1- n2 - 1 n2 + 1 2 2 = 0.76 12.26. Show how a single block of glass can be used to turn a p-polarized beam of iight through 180 , with the light suffering, in principle, zero reflective loss. The light is incident from air, and the returning beam (also in air) may be displaced sideways from the incident beam. Specify all pertinent angles and use n = 1.45 for glass. More than one design is possible here. The prism below is designed such that light enters at Brewster's angle, and once inside, is turned around using total reflection. Using the result of Example 12.9, we find that with glass, B = 55.4 , which, by the geometry, is also the incident angle for total reflection at the back of the prism. For this to work, the Brewster angle must be greater than or equal to the critical angle. This is in fact the case, since c = sin-1 (n2 /n1 ) = sin-1 (1/1.45) = 43.6 . 12.27. Using Eq. (59) in Chapter 11 as a starting point, determine the ratio of the group and phase velocities of an electromagnetic wave in a good conductor. Assume conductivity does not vary with frequency: In a good conductor: = Thus d = d d d f = 2 d 1 = d 2 2 -1/2 2 -1 =2 2 = vg and vp = = = /2 2 Therefore vg /vp = 2. 211 12.28. Over a certain frequency range, the refractive index of a certain material varies approximately linearly . with frequency: n() = na + nb ( - a ), where na , nb , and a are constants. Using = n/c: a) determine the group velocity as a function (or perhaps not a function) of frequency: vg = (d/d)-1 , where d d na nb ( - a ) 1 = + = [na + nb (2 - a )] d d c c c so that vg () = c [na + nb (2 - a )]-1 b) determine the group dispersion parameter, 2 : 2 = d 2 d2 = d 1 [na + nb (2 - a )] d c = 2nb /c 0 0 c) Discuss the implications of these results, if any, on pulse broadening: The point of this problem was to show that higher order terms (involving d 3 /d3 and higher) in the Taylor series expansion, Eq. (89), do not exist if the refractive index varies linearly with . These higher order terms would be necessary in cases involving pulses of exremely large bandwidth, or in media exhibiting complicated variations in their - curves over relatively small frequency ranges. With d 2 /d2 constant, the three-term Taylor expansion of Eq. (89) describes the phase constant of this medium exactly. The pulse will broaden and will acquire a frequency sweep (chirp) that is precisely linear with time. Additionally, a pulse of a given bandwidth will broaden by the same amount, regardless of what carrier frequency is used. 12.29. A T = 5 ps transform-limited pulse propagates in a dispersive channel for which 2 = 10 ps2 /km. Over what distance will the pulse spread to twice its initial width? After propagation, the width is T = T 2 + ( )2 = 2T . Thus = 3T , where = 2 z/T . Therefore 2 2 z 3T 3(5 ps)2 = 3T or z = = 4.3 km = T 2 10 ps2 /km 12.30. A T = 20 ps transform-limited pulse propagates through 10 km of a dispersive channel for which 2 = 12 ps2 /km. The pulse then propagates through a second 10 km channel for which 2 = -12 ps2 /km. Describe the pulse at the output of the second channel and give a physical explanation for what happened. Our theory of pulse spreading will allow for changes in 2 down the length of the channel. In fact, we may write in general: 1 L = 2 (z) dz T 0 Having 2 change sign at the midpoint, yields a zero , and so the pulse emerges from the output unchanged! Physically, the pulse acquires a positive linear chirp (frequency increases with time over the pulse envelope) during the first half of the channel. When 2 switches sign, the pulse begins to acquire a negative chirp in the second half, which, over an equal distance, will completely eliminate the chirp acquired during the first half. The pulse, if originally transform-limited at input, will emerge, again transform-limited, at its original width. More generally, complete dispersion compensation is achieved using a two-segment channel when 2 L = -2 L , assuming dispersion terms of higher order than 2 do not exist. 212 CHAPTER 13 13.1. The parameters of a certain transmission line operating at 6 108 rad/s are L = 0.4 H/m, C = 40 pF/m, G = 80 mS/m, and R = 20 /m. a) Find , , , , and Z0 : We use = ZY = (R + j L)(G + j C) = [20 + j (6 108 )(0.4 10-6 )][80 10-3 + j (6 108 )(40 10-12 )] = 2.8 + j 3.5 m-1 = + j Therefore, = 2.8 Np/m, = 3.5 rad/m, and = 2/ = 1.8 m. Finally, Z0 = Z = Y R + j L = G + j C 20 + j 2.4 102 = 44 + j 30 80 10-3 + j 2.4 10-2 b) If a voltage wave travels 20 m down the line, what percentage of the original amplitude remains, and by how many degrees is it phase shifted? First, V20 = e-L = e-(2.8)(20) = 4.8 10-25 or 4.8 10-23 percent! V0 Then the phase shift is given by L, which in degrees becomes = L 360 2 = (3.5)(20) 360 2 = 4.0 103 degrees 13.2. A lossless transmission line with Z0 = 60 is being operated at 60 MHz. The velocity on the line is 3 108 m/s. If the line is short-circuited at z = 0, find Zin at: a) z = -1m: We use the expression for input impedance (Eq. 12), under the conditions Z2 = 60 and Z3 = 0: Z3 cos(l) + j Z2 sin(l) Zin = Z2 = j 60 tan(l) Z2 cos(l) + j Z3 sin(l) where l = -z, and where the phase constant is = 2c/f = 2(3 108 )/(6 107 ) = (2/5) rad/m. Now, with z = -1 (l = 1), we find Zin = j 60 tan(2/5) = j 184.6 . b) z = -2 m: Zin = j 60 tan(4/5) = -j 43.6 c) z = -2.5 m: Zin = j 60 tan(5/5) = 0 d) z = -1.25 m: Zin = j 60 tan(/2) = j (open circuit) . If L = 0.5 H/m, find: 13.3. The characteristic impedance of a certain lossless transmission line is 72 a) C: Use Z0 = L/C, or C= L 5 10-7 = = 9.6 10-11 F/m = 96 pF/m 2 2 (72) Z0 213 13.3b) vp : 1 vp = = LC 1 (5 10-7 )(9.6 10-11 ) = 1.44 108 m/s c) if f = 80 MHz: 2 80 106 = LC = = 3.5 rad/m 1.44 108 . Find and s: s= 1 + .09 1+| | = = 1.2 1-| | 1 - .09 d) The line is terminated with a load of 60 = 60 - 72 = -0.09 60 + 72 13.4. A lossless transmission line having Z0 = 120 is operating at = 5 108 rad/s. If the velocity on m/s, the line is 2.4 108 find: a) L: With Z0 = L/C and v = 1/ LC, we find L = Z0 /v = 120/2.4 108 = 0.50 H/m. b) C: Use Z0 v = L/C/ LC C = 1/(Z0 v) = [120(2.4 108 )]-1 = 35 pF/m. c) Let ZL be represented by an inductance of 0.6 H in series with a 100- resistance. Find and s: The inductive impedance is j L = j (5 108 )(0.6 10-6 ) = j 300. So the load impedance is ZL = 100 + j 300 . Now = Then s= ZL - Z0 100 + j 300 - 120 = = 0.62 + j 0.52 = 0.808 40 ZL + Z 0 100 + j 300 + 120 1+| | 1 + 0.808 = = 9.4 1-| | 1 - 0.808 13.5. Two characteristics of a certain lossless transmission line are Z0 = 50 and = 0 + j 0.2 m-1 at f = 60 MHz. a) Find L and C for the line: We have = 0.2 = LC and Z0 = 50 = L/C. Thus 1 0.2 = 1010 = 33.3 pF/m = C C = = 6 )(50) Z0 Z0 (2 60 10 3 2 Then L = CZ0 = (33.3 10-12 )(50)2 = 8.33 10-8 H/m = 83.3 nH/m. b) A load, ZL = 60 + j 80 is located at z = 0. What is the shortest distance from the load to a point at which Zin = Rin + j 0? I will do this using two different methods: The Hard Way: We use the general expression Zin = Z0 ZL + j Z0 tan(l) Z0 + j ZL tan(l) We can then normalize the impedances with respect to Z0 and write zin = zL + j tan(l) Zin (ZL /Z0 ) + j tan(l) = = Z0 1 + j (ZL /Z0 ) tan(l) 1 + j zL tan(l) where zL = (60 + j 80)/50 = 1.2 + j 1.6. 214 13.5b. (continued) Using this, and defining x = tan(l), we find zin = 1.2 + j (1.6 + x) (1 - 1.6x) + j 1.2x (1 - 1.6x) - j 1.2x (1 - 1.6x) - j 1.2x The second bracketed term is a factor of one, composed of the complex conjugate of the denominator of the first term, divided by itself. Carrying out this product, we find zin = 1.2(1 - 1.6x) + 1.2x(1.6 + x) - j [(1.2)2 x - (1.6 + x)(1 - 1.6x)] (1 - 1.6x)2 + (1.2)2 x 2 We require the imaginary part to be zero. Thus (1.2)2 x - (1.6 + x)(1 - 1.6x) = 0 1.6x 2 + 3x - 1.6 = 0 So x = tan(l) = We take the positive root, and find l = tan-1 (.433) = 0.409 l = The Easy Way: We find = 60 + j 80 - 50 = 0.405 + j 0.432 = 0.59 0.818 60 + j 80 + 50 0.409 = 0.65 m = 65 cm 0.2 -3 9 + 4(1.6)2 = (.433, -2.31) 2(1.6) Thus = 0.818 rad, and we use the fact that the input impedance will be purely real at the location of a voltage minimum or maximum. The first voltage maximum will occur at a distance in front of the load given by 0.818 zmax = = = 0.65 m 2 2(0.2) 13.6. The propagation constant of a lossy transmission line is 1 + j 2 m-1 , and its characteristic impedance is 20 + j 0 at = 1 Mrad/s. Find L, C, R, and G for the line: Begin with Z0 = Then 2 = (R + j L)(G + j C) = (1 + j 2)2 400(G + j C)2 = (1 + j 2)2 (2) where (1) has been used. Eq. 2 now becomes G + j C = (1 + j 2)/20. Equating real and imaginary parts leads to G = .05 S/m and C = 1/(10) = 10-7 = 0.1 F/m. R + j L = 20 R + j L = 400(G + j C) G + j L (1) 215 13.6. (continued) Now, (1) becomes 20 = R + j L R + j L 20 + j 40 = R + j L 20 20 = 1 + j2 1 + j2 Again, equating real and imaginary parts leads to R = 20 /m and L = 40/ = 40 H/m. 13.7. The dimensions of the outer conductor of a coaxial cable are b and c, c > b. Assume = c and let = 0 . Find the magnetic energy stored per unit length in the region b < r < c for a uniformly distributed total current I flowing in opposite directions in the inner and outer conductors: First, from the inner conductor, the magnetic field will be H1 = I a 2 The contribution from the outer conductor to the magnetic field within that conductor is found from Ampere's circuital law to be: I 2 - b2 H2 = - a 2 c2 - b2 The total magnetic field within the outer conductor will be the sum of the two fields, or HT = H1 + H2 = The energy density is 1 0 I 2 2 wm = 0 HT = 2 8 2 c2 - 2 c2 - b2 2 I 2 c2 - 2 c2 - b2 a J/m3 The stored energy per unit length in the outer conductor is now Wm = 1 0 0 2 b c 0 I 2 8 2 c2 - 2 c2 - b2 2 d d dz = J 0 I 2 4(c2 - b2 )2 c b c4 - 2c2 + 3 d 0 I 2 = 4 c c4 b2 - (3/4)c2 ln + (c2 - b2 )2 b (c2 - b2 ) 13.8. The conductors of a coaxial transmission line are copper (c = 5.8 10-7 S/m) and the dielectric is polyethylene ( R = 2.26, / = 0.0002). If the inner radius of the outer conductor is 4 mm, find the radius of the inner conductor so that (assuming a