sadiku_chap7_probs

sadiku_chap7_probs - 310 ; : L‘PRLOB‘LLEMS CHAPTER '7...

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Unformatted text preview: 310 ; : L‘PRLOB‘LLEMS CHAPTER '7 MAGNETDSTATIC FIELDS BEETIDN 7.2 BIDT*‘SAVARTIS LAW 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 10A Figure 7.27 For Problem 7.3. (a) State Biot—Savart’s law. (b) The y- and z—axes, respectively, carry filamentary currents 10 A along ay and 20 A along ~31. Find H at (—3, 4, 5). An infinitely long conducting filament is placed along the x-axis and carries current 10 mA in the ax direction. Find H at (—2, 3, 4). Two infinitely long filaments are placed parallel to the x-axis as shown in Figure 7.27. (a) Find H at the origin. (b) Determine H at (— l, 2, 2). A current element I d1 = 43x A - m is located at the origin. Determine its contribution to the magnetic field intensity at (a) (l, 0, 0), (b) (0, 1,0), (0) (0, 0, l), (d) (1, l, 1). A conducting filament carries current I from point A(0, 0, a) to point B(0, 0, b). Show that at point P(x, y, 0), I ‘ Mm H b a 2 2 2 2 2 2 ad’ Vx +y +19 Vx+y+a Consider AB in Figure 7.28 as part of an electric circuit. Find H at the origin due to AB. Repeat Problem 7.6 for the conductor AB in Figure 7.29. Line x = 0, y = 0, 0 S 2 S 10 m carries current 2A along 32. Calculate H at points (a) (5, 0,0) (C) (5,15, 0) (b) (5, 5, 0) (d) (5, —15,0) y 1 A 6A B x 10A 0 1 Figure 7.28 For Problem 7.6. P R D B L E M s 3 1 1 y y 2 A 4 A 5 A B x . , x O