lossless line): a) Z0 = 50 : Use Z0 = 1 2 b a = 50 ln b a = 2 R (50) ln 377 = 1.25 Thus b/a = e1.25 = 3.50, or a = 4/3.50 = 1.142 mm 216 13.8b. C = 100 pF/m: Begin with C= 2 b = 10-10 ln ln(b/a) a = 2(2.26)(8.854 10-2 ) = 1.257 So b/a = e1.257 = 3.51, or a = 4/3.51 = 1.138 mm. c) L = 0.2 H/m: Use L= 0 b ln 2 a = 0.2 10-6 ln b a = 2(0.2 10-6 ) =1 4 10-7 Thus b/a = e1 = 2.718, or a = b/2.718 = 1.472 mm. 13.9. Two aluminum-clad steel conductors are used to construct a two-wire transmission line. Let Al = 3.8 107 S/m, St = 5 106 S/m, and St = 100 H/m. The radius of the steel wire is 0.5 in., and the aluminum coating is 0.05 in. thick. The dielectric is air, and the center-to-center wire separation is 4 in. Find C, L, G, and R for the line at 10 MHz: The first question is whether we are in the high frequency or low frequency regime. Calculation of the skin depth, , will tell us. We have, for aluminum, 1 = = f 0 Al 1 (107 )(4 10-7 )(3.8 107 ) = 2.58 10-5 m so we are clearly in the high frequency regime, where uniform current distributions cannot be assumed. Furthermore, the skin depth is considerably less than the aluminum layer thickness, so the bulk of the current resides in the aluminum, and we may neglect the steel. Assuming solid aluminum wires of radius a = 0.5 + 0.05 = 0.55 in = 0.014 m, the resistance of the two-wire line is now R= 1 1 = 0.023 = aAl (.014)(2.58 10-5 )(3.8 107 ) /m Next, since the dielectric is air, no leakage will occur from wire to wire, and so G = 0 mho/m. Now the capacitance will be C= 0 cosh-1 (d/2a) = 8.85 10-12 = 1.42 10-11 F/m = 14.2 pF/m cosh-1 (4/(2 0.55)) Finally, the inductance per unit length will be L= 0 4 10-7 cosh(d/2a) = cosh (4/(2 0.55)) = 7.86 10-7 H/m = 0.786 H/m 217 13.10. Each conductor of a two-wire transmission line has a radius of 0.5mm; their center-to-center distance is 0.8cm. Let f = 150MH z and assume = 0 and c (note error in problem statement). Find the dielectric constant of the insulating medium if a) Z0 = 300 : Use 300 = 1 0 R 0 cosh-1 d 2a R = 120 cosh-1 300 8 2(.5) = 1.107 R = 1.23 b) C = 20 pF/m: Use 20 10-12 = c) vp = 2.6 108 m/s: = vp = LC 1 1 0 0 R cosh-1 (d/2a) R = 20 10-12 cosh-1 (8) = 1.99 0 = c R R = 3.0 108 2.6 108 2 = 1.33 13.11. Pertinent dimensions for the transmission line shown in Fig. 13.4 are b = 3 mm, and d = 0.2 mm. The conductors and the dielectric are non-magnetic. a) If the characteristic impedance of the line is 15 , find R : We use Z0 = d b = 15 R = 377 15 2 .04 = 2.8 9 b) Assume copper conductors and operation at 2 108 rad/s. If RC = GL, determine the loss tangent of the dielectric: For copper, c = 5.8 107 S/m, and the skin depth is = Then R= Now C= and L= Then, with RC = GL, G= 2 = 0 c (2 108 )(4 2 = 1.2 10-5 m -7 )(5.8 107 ) 10 /m 2 2 = 0.98 = 7 )(1.2 10-5 )(.003) c b (5.8 10 (2.8)(8.85 10-12 )(3) b = = 3.7 10-10 F/m d 0.2 0 d (4 10-7 )(0.2) = = 8.4 10-8 H/m b 3 RC d b (.98)(3.7 10-10 ) = = 4.4 10-3 mho/m = -8 ) L (8.4 10 d 2.9 10-4 d = = 5.85 10-2 (2 108 )(2.8)(8.85 10-12 ) 218 Thus d = (4.4 10-3 )(0.2/3) = 2.9 10-4 S/m. The loss tangent is l.t. = 13.12. A transmission line constructed from perfect conductors and an air dielectric is to have a maximum dimension of 8mm for its cross-section. The line is to be used at high frequencies. Specify its dimensions if it is: a) a two-wire line with Z0 = 300 : With the maximum dimension of 8mm, we have, using (27): Z0 = 1 cosh-1 8 - 2a 2a = 300 8 - 2a 300 = cosh 2a 120 = 6.13 Solve for a to find a = 0.56 mm. Then d = 8 - 2a = 6.88 mm. b) a planar line with Z0 = 15 : In this case our maximum dimension dictates that So, using (34), we write Z0 = 64 - b2 = 15 b 64 - b2 = 15 b 377 d 2 + b2 = 8. Solving, we find b = 7.99 mm and d = 0.32 mm. c) a 72 coax having a zero-thickness outer conductor: With a zero-thickness outer conductor, we note that the outer radius is b = 8/2 = 4mm. Using (18), we write Z0 = 1 2 ln b a = 72 ln b a = 2(72) = 1.20 a = be-1.20 = 4e-1.20 = 1.2 120 Summarizing, a = 1.2 mm and b = 4 mm. 13.13. The incident voltage wave on a certain lossless transmission line for which Z0 = 50 m/s is V + (z, t) = 200 cos(t - z) V. a) Find : We know = = /vp , so = (2 108 ) = 6.28 108 rad/s. b) Find I + (z, t): Since Z0 is real, we may write I + (z, t) = V + (z, t) = 4 cos(t - z) A Z0 and vp = 2108 The section of line for which z > 0 is replaced by a load ZL = 50 + j 30 c) L : This will be L at z = 0. Find = 50 + j 30 - 50 = .0825 + j 0.275 = 0.287 1.28 rad 50 + j 30 + 50 = 0.287(200)ej z ej 1.28 = 57.5ej ( z+1.28) d) Vs- (z) = + j 2z L Vs (z)e e) Vs at z = -2.2 m: Vs (-2.2) = Vs+ (-2.2) + Vs- (-2.2) = 200ej 2.2 + 57.5e-j (2.2 -1.28) = 257.5ej 0.63 = 257.5 36 219 13.14. Coaxial lines 1 and 2 have the following parameters: 1 = 2 = 0 , 1 = 2 = 0, R2 = 4, a1 = a2 = 0.8mm, b1 = 6mm, b2 = 3mm, ZL2 = Z02 , and ZL1 is Zin2 . a) Find Z01 and Z02 . For either line, we have Z0 = 1 2 ln b a = 377 2 and R R1 = 2.25, ln b a Z02 = leading to Z01 = 377 6 ln .8 2 2.25 = 80.6 377 3 ln .8 2 4 = 39.7 b) Find s on line 1: Line 1's load is line 2's input impedance (they are connected end-to-end). Also, since line 2 is matched, its input impedance is just it's characteristic impedance. Therefore, ZL1 = Zin2 = Z02 . The reflection coefficient encountered by waves incident on ZL1 from line 1 can now be found, along with the standing wave ratio: 12 = 1 + .34 39.7 - 80.6 = -0.34 s = = 2.03 39.7 + 80.6 1 - .34 c) If a 20cm length of line 1 is inserted immediately in front of ZL2 and f = 300MHz, find s on line 2: The line 1 length now has a load impedance of 39.7 and it is 20cm long. We need to find its input impedance. At 300 MHz, the free space wavelength is 1m. In line 1, having a dielectric constant of 2.25, the wavelength is = 1m/ 2.25 = 0.67m. Therefore l = 2l/ = 2(20)/(67) = 1.87. We now find the input impedance for this situation through Zin = Z01 ZL2 cos(l) + j Z01 sin(l) 39.7 cos(1.87) + j 80.6 sin(1.87) = 80.6 Z01 cos(l) + j ZL2 sin(l) 80.6 cos(1.87) + j 39.7 sin(1.87) = 128.7 - j 55.8 = 140.3 - 23.4 Now for waves incident at the line 1 - line 2 junction from line 2, the reflection coefficient will be 21 = Zin - Z02 128.7 - 39.7 - j 55.8 = = 0.58 - j 0.14 = 0.59 - 13.7 Zin + Z02 128.7 + 39.7 - j 55.8 s= 1 + .59 = 3.9 1 - .59 The standing wave ratio is now 220 13.15. For the transmission line represented in Fig. 13.26, find Vs,out if f =: a) 60 Hz: At this frequency, = 2 60 = 1.9 10-6 rad/m So l = (1.9 10-6 )(80) = 1.5 10-4 << 1 = vp (2/3)(3 108 ) . Therefore . The line is thus essentially a lumped circuit, where Zin = ZL = 80 Vs,out = 120 b) 500 kHz: In this case = Now Zin = 50 80 = 104 V 12 + 80 2 5 105 = 1.57 10-2 rad/s So l = 1.57 10-2 (80) = 1.26 rad 2 108 80 cos(1.26) + j 50 sin(1.26) = 33.17 - j 9.57 = 34.5 - .28 50 cos(1.26) + j 80 sin(1.26) The equivalent circuit is now the voltage source driving the series combination of Zin and the 12 ohm resistor. The voltage across Zin is thus Vin = 120 Zin = 120 12 + Zin 33.17 - j 9.57 = 89.5 - j 6.46 = 89.7 - .071 12 + 33.17 - j 9.57 The voltage at the line input is now the sum of the forward and backward-propagating waves just to the right of the input. We reference the load at z = 0, and so the input is located at z = -80 m. + - In general we write Vin = V0 e-jz + V0 ejz , where - V0 = + L V0 = 80 - 50 + 3 + V0 = V 80 + 50 13 0 At z = -80 m we thus have + Vin = V0 ej 1.26 + 3 -j 1.26 e 13 + V0 = ej 1.26 89.5 - j 6.46 = 42.7 - j 100 V + (3/13)e-j 1.26 Now + Vs,out = V0 (1 + L) = (42.7 - j 100)(1 + 3/(13)) = 134 - 1.17 rad = 52.6 - j 123 V As a check, we can evaluate the average power reaching the load: Pavg,L = 1 |Vs,out |2 1 (134)2 = 112 W = 2 RL 2 80 This must be the same power that occurs at the input impedance: Pavg,in = 1 1 Re Vin Iin = Re {(89.5 - j 6.46)(2.54 + j 0.54)} = 112 W 2 2 where Iin = Vin /Zin = (89.5 - j 6.46)/(33.17 - j 9.57) = 2.54 + j 0.54. 221 13.16. A 300 ohm transmission line is 0.8 m long and is terminated with a short circuit. The line is operating in air with a wavelength of 0.3 m (incorrectly stated as 0.8 m in early printings) and is lossless. a) If the input voltage amplitude is 10V, what is the maximum voltage amplitude at any point on the line? The net voltage anywhere on the line is the sum of the forward and backward wave voltages, + - and is written as V (z) = V0 e-jz + V0 ejz . Since the line is short-circuited at the load end - + (z = 0), we have V0 = -V0 , and so + + V (z) = V0 e-jz - ejz = -2j V0 sin(jz) We now evaluate the voltage at the input, where z = -0.8m, and = 0.3m. + Vin = -2j V0 sin 2(-0.8) 0.3 + = -j 1.73V0 + The magnitude of Vin is given as 10V, so we find V0 = 10/1.73 = 5.78V. The maximum voltage amplitude on the line will be twice this value (where the sine function is unity), so |V |max = 2(5.78) = 11.56 V. b) What is the current amplitude in the short circuit? At the shorted end, the current will be IL = + + V0 V- 2V0 11.56 = 0.039A = 39 mA - 0 = = Z0 Z0 Z0 300 13.17. Determine the average power absorbed by each resistor in Fig. 13.27: The problem is made easier by first converting the current source/100 ohm resistor combination to its Thevenin equivalent. This is a 50 0 V voltage source in series with the 100 ohm resistor. The next step is to determine the input impedance of the 2.6 length line, terminated by the 25 ohm resistor: We use l = (2/)(2.6) = 16.33 rad. This value, modulo 2 is (by subtracting 2 twice) 3.77 rad. Now Zin = 50 25 cos(3.77) + j 50 sin(3.77) = 33.7 + j 24.0 50 cos(3.77) + j 25 sin(3.77) The equivalent circuit now consists of the series combination of 50 V source, 100 ohm resistor, and Zin , as calculated above. The current in this circuit will be I= 50 = 0.368 - .178 100 + 33.7 + j 24.0 The power dissipated by the 25 ohm resistor is the same as the power dissipated by the real part of Zin , or 1 1 P25 = P33.7 = |I |2 R = (.368)2 (33.7) = 2.28 W 2 2 To find the power dissipated by the 100 ohm resistor, we need to return to the Norton configuration, with the original current source in parallel with the 100 ohm resistor, and in parallel with Zin . The voltage across the 100 ohm resistor will be the same as that across Zin , or V = I Zin = (.368 - .178)(33.7 + j 24.0) = 15.2 0.44. The power dissipated by the 100 ohm resistor is now 1 |V |2 1 (15.2)2 P100 = = = 1.16 W 2 R 2 100 222 13.18 The line shown in Fig. 13.28 is lossless. Find s on both sections 1 and 2: For section 2, we consider the propagation of one forward and one backward wave, comprising the superposition of all reflected waves from both ends of the section. The ratio of the backward to the forward wave amplitude is given by the reflection coefficient at the load, which is L = 50 - j 100 - 50 -j 1 = = (1 - j ) 50 - j 100 + 50 1-j 2 Then | L| = (1/2) (1 - j )(1 + j ) = 1/ 2. Finally 1+| s2 = 1-| 1 + 1/ 2 = = 5.83 1 - 1/ 2 L| L| For section 1, we need the reflection coefficient at the junction (location of the 100 resistor) seen by waves incident from section 1: We first need the input impedance of the .2 length of section 2: Zin2 = 50 (50 - j 100) cos(2 l) + j 50 sin(2 l) (1 - j 2)(0.309) + j 0.951 = 50 50 cos(2 l) + j (50 - j 100) sin(2 l) 0.309 + j (1 - j 2)(0.951) = 8.63 + j 3.82 = 9.44 0.42 rad Now, this impedance is in parallel with the 100 resistor, leading to a net junction impedance found by 1 ZinT = 1 1 + ZinT = 8.06 + j 3.23 = 8.69 0.38 rad 100 8.63 + j 3.82 The reflection coefficient will be j = ZinT - 50 = -0.717 + j 0.096 = 0.723 3.0 rad ZinT + 50 and the standing wave ratio is s1 = (1 + 0.723)/(1 - 0.723) = 6.22. 13.19. A lossless transmission line is 50 cm in length and operating at a frequency of 100 MHz. The line parameters are L = 0.2 H/m and C = 80 pF/m. The line is terminated by a short circuit at z = 0, and there is a load, ZL = 50 + j 20 ohms across the line at location z = -20 cm. What average power is delivered to ZL if the input voltage is 100 0 V? With the given capacitance and inductance, we find Z0 = and L = C 2 10-7 = 50 8 10-11 1 (2 10-7 )(9 10-11 ) = 2.5 108 m/s 1 vp = = LC Now = /vp = (2 108 )/(2.5 108 ) = 2.5 rad/s. We then find the input impedance to the shorted line section of length 20 cm (putting this impedance at the location of ZL , so we can combine them): We have l = (2.5)(0.2) = 0.50, and so, using the input impedance formula with a zero load impedance, we find Zin1 = j 50 tan(0.50) = j 27.4 ohms. 223 13.19 (continued) Now, at the location of ZL , the net impedance there is the parallel combination of ZL and Zin1 : Znet = (50 + j 20)||(j 27.4) = 7.93 + j 19.9. We now transform this impedance to the line input, 30 cm to the left, obtaining (with l = (2.5)(.3) = 0.75): Zin2 = 50 (7.93 + j 19.9) cos(.75) + j 50 sin(.75) = 35.9 + j 98.0 = 104.3 1.22 50 cos(.75) + j (7.93 + j 19.9) sin(.75) The power delivered to ZL is the same as the power delivered to Zin2 : The current magnitude is |I | = (100)/(104.3) = 0.96 A. So finally, P = 1 1 2 |I | R = (0.96)2 (35.9) = 16.5 W 2 2 13.20. This problem was originally posed incorrectly. The corrected version should have an inductor in the input circuit instead of a capacitor. I will proceed with this replacement understood, and will change the wording as appropriate in parts c and d: a) Determine s on the transmission line of Fig. 13.29. Note that the dielectric is air: The reflection coefficient at the load is L = 40 + j 30 - 50 = j 0.333 = 0.333 1.57 rad 40 + j 30 + 50 Then s = 1 + .333 = 2.0 1 - .333 b) Find the input impedance: With the length of the line at 2.7, we have l = (2)(2.7) = 16.96 rad. The input impedance is then Zin = 50 (40 + j 30) cos(16.96) + j 50 sin(16.96) -1.236 - j 5.682 = 50 = 61.8 - j 37.5 50 cos(16.96) + j (40 + j 30) sin(16.96) 1.308 - j 3.804 c) If L = 10 , find Is : The source drives a total impedance given by Znet = 20 + j L + Zin = 20+j 10+61.8-j 37.5 = 81.8-j 27.5. The current is now Is = 100/(81.8-j 27.5) = 1.10 + j 0.37 A. d) What value of L will produce a maximum value for |Is | at = 1 Grad/s? To achieve this, the imaginary part of the total impedance of part c must be reduced to zero (so we need an inductor). The inductor impedance must be equal to negative the imaginary part of the line input impedance, or L = 37.5, so that L = 37.5/ = 37.5 nH. Continuing, for this value of L, calculate the average power: e) supplied by the source: Ps = (1/2)Re{Vs Is } = (1/2)(100)2 /(81.8) = 61.1 W. f) delivered to ZL = 40 + j 30 : The power delivered to the load will be the same as the power delivered to the input impedance. We write PL = 1 1 Re{Zin }|Is |2 = (61.8)(1.22)2 = 46.1 W 2 2 224 13.21. A lossless line having an air dielectric has a characteristic impedance of 400 . The line is operating at 200 MHz and Zin = 200 - j 200 . Use analytic methods or the Smith chart (or both) to find: (a) s; (b) ZL if the line is 1 m long; (c) the distance from the load to the nearest voltage maximum: I will first use the analytic approach. Using normalized impedances, Eq. (13) becomes zin = Solve for zL : zL = zL + j tan(L) Zin zL cos(L) + j sin(L) = = Z0 cos(L) + j zL sin(L) 1 + j zL tan(L) zin - j tan(L) 1 - j zin tan(L) where, with = c/f = 3 108 /2 108 = 1.50 m, we find L = (2)(1)/(1.50) = 4.19, and so tan(L) = 1.73. Also, zin = (200 - j 200)/400 = 0.5 - j 0.5. So zL = 0.5 - j 0.5 - j 1.73 = 2.61 + j 0.174 1 - j (0.5 - j 0.5)(1.73) . Next Finally, ZL = zL (400) = 1.04 103 + j 69.8 = Now s= Finally zmax = - ZL - Z0 6.42 102 + j 69.8 = .446 + j 2.68 10-2 = .447 6.0 10-2 rad = ZL + Z 0 1.44 103 + j 69.8 1 + .447 1+| | = = 2.62 1-| | 1 - .447 (6.0 10-2 )(1.50) =- =- = -7.2 10-3 m = -7.2 mm 2 4 4 We next solve the problem using the Smith chart. Referring to the figure on the next page, we first locate and mark the normalized input impedance, zin = 0.5 - j 0.5. A line drawn from the origin through this point intersects the outer chart boundary at the position 0.0881 on the wavelengths toward load (WTL) scale. With a wavelength of 1.5 m, the 1 meter line is 0.6667 wavelengths long. On the WTL scale, we add 0.6667, or equivalently, 0.1667 (since 0.5 is once around the chart), obtaining (0.0881+0.1667)) = 0.2548, which is the position of the load. A straight line is now drawn from the origin though the 0.2548 position. A compass is then used to measure the distance between the origin and zin . With this distance set, the compass is then used to scribe off the same distance from the origin to the load impedance, along the line between the origin and the 0.2548 position. That point is the normalized load impedance, which is read to be zL = 2.6 + j 0.18. Thus ZL = zL (400) = 1040 + j 72. This is in reasonable agreement with the analytic result of 1040 + j 69.8. The difference in imaginary parts arises from uncertainty in reading the chart in that region. In transforming from the input to the load positions, we cross the r > 1 real axis of the chart at r=2.6. This is close to the value of the VSWR, as we found earlier. We also see that the r > 1 real axis (at which the first Vmax occurs) is a distance of 0.0048 (marked as .005 on the chart) in front of the load. The actual distance is zmax = -0.0048(1.5) m = -0.0072 m = -7.2 mm. 225 . 13.22. A lossless two-wire line has a characteristic impedance of 300 and a capacitance of 15 pF/m. The load at z = 0 consists of a 600- resistor in parallel with a 10-pF capacitor. If = 108 rad/s and the line is 20m long, use the Smith chart to find a) | L |; b) s; c) Zin . First, the wavelength on the line is found using = 2vp /, where vp = 1/(CZ0 ). Assuming higher accuracy in the given values than originally stated, we obtain = 2 2 = = 13.96 m 8 )(15 10-12 )(300) CZ0 (10 The line length in wavelengths is therefore 20/13.96 = 1.433. The normalized load admittance is now 1 1 + j (108 )(10-11 ) = 0.50 + j 0.30 + j C = 300 yL = YL Z0 = Z0 RL 600 226 The yL value is plotted on the chart and labeled as yL . Next, yL is inverted to find zL by transforming the point halfway around the chart, using the compass and a straight edge. The result, labeled zL on the chart is read to be zL = 1.5 - j 0.87. This is close to the computed inverse of yL , which is 1.47 - j 0.88. Scribing the compass arc length along the bottom scale for reflection coefficient yields | L | = 0.38. The VSWR is found by scribing the compass arc length either along the bottom SWR scale or along the positive real axis of the chart, both methods yielding s = 2.2. Now, the position of zL is read on the outer edge of the chart as 0.308 on the WTG scale. The point is now transformed through the line length distance of 1.433 toward the generator (the net chart distance will be 0.433, since a full wavelength is two complete revolutions). The final reading on the WTG scale after the transformation is found through (0.308 + 0.433 - 0.500) = 0.241. Drawing a line between this mark on the WTG scale and the chart center, and scribing the compass arc length on this line, yields the normalized input impedance. This is read as zin = 2.2 + j 0.21 (the computed value found through the analytic solution is zin = 2.21 + j 0.219. The input impedance is now found by multiplying the chart reading by 300, or Zin = 660 + j 63 . 227 13.23. The normalized load on a lossless transmission line is zL = 2 + j 1. Let l = 20 m (there was a missprint in the problem statement, since = 20 m should have been stated. I will specify answers in terms of wavelength). Make use of the Smith chart to find: a) the shortest distance from the load to the point at which zin = rin + j 0, where rin > 1 (not greater than 0 as stated): Referring to the figure below, we start by marking the given zL on the chart and drawing a line from the origin through this point to the outer boundary. On the WTG scale, we read the zL location as 0.213. Moving from here toward the generator, we cross the positive R axis (at which the impedance is purely real and greater than 1) at 0.250. The distance is then (0.250 - 0.213) = 0.037 from the load. If we use = 20 m, the actual distance would be 20(0.037) = 0.74 m. b) Find zin at the point found in part a: Using a compass, we set its radius at the distance between the origin and zL . We then scribe this distance along the real axis to find zin = rin = 2.61. c) The line is cut at this point and the portion containing zL is thrown away. A resistor r = rin of part a is connected across the line. What is s on the remainder of the line? This will be just s for the line as it was before. As we know, s will be the positive real axis value of the normalized impedance, or s = 2.61. d) What is the shortest distance from this resistor to a point at which zin = 2 + j 1? This would return us to the original point, requiring a complete circle around the chart (one-half wavelength distance). The distance from the resistor will therefore be: d = 0.500 - 0.037 = 0.463 . With = 20 m, the actual distance would be 20(0.463) = 9.26 m. 228 13.24. With the aid of the Smith chart, plot a curve of |Zin | vs. l for the transmission line shown in Fig. 13.30. Cover the range 0 < l/ < 0.25. The required input impedance is that at the actual line input (to the left of the two 20 resistors. The input to the line section occurs just to the right of the 20 resistors, and the input impedance there we first find with the Smith chart. This impedance is in series with the two 20 resistors, so we add 40 to the calculated impedance from the Smith chart to find the net line input impedance. To begin, the 20 load resistor represents a normalized impedance of zl = 0.4, which we mark on the chart (see below). Then, using a compass, draw a circle beginning at zL and progressing clockwise to the positive real axis. The circle traces the locus of zin values for line lengths over the range 0 < l < /4. On the chart, radial lines are drawn at positions corresponding to .025 increments on the WTG scale. The intersections of the lines and the circle give a total of 11 zin values. To these we add normalized impedance of 40/50 = 0.8 to add the effect of the 40 resistors and obtain the normalized impedance at the line input. The magnitudes of these values are then found, and the results are multiplied by 50 . The table below summarizes the results. l/ 0 .025 .050 .075 .100 .125 .150 .175 .200 .225 .250 zinl (to right of 40 ) 0.40 0.41 + j.13 0.43 + j.27 0.48 + j.41 0.56 + j.57 0.68 + j.73 0.90 + j.90 1.20 + j1.05 1.65 + j1.05 2.2 + j.7 2.5 zin = zinl + 0.8 1.20 1.21 + j.13 1.23 + j.27 1.28 + j.41 1.36 + j.57 1.48 + j.73 1.70 + j.90 2.00 + j1.05 2.45 + j1.05 3.0 + j.7 3.3 229 |Zin | = 50|zin | 60 61 63 67 74 83 96 113 134 154 165 13.24. (continued) As a check, the line input input impedance can be found analytically through Zin = 40 + 50 from which 36 cos2 (2l/) + 43.6 sin2 (2l/) |Zin | = 50 25 cos2 (2l/) + 4 sin2 (2l/) 20 cos(2l/) + j 50 sin(2l/) 60 cos(2l/) + j 66 sin(2l/) = 50 50 cos(2l/) + j 20 sin(2l/) 50 cos(2l/) + j 20 sin(2l/) 1/2 This function is plotted below along with the results obtained from the Smith chart. A fairly good comparison is obtained. 230 13.25. A 300-ohm transmission line is short-circuited at z = 0. A voltage maximum, |V |max = 10 V, is found at z = -25 cm, and the minimum voltage, |V |min = 0, is found at z = -50 cm. Use the Smith chart to find ZL (with the short circuit replaced by the load) if the voltage readings are: a) |V |max = 12 V at z = -5 cm, and vertV |min = 5 V: First, we know that the maximum and minimum voltages are spaced by /4. Since this distance is given as 25 cm, we see that = 100 cm = 1 m. Thus the maximum voltage location is 5/100 = 0.05 in front of the load. The standing wave ratio is s = |V |max /|V |min = 12/5 = 2.4. We mark this on the positive real axis of the chart (see next page). The load position is now 0.05 wavelengths toward the load from the |V |max position, or at 0.30 on the WTL scale. A line is drawn from the origin through this point on the chart, as shown. We next set the compass to the distance between the origin and the z = r = 2.4 point on the real axis. We then scribe this same distance along the line drawn through the .30 position. The intersection is the value of zL , which we read as zL = 1.65 + j.97. The actual load impedance is then ZL = 300zL = 495 + j 290 . b) |V |max = 17 V at z = -20 cm, and |V |min = 0. In this case the standing wave ratio is infinite, which puts the starting point on the r point on the chart. The distance of 20 cm corresponds to 20/100 = 0.20 , placing the load position at 0.45 on the WTL scale. A line is drawn from the origin through this location on the chart. An infinite standing wave ratio places us on the outer boundary of the chart, so we read zL = j 0.327 at the 0.45 WTL position. Thus . ZL = j 300(0.327) = j 98 . 231 13.26. A lossless 50 transmission line operates with a velocity that is 3/4c. A load, ZL = 60 + j 30 is located at z = 0. Use the Smith chart to find: a) s: First we find the normalized load impedance, zL = (60 + j 30)/50 = 1.2 + j 0.6, which is then marked on the chart (see below). Drawing a line from the chart center through this point yields its location at 0.328 on the WTL scale. The distance from the origin to the load impedance point is now set on the compass; the standing wave ratio is then found by scribing this distance along the positive real axis, yielding s = 1.76, as shown. Alternately, use the s scale at the bottom of the chart, setting the compass point at the center, and scribing the distance on the scale to the left. b) the distance from the load to the nearest voltage minimum if f = 300 MHz: This distance is found by transforming the load impedance clockwise around the chart until the negative real axis is reached. This distance in wavelengths is just the load position on the WTL scale, since the starting point for this scale is the negative real axis. So the distance is 0.328. The wavelength is = v (3/4)c 3(3 108 ) = 0.75 m = = f 300MHz 4(3 108 ) So the actual distance to the first voltage minimum is dmin = 0.328(0.75) m = 24.6 cm. c) the input impedance if f = 200 MHz and the input is at z = -110cm: The wavelength at this frequency is = (3/4)(3 108 )/(2 108 ) = 1.125 m. The distance to the input in wavelengths is then din = (1.10)/(1.125) = 0.9778. Transforming the load through this distance toward the generator involves revolution once around the chart (0.500) plus the remainder of 0.4778, . which leads to a final position of 0.1498 = 0.150 on the WTG scale, or 0.350 on the WTL scale. A line is drawn between this point and the chart center. Scribing the compass arc length through this line yields the normalized input impedance, read as zin = 1.03 + j 0.56. The actual input impedance is Zin = zin 50 = 51.5 + j 28.0 . 232 13.27. The characteristic admittance (Y0 = 1/Z0 ) of a lossless transmission line is 20 mS. The line is terminated in a load YL = 40 - j 20 mS. Make use of the Smith chart to find: a) s: We first find the normalized load admittance, which is yL = YL /Y0 = 2 - j 1. This is plotted on the Smith chart below. We then set on the compass the distance between yL and the origin. The same distance is then scribed along the positive real axis, and the value of s is read as 2.6. b) Yin if l = 0.15 : First we draw a line from the origin through zL and note its intersection with the WTG scale on the chart outer boundary. We note a reading on that scale of about 0.287 . To this we add 0.15 , obtaining about 0.437 , which we then mark on the chart (0.287 is not the precise value, but I have added 0.15 to that mark to obtain the point shown on the chart that is near to 0.437 . This "eyeballing" method increases the accuracy a little). A line drawn from the 0.437 position on the WTG scale to the origin passes through the input admittance. Using the compass, we scribe the distance found in part a across this line to find yin = 0.56 - j 0.35, or Yin = 20yin = 11 - j 7.0 mS. c) the distance in wavelengths from YL to the nearest voltage maximum: On the admittance chart, the Vmax position is on the negative r axis. This is at the zero position on the WTL scale. The load is at the approximate 0.213 point on the WTL scale, so this distance is the one we want. 233 13.28. The wavelength on a certain lossless line is 10cm. If the normalized input impedance is zin = 1 + j 2, use the Smith chart to determine: a) s: We begin by marking zin on the chart (see below), and setting the compass at its distance from the origin. We then use the compass at that setting to scribe a mark on the positive real axis, noting the value there of s = 5.8. b) zL , if the length of the line is 12 cm: First, use a straight edge to draw a line from the origin through zin , and through the outer scale. We read the input location as slightly more than 0.312 on the WTL scale (this additional distance beyond the .312 mark is not measured, but is instead used to add a similar distance when the impedance is transformed). The line length of 12cm corresponds to 1.2 wavelengths. Thus, to transform to the load, we go counter-clockwise twice around the chart, plus 0.2, finally arriving at (again) slightly more than 0.012 on the WTL scale. A line is drawn to the origin from that position, and the compass (with its previous setting) is scribed through the line. The intersection is the normalized load impedance, which we read as zL = 0.173 - j 0.078. c) xL , if zL = 2 + j xL , where xL > 0. For this, use the compass at its original setting to scribe through the r = 2 circle in the upper half plane. At that point we read xL = 2.62. 234 13.29. A standing wave ratio of 2.5 exists on a lossless 60 line. Probe measurements locate a voltage minimum on the line whose location is marked by a small scratch on the line. When the load is replaced by a short circuit, the minima are 25 cm apart, and one minimum is located at a point 7 cm toward the source from the scratch. Find ZL : We note first that the 25 cm separation between minima imply a wavelength of twice that, or = 50 cm. Suppose that the scratch locates the first voltage minimum. With the short in place, the first minimum occurs at the load, and the second at 25 cm in front of the load. The effect of replacing the short with the load is to move the minimum at 25 cm to a new location 7 cm toward the load, or at 18 cm. This is a possible location for the scratch, which would otherwise occur at multiples of a half-wavelength farther away from that point, toward the generator. Our assumed scratch position will be 18 cm or 18/50 = 0.36 wavelengths from the load. Using the Smith chart (see below) we first draw a line from the origin through the 0.36 point on the wavelengths toward load scale. We set the compass to the length corresponding to the s = r = 2.5 point on the chart, and then scribe this distance through the straight line. We read zL = 0.79 + j 0.825, from which ZL = 47.4 + j 49.5 . As a check, I will do the problem analytically. First, we use 4(18) 1 - 1 = 1.382 rad = 79.2 zmin = -18 cm = - ( + ) = 2 50 Now s-1 2.5 - 1 | L| = = = 0.4286 s+1 2.5 + 1 and so L = 0.4286 1.382. Using this, we find 1+ L = 0.798 + j 0.823 zL = 1- L and thus ZL = zL (60) = 47.8 + j 49.3 . 235 13.30. A 2-wire line, constructed of lossless wire of circular cross-section is gradually flared into a coupling loop that looks like an egg beater. At the point X, indicated by the arrow in Fig. 13.31, a short circuit is placed across the line. A probe is moved along the line and indicates that the first voltage minimum to the left of X is 16cm from X. With the short circuit removed, a voltage minimum is found 5cm to the left of X, and a voltage maximum is located that is 3 times voltage of the minimum. Use the Smith chart to determine: a) f : No Smith chart is needed to find f , since we know that the first voltage minimum in front of a short circuit is one-half wavelength away. Therefore, = 2(16) = 32cm, and (assuming an air-filled line), f = c/ = 3 108 /0.32 = 0.938 GHz. b) s: Again, no Smith chart is needed, since s is the ratio of the maximum to the minimum voltage amplitudes. Since we are given that Vmax = 3Vmin , we find s = 3. c) the normalized input impedance of the egg beater as seen looking the right at point X: Now we need the chart. From the figure below, s = 3 is marked on the positive real axis, which determines the compass radius setting. This point is then transformed, using the compass, to the negative real axis, which corresponds to the location of a voltage minimum. Since the first Vmin is 5cm in front of X, this corresponds to (5/32) = 0.1563 to the left of X. On the chart, we now move this distance from the Vmin location toward the load, using the WTL scale. A line is drawn from the origin through the 0.1563 mark on the WTL scale, and the compass is used to scribe the original radius through this line. The intersection is the normalized input impedance, which is read as zin = 0.86 - j 1.06. 236 13.31. In order to compare the relative sharpness of the maxima and minima of a standing wave, assume a load zL = 4 + j 0 is located at z = 0. Let |V |min = 1 and = 1 m. Determine the width of the a) minimum, where |V | < 1.1: We begin with the general phasor voltage in the line: V (z) = V + (e-jz + ejz ) With zL = 4 + j 0, we recognize the real part as the standing wave ratio. Since the load impedance is real, the reflection coefficient is also real, and so we write =| |= The voltage magnitude is then |V (z)| = V (z)V (z) = V + (e-jz + ejz )(ejz + e-jz ) 2 1/2 1/2 4-1 s-1 = = 0.6 s+1 4+1 = V + 1 + 2 cos(2z) + Note that with cos(2z) = 1, we obtain |V | = V + (1 ) as expected. With s = 4 and with |V |min = 1, we find |V |max = 4. Then with = 0.6, it follows that V + = 2.5. The net expression for |V (z)| is then V (z) = 2.5 1.36 + 1.2 cos(2z) To find the width in z of the voltage minimum, defined as |V | < 1.1, we set |V (z)| = 1.1 and solve for z: We find 1.1 2.5 2 = 1.36 + 1.2 cos(2z) 2z = cos-1 (-0.9726) Thus 2z = 2.904. At this stage, we note the the |V |min point will occur at 2z = . We therefore compute the range, z, over which |V | < 1.1 through the equation: 2( z) = 2( - 2.904) where = 1 m has been used. b) Determine the width of the maximum, where |V | > 4/1.1: We use the same equation for |V (z)|, which in this case reads: 4/1.1 = 2.5 1.36 + 1.2 cos(2z) cos(2z) = 0.6298 Since the maximum corresponds to 2z = 0, we find the range through 2 z = 2 cos-1 (0.6298) z= 0.8896 = 0.142 m = 14.2 cm 2/1 z= - 2.904 = 0.0378 m = 3.8 cm 2/1 237 13.32. A lossless line is operating with Z0 = 40 , f = 20 MHz, and = 7.5 rad/m. With a short circuit replacing the load, a minimum is found at a point on the line marked by a small spot of puce paint. With the load installed, it is found that s = 1.5 and a voltage minimum is located 1m toward the source from the puce dot. a) Find ZL : First, the wavelength is given by = 2/ = 2/7.5 = 0.2667m. The 1m distance is therefore 3.75. With the short installed, the Vmin positions will be at multiples of /2 to the left of the short. Therefore, with the actual load installed, the Vmin position as stated would be 3.75 + n/2, which means that a maximum voltage occurs at the load. This being the case, the normalized load impedance will lie on the positive real axis of the Smith chart, and will be equal to the standing wave ratio. Therefore, ZL = 40(1.5) = 60 . b) What load would produce s = 1.5 with |V |max at the paint spot? With |V |max at the paint spot and with the spot an integer multiple of /2 to the left of the load, |V |max must also occur at the load. The answer is therefore the same as part a, or ZL = 60 . 13.33. In Fig. 13.14, let ZL = 40 - j 10 , Z0 = 50 , f = 800 MHz, and v = c. a) Find the shortest length, d1 , of a short-circuited stub, and the shortest distance d that it may be located from the load to provide a perfect match on the main line to the left of the stub: The Smith chart construction is shown on the next page. First we find zL = (40 - j 10)/50 = 0.8 - j 0.2 and plot it on the chart. Next, we find yL = 1/zL by transforming this point halfway around the chart, where we read yL = 1.17 + j 0.30. This point is to be transformed to a location at which the real part of the normalized admittance is unity. The g = 1 circle is highlighted on the chart; yL transforms to two locations on it: yin1 = 1 - j 0.32 and yin2 = 1 + j 0.32. The stub is connected at either of these two points. The stub input admittance must cancel the imaginary part of the line admittance at that point. If yin2 is chosen, the stub must have input admittance of -j 0.32. This point is marked on the outer circle and occurs at 0.452 on the WTG scale. The length of the stub is found by computing the distance between its input, found above, and the short-circuit position (stub load end), marked as Psc . This distance is d1 = (0.452 - 0.250) = 0.202 . With f = 800 MHz and v = c, the wavelength is = (3 108 )/(8 108 ) = 0.375 m. The distance is thus d1 = (0.202)(0.375) = 0.758 m = 7.6 cm. This is the shortest of the two possible stub lengths, since if we had used yin1 , we would have needed a stub input admittance of +j 0.32, which would have required a longer stub length to realize. The length of the main line between its load and the stub attachment point is found on the chart by measuring the distance between yL and yin2 , in moving clockwise (toward generator). This distance will be d = [0.500 - (0.178 - 0.138)] = 0.46 . The actual length is then d = (0.46)(0.375) = 0.173m = 17.3 cm. 238 13.33b) Repeat for an open-circuited stub: In this case, everything is the same, except for the load-end position of the stub, which now occurs at the Poc point on the chart. To use the shortest possible stub, we need to use yin1 = 1 - j 0.32, requiring ys = +j 0.32. We find the stub length by moving from Poc to the point at which the admittance is j 0.32. This occurs at 0.048 on the WTG scale, which thus determines the required stub length. Now d1 = (0.048)(0.375) = 0.18 m = 1.8 cm. The attachment point is found by transforming yL to yin1 , where the former point is located at 0.178 on the WTG scale, and the latter is at 0.362 on the same scale. The distance is then d = (0.362 - 0.178) = 0.184. The actual length is d = (0.184)(0.375) = 0.069 m = 6.9 cm. 239 13.34. The lossless line shown in Fig. 13.32 is operating with = 100cm. If d1 = 10cm, d = 25cm, and the line is matched to the left of the stub, what is ZL ? For the line to be matched, it is required that the sum of the normalized input admittances of the shorted stub and the main line at the point where the stub is connected be unity. So the input susceptances of the two lines must cancel. To find the stub input susceptance, use the Smith chart to transform the short circuit point 0.1 toward the generator, and read the input value as bs = -1.37 (note that the stub length is one-tenth of a wavelength). The main line input admittance must now be yin = 1 + j 1.37. This line is one-quarter wavelength long, so the normalized load impedance is equal to the normalized input admittance. Thus zL = 1 + j 1.37, so that ZL = 300zL = 300 + j 411 . 240 13.35. A load, ZL = 25 + j 75 , is located at z = 0 on a lossless two-wire line for which Z0 = 50 and v = c. a) If f = 300 MHz, find the shortest distance d (z = -d) at which the input impedance has a real part equal to 1/Z0 and a negative imaginary part: The Smith chart construction is shown below. We begin by calculating zL = (25 + j 75)/50 = 0.5 + j 1.5, which we then locate on the chart. Next, this point is transformed by rotation halfway around the chart to find yL = 1/zL = 0.20 - j 0.60, which is located at 0.088 on the WTL scale. This point is then transformed toward the generator until it intersects the g = 1 circle (shown highlighted) with a negative imaginary part. This occurs at point yin = 1.0 - j 2.23, located at 0.308 on the WTG scale. The total distance between load and input is then d = (0.088 + 0.308) = 0.396. At 300 MHz, and with v = c, the wavelength is = 1 m. Thus the distance is d = 0.396 m = 39.6 cm. b) What value of capacitance C should be connected across the line at that point to provide unity standing wave ratio on the remaining portion of the line? To cancel the input normalized susceptance of -2.23, we need a capacitive normalized susceptance of +2.23. We therefore write C = 2.23 2.23 = 2.4 10-11 F = 24 pF C= Z0 (50)(2 3 108 ) 241 13.36. The two-wire lines shown in Fig. 13.33 are all lossless and have Z0 = 200 . Find d and the shortest possible value for d1 to provide a matched load if = 100cm. In this case, we have a series combination of the loaded line section and the shorted stub, so we use impedances and the Smith chart as an impedance diagram. The requirement for matching is that the total normalized impedance at the junction (consisting of the sum of the input impedances to the stub and main loaded section) is unity. First, we find zL = 100/200 = 0.5 and mark this on the chart (see below). We then transform this point toward the generator until we reach the r = 1 circle. This happens at two possible points, indicated as zin1 = 1 + j.71 and zin2 = 1 - j.71. The stub input impedance must cancel the imaginary part of the loaded section input impedance, or zins = j.71. The shortest stub length that accomplishes this is found by transforming the short circuit point on the chart to the point zins = +j 0.71, which yields a stub length of d1 = .098 = 9.8 cm. The length of the loaded section is then found by transforming zL = 0.5 to the point zin2 = 1 - j.71, so that zins + zin2 = 1, as required. This transformation distance is d = 0.347 = 37.7 cm. 242 13.37. In the transmission line of Fig. 13.17, RL = Z0 = 50 . Determine and plot the voltage at the load resistor and the current in the battery as functions of time by constructing appropriate voltage and current reflection diagrams: Referring to the figure, closing the switch launches a voltage wave whose value is given by Eq. (50): V0 Z0 50 2 + V0 = V0 = V1 = Rg + Z 0 75 3 We note that L = 0, since the load impedance is matched to that of the line. So the voltage wave traverses the line and does not reflect. The voltage reflection diagram would be that shown in Fig. 13.18a, except that no waves are present after time t = l/v. Likewise, the current reflection diagram is that of Fig. 13.19a, except, again, no waves exist after t = l/v. The voltage at the load will be just + V1 = (2/3)V0 for times beyond l/v. The current through the battery is found through + V1 V0 = A Z0 75 This current initiates at t = 0, and continues indefinitely. + I1 = 13.38. Repeat Problem 37, with Z0 = 50 , and RL = Rg = 25 . Carry out the analysis for the time period 0 < t < 8l/v. At the generator end, we have g = -1/3, as before. The difference is at the load end, where L = -1/3, whereas in Problem 37, the load was matched. The initial wave, as in the last problem, is of magnitude V + = (2/3)V0 . Using these values, voltage and current reflection diagrams are constructed, and are shown below: 243 13.38. (continued) From the diagrams, voltage and current plots are constructed. First, the load voltage is found by adding voltages along the right side of the voltage diagram at the indicated times. Second, the current through the battery is found by adding currents along the left side of the current reflection diagram. Both plots are shown below, where currents and voltages are expressed to three significant figures. The steady state values, VL = 0.5V and IB = 0.02A, are expected as t . 13.39. In the transmission line of Fig. 13.17, Z0 = 50 and RL = Rg = 25 . The switch is closed at t = 0 and is opened again at time t = l/4v, thus creating a rectangular voltage pulse in the line. Construct an appropriate voltage reflection diagram for this case and use it to make a plot of the voltage at the load resistor as a function of time for 0 < t < 8l/v (note that the effect of opening the switch is to initiate a second voltage wave, whose value is such that it leaves a net current of zero in its wake): The value of the initial voltage wave, formed by closing the switch, will be V+ = Z0 50 2 V0 = V0 V0 = Rg + Z 0 25 + 50 3 On opening the switch, a second wave, V + , is generated which leaves a net current behind it of zero. This means that V + = -V + = -(2/3)V0 . Note also that when the switch is opened, the reflection coefficient at the generator end of the line becomes unity. The reflection coefficient at the load end is L = (25 - 50)/(25 + 50) = -(1/3). The reflection diagram is now constructed in the usual manner, and is shown on the next page. The path of the second wave as it reflects from either end is shown in dashed lines, and is a replica of the first wave path, displaced later in time by l/(4v).a All values for the second wave after each reflection are equal but of opposite sign to the immediately preceding first wave values. The load voltage as a function of time is found by accumulating voltage values as they are read moving up along the right hand boundary of the chart. The resulting function, plotted just below the reflection diagram, is found to be a sequence of pulses that alternate signs. The pulse amplitudes are calculated as follows: 244 13.39. (continued) l 1 5l <t < : V1 = 1 - V + = 0.44 V0 v 4v 3 13l 1 3l 1 <t < : V2 = - 1- V + = -0.15 V0 v 4v 3 3 21l 5l <t < : V3 = v 4v 1 3 2 1- 3 1 V + = 0.049 V0 3 1 V + = -0.017 V0 3 1 29l 7l <t < : V4 = - v 4v 3 1- 245 13.40. In the charged line of Fig. 13.22, the characteristic impedance is Z0 = 100 , and Rg = 300 . The line is charged to initial voltage V0 = 160 V, and the switch is closed at t = 0. Determine and plot the voltage and current through the resistor for time 0 < t < 8l/v (four round trips). This problem accompanies Example 13.6 as the other special case of the basic charged line problem, in which now Rg > Z0 . On closing the switch, the initial voltage wave is V + = -V0 Z0 100 = -40 V = -160 Rg + Z0 400 Now, with g = 1/2 and L = 1, the voltage and current reflection diagrams are constructed as shown below. Plots of the voltage and current at the resistor are then found by accumulating values from the left sides of the two charts, producing the plots as shown. 246 13.41. In the transmission line of Fig. 13.34, the switch is located midway down the line, and is closed at t = 0. Construct a voltage reflection diagram for this case, where RL = Z0 . Plot the load resistor voltage as a function of time: With the left half of the line charged to V0 , closing the switch initiates (at the switch location) two voltage waves: The first is of value -V0 /2 and propagates toward the left; the second is of value V0 /2 and propagates toward the right. The backward wave reflects at the battery with g = -1. No reflection occurs at the load end, since the load is matched to the line. The reflection diagram and load voltage plot are shown below. The results are summarized as follows: l : VL = 0 2v 3l V0 l <t < : VL = 2v 2v 2 3l : VL = V0 t> 2v 0<t < 247 13.42. A simple frozen wave generator is shown in Fig. 13.35. Both switches are closed simultaneously at t = 0. Construct an appropriate voltage reflection diagram for the case in which RL = Z0 . Determine and plot the load voltage as a function of time: Closing the switches sets up a total of four voltage waves as shown in the diagram below. Note that the first and second waves from the left are of magnitude V0 , since in fact we are superimposing voltage waves from the -V0 and +V0 charged sections acting alone. The reflection diagram is drawn and is used to construct the load voltage with time by accumulating voltages up the right hand vertical axis. 248 CHAPTER 14 14.1. A parallel-plate waveguide is known to have a cutoff wavelength for the m = 1 TE and TM modes of c1 = 0.4 cm. The guide is operated at wavelength = 1 mm. How many modes propagate? The cutoff wavelength for mode m is cm = 2nd/m, where n is the refractive index of the guide interior. For the first mode, we are given c1 = 0.4 0.2 2nd = 0.4 cm d = = cm 1 2n n Now, for mode m to propagate, we require 2nd 0.4 0.4 0.4 = m = =4 m m 0.1 So, accounting for 2 modes (TE and TM) for each value of m, and the single TEM mode, we will have a total of 9 modes. 14.2. A parallel-plate guide is to be constructed for operation in the TEM mode only over the frequency range 0 < f < 3 GHz. The dielectric between plates is to be teflon ( R = 2.1). Determine the maximum allowable plate separation, d: We require that f < fc1 , which, using (7), becomes f < c 3 108 c dmax = = = 3.45 cm 2nd 2nfmax 2 2.1 (3 109 ) 14.3. A lossless parallel-plate waveguide is known to propagate the m = 2 TE and TM modes at frequencies as low as 10GHz. If the plate separation is 1 cm, determine the dielectric constant of the medium between plates: Use fc2 = c 3 1010 = = 1010 n = 3 or nd n(1) R =9 14.4. A d = 1 cm parallel-plate guide is made with glass (n = 1.45) between plates. If the operating frequency is 32 GHz, which modes will propagate? For a propagating mode, we require f > fcm Using (7) and the given values, we write f > 2f nd 2(32 109 )(1.45)(.01) mc m< = = 3.09 2nd c 3 108 The maximum allowed m in this case is thus 3, and the propagating modes will be TM1 , TE1 , TM2 , TE2 , TM3 , and TE3 . 14.5. For the guide of Problem 14.4, and at the 32 GHz frequency, determine the difference between the group delays of the highest order mode (TE or TM) and the TEM mode. Assume a propagation distance of 10 cm: From Problem 14.4, we found mmax = 3. The group velocity of a TE or TM mode for m = 3 is vg3 = c 1- n fc3 f 2 where fc3 = 3(3 1010 ) = 3.1 1010 = 31 GHz 2(1.45)(1) 249 14.5. (continued) Thus vg3 = 3 1010 1.45 1- 31 32 2 = 5.13 109 cm/s For the TEM mode (assuming no material dispersion) vg,T EM = c/n = 3 1010 /1.45 = 2.07 1010 cm/s. The group delay difference is now tg = z 1 1 - vg3 vg,T EM = 10 1 1 - 9 5.13 10 2.07 1010 = 1.5 ns 14.6. The cutoff frequency of the m = 1 TE and TM modes in a parallel-plate guide is known to be fc1 = 7.5 GHz. The guide is used at wavelength = 1.5 cm. Find the group velocity of the m = 2 TE and TM modes. First we know that fc2 = 2fc1 = 15 GHz. Then f = c/ = 3 108 /.015 = 20 GHz. Now, using (23), vg2 = c 1- n fc2 f 2 = c 1- n 15 20 2 = 2 108 /n m/s n was not specified in the problem. 14.7. A parallel-plate guide is partially filled with two lossless dielectrics (Fig. 14.23) where R1 = 4.0, R2 = 2.1, and d = 1 cm. At a certain frequency, it is found that the TM1 mode propagates through the guide without suffering any reflective loss at the dielectric interface. a) Find this frequency: The ray angle is such that the wave is incident on the interface at Brewster's angle. In this case 2.1 B = tan-1 = 35.9 4.0 The ray angle is thus = 90 - 35.9 = 54.1 . The cutoff frequency for the m = 1 mode is fc1 = 2d c R1 = 3 1010 = 7.5 GHz 2(1)(2) The frequency is thus f = fc1 / cos = 7.5/ cos(54.1 ) = 12.8 GHz. b) Is the guide operating at a single TM mode at the frequency found in part a? The cutoff frequency for the next higher mode, TM2 is fc2 = 2fc1 = 15 GHz. The 12.8 GHz operating frequency is below this, so TM2 will not propagate. So the answer is yes. 14.8. In the guide of Problem 14.7, it is found that m = 1 modes propagating from left to right totally reflect at the interface, so that no power is transmitted into the region of dielectric constant R2 . a) Determine the range of frequencies over which this will occur: For total reflection, the ray angle measured from the normal to the interface must be greater than or equal to the critical angle, c , where sin c = ( R2 / R1 )1/2 . The minimum mode ray angle is then 1 min = 90 - c . Now, using (5), we write 90 - c = cos-1 kmin d = cos-1 c = cos-1 c 4dfmin 2fmin d 4 250 14.8a. (continued) Now cos(90 - c ) = sin c = R2 R1 = Therefore fmin = c/(2 2.1d) = (3 108 )/(2 2.1(.01)) = 10.35 GHz. The frequency range is thus f > 10.35 GHz. b) Does your part a answer in any way relate to the cutoff frequency for m = 1 modes in any region? We note that fmin = c/(2 2.1d) = fc1 in guide 2. To summarize, as frequency is lowered, the ray angle in guide 1 decreases, which leads to the incident angle at the interface increasing to eventually reach and surpass the critical angle. At the critical angle, the refracted angle in guide 2 is 90 , which corresponds to a zero degree ray angle in that guide. This defines the cutoff condition in guide 2. So it would make sense that fmin = fc1 (guide 2). 14.9. A rectangular waveguide has dimensions a = 6 cm and b = 4 cm. a) Over what range of frequencies will the guide operate single mode? The cutoff frequency for mode mp is, using Eq. (54): c p 2 m 2 fc,mn = + 2n a b where n is the refractive index of the guide interior. We require that the frequency lie between the cutoff frequencies of the T E10 and T E01 modes. These will be: fc10 = fc01 = 3 108 2.5 109 c = = 2na 2n(.06) n 3 108 3.75 109 c = = 2nb 2n(.04) n c 4dfmin Thus, the range of frequencies over which single mode operation will occur is 3.75 2.5 GHz < f < GHz n n b) Over what frequency range will the guide support both T E10 and T E01 modes and no others? We note first that f must be greater than fc01 to support both modes, but must be less than the cutoff frequency for the next higher order mode. This will be fc11 , given by fc11 = c 2n 1 .06 2 + 1 .04 2 = 4.5 109 30c = 2n n The allowed frequency range is then 3.75 4.5 GHz < f < GHz n n 251 14.10. Two rectangular waveguides are joined end-to-end. The guides have identical dimensions, where a = 2b. One guide is air-filled; the other is filled with a lossless dielectric characterized by R . a) Determine the maximum allowable value of R such that single mode operation can be simultaneously ensured in both guides at some frequency: Since a = 2b, the cutoff frequency for any mode in either guide is written using (54): fcmp = mc 4nb 2 + pc 2nb 2 where n = 1 in guide 1 and n = R in guide 2. We see that, with a = 2b, the next modes (having the next higher cutoff frequency) above TE10 with be TE20 and TE01 . We also see that in general, fcmp (guide 2) < fcmp (guide 1). To assure single mode operation in both guides, the operating frequency must be above cutoff for TE10 in both guides, and below cutoff for the next mode in both guides. The allowed frequency range is therefore fc10 (guide 1) < f < fc20 (guide 2). This leads to c/(2a) < f < c/(a R ). For this range to be viable, it is required that R < 4. b) Write an expression for the frequency range over which single mode operation will occur in both guides; your answer should be in terms of R , guide dimensions as needed, and other known constants: This was already found in part a: c <f < 2a where < 4. c R a R 14.11. An air-filled rectangular waveguide is to be constructed for single-mode operation at 15 GHz. Specify the guide dimensions, a and b, such that the design frequency is 10/while being 10% lower than the cutoff frequency for the next higher-order mode: For an air-filled guide, we have fc,mp = mc 2a 2 + pc 2b 2 For TE10 we have fc10 = c/2a, while for the next mode (TE01 ), fc01 = c/2b. Our requirements state that f = 1.1fc10 = 0.9fc01 . So fc10 = 15/1.1 = 13.6 GHz and fc01 = 15/0.9 = 16.7 GHz. The guide dimensions will be a= 3 1010 c 3 1010 c = = = 1.1 cm and b = = 0.90 cm 2fc10 2(13.6 109 ) 2fc01 2(16.7 109 ) 14.12. Using the relation Pav = (1/2)Re{Es Hs }, and Eqs. (44) through (46), show that the average power density in the TE10 mode in a rectangular waveguide is given by Pav = 10 2 2 E sin (10 x) az 2 0 W/m2 (note that the sin term is erroneously to the first power in the original problem statement). Inspecting (44) through (46), we see that (46) includes a factor of j , and so would lead to an imaginary part of the 252 total power when the cross product with Ey is taken. Therefore, the real power in this case is found through the cross product of (44) with the complex conjugate of (45), or 10 2 2 1 Pav = Re Eys Hxs = E sin (10 x) az W/m2 2 2 0 14.13. Integrate the result of Problem 14.12 over the guide cross-section 0 < x < a, 0 < y < b, to show that the power in Watts transmitted down the guide is given as 10 ab 2 ab 2 P = E0 = E sin 10 W 4 4 0 where = / (note misprint in problem statement), and 10 is the wave angle associated with the TE10 mode. Interpret. First, the integration: P = 0 b Next, from (20), we have 10 10 ab 2 10 2 2 E0 sin (10 x) az az dx dy = E 4 0 0 2 = sin 10 , which, on substitution, leads to P = a ab 2 E0 sin 10 W with = 4 The sin 10 dependence demonstrates the principle of group velocity as energy velocity (or power). This was considered in the discussion leading to Eq. (23). 14.14. Show that the group dispersion parameter, d 2 /d2 , for given mode in a parallel-plate or rectangular waveguide is given by c 2 -3/2 n c 2 d 2 1- =- d2 c where c is the radian cutoff frequency for the mode in question (note that the first derivative form was already found, resulting in Eq. (23)). First, taking the reciprocal of (23), we find n c 2 -1/2 d = 1- d c Taking the derivative of this equation with respect to leads to d 2 1 n - = 2 d c 2 c 1- 2 -3/2 2 2c 3 n c =- c 2 c 1- 2 -3/2 14.15. Consider a transform-limited pulse of center frequency f = 10 GHz and of full-width 2T = 1.0 ns. The pulse propagates in a lossless single mode rectangular guide which is air-filled and in which the 10 GHz operating frequency is 1.1 times the cutoff frequency of the T E10 mode. Using the result of Problem 14.14, determine the length of the guide over which the pulse broadens to twice its initial width: The broadened pulse will have width given by T = T 2 + ( )2 , where = 2 L/T for a transform limited pulse (assumed gaussian). 2 is the Problem 14.14 result evaluated at the operating frequency, or d 2 1 | =- 2 = 2 =10 GHz 10 )(3 108 ) d (2 10 = 6.1 10-19 s2 /m = 0.61 ns2 /m Now = 0.61L/0.5 = 1.2L ns. For the pulse width to double, we have T = 1 ns, and (.05)2 + (1.2L)2 = 1 L = 0.72 m = 72 cm 253 1 1.1 2 1- 1 1.1 2 -3/2 14.15. (continued) What simple step can be taken to reduce the amount of pulse broadening in this guide, while maintaining the same initial pulse width? It can be seen that 2 can be reduced by increasing the operating frequency relative to the cutoff frequency; i.e., operate as far above cutoff as possible, without allowing the next higher-order modes to propagate. 14.16. A symmetric dielectric slab waveguide has a slab thickness d = 10 m, with n1 = 1.48 and n2 = 1.45. If the operating wavelength is = 1.3 m, what modes will propagate? We use the condition expressed through (77): k0 d n2 - n2 (m - 1) . Since k0 = 2/, the condition becomes 1 2 2d Therefore, mmax total). 2(10) (1.48)2 - (1.45)2 = 4.56 m - 1 1.3 = 5, and we have TE and TM modes for which m = 1, 2, 3, 4, 5 propagating (ten n2 - n2 (m - 1) 1 2 14.17. A symmetric slab waveguide is known to support only a single pair of TE and TM modes at wavelength = 1.55 m. If the slab thickness is 5 m, what is the maximum value of n1 if n2 = 3.3 (assume 3.30)? Using (78) we have 2d n2 - n2 < n1 < 1 2 + n2 = 2 2d 1.55 + (3.30)2 = 3.32 2(5) 14.18. n1 = 1.50, n2 = 1.45, and d = 10 m in a symmetric slab waveguide (note that the index values were reversed in the original problem statement). a) What is the phase velocity of the m = 1 TE or TM mode at cutoff? At cutoff, the mode propagates in the slab at the critical angle, which means that the phase velocity will be equal to that of a plane wave in the upper or lower media of index n2 . Phase velocity will therefore be vp (cutoff) = c/n2 = 3 108 /1.45 = 2.07 108 m/s. b) What is the phase velocity of the m = 2 TE or TM modes at cutoff? The reasoning of part a applies to all modes, so the answer is the same, or 2.07 108 m/s. 14.19. An asymmetric slab waveguide is shown in Fig. 14.24. In this case, the regions above and below the slab have unequal refractive indices, where n1 > n3 > n2 (note error in problem statement). a) Write, in terms of the appropriate indices, an expression for the minimum possible wave angle, 1 , that a guided mode may have: The wave angle must be equal to or greater than the critical angle of total reflection at both interfaces. The minimum wave angle is thus determined by the greater of the two critical angles. Since n3 > n2 , we find min = c,13 = sin-1 (n3 /n1 ). b) Write an expression for the maximum phase velocity a guided mode may have in this structure, using given or known parameters: We have vp,max = /min , where min = n1 k0 sin 1,min = n1 k0 n3 /n1 = n3 k0 . Thus vp,max = /(n3 k0 ) = c/n3 . 14.20. A step index optical fiber is known to be single mode at wavelengths > 1.2 m. Another fiber is to be fabricated from the same materials, but is to be single mode at wavelengths > 0.63 m. By what percentage must the core radius of the new fiber differ from the old one, and should it be larger or smaller? We use the cutoff condition, given by (80): 2a > n2 - n2 2 2.405 1 254 14.20. (continued) With reduced, the core radius, a, must also be reduced by the same fraction. Therefore, the percentage reduction required in the core radius will be %= 1.2 - .63 100 = 47.5% 1.2 14.21. A short dipole carrying current I0 cos t in the az direction is located at the origin in free space. a) If = 1 rad/m, r = 2 m, = 45 , = 0, and t = 0, give a unit vector in rectangular components that shows the instantaneous direction of E: In spherical coordinates, the components of E are given by (82) and (83): Er = E = I0 d cos e-j 2 r/ 2 1 + 2 r j 2r 3 1 2 I0 d sin e-j 2r/ j + 2+ 4 r r j 2r 3 Since we want a unit vector at t = 0, we need only the relative amplitudes of the two components, but we need the absolute phases. Since = 45 , sin = cos = 1/ 2. Also, with = 1 = 2/, it follows that = 2 m. The above two equations can be simplified by these substitutions, while dropping all amplitude terms that are common to both. Obtain Ar = A = Now with r = 2 m, we obtain Ar = A = j 1 -j 2 1 1 -j e = (1.12)e-j 26.6 e-j 2 4 8 4 1 1 1 1 + -j e-j 2 = (0.90)ej 56.3 e-j 2 4 8 16 4 1 1 + 3 2 r jr e-j r e-j r 1 1 1 1 j + 2+ 3 2 r r jr The total vector is now A = Ar ar + A a . We can normalize the vector by first finding the magnitude: 1 (1.12)2 + (0.90)2 = 0.359 |A| = A A = 4 Dividing the field vector by this magnitude and converting 2 rad to 114.6 , we write the normalized vector as AN s = 0.780e-j 141.2 ar + 0.627e-58.3 a In real instantaneous form, this becomes AN (t) = Re AN s ej t = 0.780 cos(t - 141.2 )ar + 0.627 cos(t - 58.3 )a We evaluate this at t = 0 to find AN (0) = 0.780 cos(141.2 )ar + 0.627 cos(58.3 )a = -0.608ar + 0.330a 255 14.21a. (continued) Dividing by the magnitude, (0.608)2 + (0.330)2 = 0.692, we obtain the unit vector at t = 0: aN (0) = -0.879ar + 0.477a . We next convert this to cartesian components: 1 aN x = aN (0) ax = -0.879 sin cos + 0.477 cos cos = (-0.879 + 0.477) = -0.284 2 aNy = aN (0) ay = -0.879 sin sin + 0.477 cos sin = 0 since = 0 1 aN z = aN (0) az = -0.879 cos - 0.477 sin = (-0.879 - 0.477) = -0.959 2 The final result is then aN (0) = -0.284ax - 0.959az b) What fraction of the total average power is radiated in the belt, 80 < < 100 ? We use the far-zone phasor fields, (84) and (85), and first find the average power density: Pavg = I 2d 2 1 Re[Es Hs ] = 0 2 2 sin2 W/m2 2 8 r We integrate this over the given belt, an at radius r: Pbelt = 2 0 100 80 2 2 I0 d 2 2 2 I0 d 2 sin r sin d d = 82 r 2 42 100 80 sin3 d Evaluating the integral, we find Pbelt = 2 I0 d 2 1 - cos sin2 + 2 2 4 3 100 = (0.344) 80 2 I0 d 2 42 The total power is found by performing the same integral over , where 0 < < 180 . Doing this, it is found that I 2 d 2 Ptot = (1.333) 0 2 4 The fraction of the total power in the belt is then f = 0.344/1.333 = 0.258. 14.22. Prepare a curve, r vs. in polar coordinates, showing the locus in the = 0 plane where: a) the radiation field |E s | is one-half of its value at r = 104 m, = /2: Assuming the far field approximation, we use (84) to set up the equation: |E s | = 1 I0 d I0 d sin = r = 2 104 sin 2r 2 2 104 b) the average radiated power density, Pr,av , is one-half of its value at r = 104 m, = /2. To find the average power, we use (84) and (85) in Pr,av = 2 2 1 1 I0 d 2 2 1 1 I0 d 2 sin = Re{E s Hs } = r = 2 104 sin 2r 2 2 (108 ) 2 2 4 2 2 4 256 14.22. (continued) The polar plots for field (r = 2 104 sin ) and power (r = below. Both are circles. 2 104 sin ) are shown 14.23. Two short antennas at the origin in free space carry identical currents of 5 cos t A, one in the az direction, one in the ay direction. Let = 2 m and d = 0.1 m. Find Es at the distant point: a) (x = 0, y = 1000, z = 0): This point lies along the axial direction of the ay antenna, so its contribution to the field will be zero. This leaves the az antenna, and since = 90 , only the E s component will be present (as (82) and (83) show). Since we are in the far zone, (84) applies. We use = 90 , d = 0.1, = 2, = 0 = 120 , and r = 1000 to write: Es = E s a = j I0 d 5(0.1)(120) -j 1000 sin e-j 2 r/ a = j e a 2r 4(1000) = j (1.5 10-2 )e-j 1000 a = -j (1.5 10-2 )e-j 1000 az V/m b) (0, 0, 1000): Along the z axis, only the ay antenna will contribute to the field. Since the distance is the same, we can apply the part a result, modified such the the field direction is in -ay : Es = -j (1.5 10-2 )e-j 1000 ay V/m c) (1000, 0, 0): Here, both antennas will contribute. Applying the results of parts a and b, we find Es = -j (1.5 10-2 )(ay + az ). d) Find E at (1000, 0, 0) at t = 0: This is found through E(t) = Re Es ej t = (1.5 10-2 ) sin(t - 1000)(ay + az ) Evaluating at t = 0, we find E(0) = (1.5 10-2 )[- sin(1000)](ay + az ) = -(1.24 10-2 )(ay + az ) V/m. e) Find |E| at (1000, 0, 0) at t = 0: Taking the magnitude of the part d result, we find |E| = 1.75 10-2 V/m. 257 14.24. A short current element has d = 0.03. Calculate the radiation resistance for each of the following current distributions: a) uniform: In this case, (86) applies directly and we find Rrad = 80 2 d 2 = 80 2 (.03)2 = 0.711 b) linear, I (z) = I0 (0.5d - |z|)/0.5d: Here, the average current is 0.5I0 , and so the average power drops by a factor of 0.25. The radiation resistance therefore is down to one-fourth the value found in part a, or Rrad = (0.25)(0.711) = 0.178 . c) step, I0 for 0 < |z| < 0.25d and 0.5I0 for 0.25d < |z| < 0.5d: In this case the average current on the wire is 0.75I0 . The radiated power (and radiation resistance) are down to a factor of (0.75)2 times their values for a uniform current, and so Rrad = (0.75)2 (0.711) = 0.400 . 14.25. A dipole antenna in free space has a linear current distribution. If the length is 0.02, what value of I0 is required to: a) provide a radiation-field amplitude of 100 mV/m at a distance of one mile, at = 90 : With a linear current distribution, the peak current, I0 , occurs at the center of the dipole; current decreases linearly to zero at the two ends. The average current is thus I0 /2, and we use Eq. (84) to write: |E | = I0 d0 I0 (0.02)(120) sin(90 ) = = 0.1 I0 = 85.4 A 4r (4)(5280)(12)(0.0254) b) radiate a total power of 1 watt? We use Pavg = 1 4 1 2 I Rrad 2 0 where the radiation resistance is given by Eq. (86), and where the factor of 1/4 arises from the 2 average current of I0 /2: We obtain Pavg = 10 2 I0 (0.02)2 = 1 I0 = 5.03 A. 14.26. A monopole antenna in free space, extending vertically over a perfectly conducting plane, has a linear current distribution. If the length of the antenna is 0.01, what value of I0 is required to a) provide a radiation field amplitude of 100 mV/m at a distance of 1 mi, at = 90 : The image antenna below the plane provides a radiation pattern that is identical to a dipole antenna of length 0.02. The radiation field is thus given by (84) in free space, where = 90 , and with an additional factor of 1/2 included to account for the linear current distribution: |E | = 1 I0 d0 4r|E | 4(5289)(12 .0254)(100 10-3 ) = I0 = = 85.4 A 2 2r (d/)0 (.02)(377) b) radiate a total power of 1W: For the monopole over the conducting plane, power is radiated only over the upper half-space. This reduces the radiation resistance of the equivalent dipole antenna by a factor of one-half. Additionally, the linear current distribution reduces the radiation resistance of a dipole having uniform current by a factor of one-fourth. Therefore, Rrad is one-eighth the value obtained from (86), or Rrad = 10 2 (d/)2 . The current magnitude is now 1/2 2 2Pav 1/2 2(1) I0 = = = = 7.1 A 2 (d/)2 Rrad 10 10 (.02) 258 14.27. The radiation field of a certain short vertical current element is Es = (20/r) sin e-j 10r V/m if it is located at the origin in free space. a) Find E s at P (r = 100, = 90 , = 30 ): Substituting these values into the given formula, find E s = 20 sin(90 )e-j 10(100) = 0.2e-j 1000 V/m 100 b) Find E s at P if the vertical element is located at A(0.1, 90 , 90 ): This places the element on the y axis at y = 0.1. As a result of moving the antenna from the origin to y = 0.1, the change in distance to point P is negligible when considering the change in field amplitude, but is not when considering the change in phase. Consider lines drawn from the origin to P and from y = 0.1 to P . These lines can be considered essentially parallel, and so the difference in their lengths is . l = 0.1 sin(30 ), with the line from y = 0.1 being shorter by this amount. The construction and arguments are similar to those used in the discussion of the electric dipole in Sec. 4.7. The electric field is now the result of part a, modified by including a shorter distance, r, in the phase term only. We show this as an additional phase factor: E s = 0.2e-j 1000 ej 10(0.1 sin 30 = 0.2e-j 1000 ej 0.5 V/m c) Find E s at P if identical elements are located at A(0.1, 90 , 90 ) and B(0.1, 90 , 270 ): The original element of part b is still in place, but a new one has been added at y = -0.1. Again, constructing a line between B and P , we find, using the same arguments as in part b, that the length of this line is approximately 0.1 sin(30 ) longer than the distance from the origin to P . The part b result is thus modified to include the contribution from the second element, whose field will add to that of the first: E s = 0.2e-j 1000 ej 0.5 + e-j 0.5 = 0.2e-j 1000 2 cos(0.5) = 0 The two fields are out of phase at P under the approximations we have used. 259 ...
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Solutions%20Of%20Engineering%20Electromagnetics%207th%20ed-Hayt%20(2002)

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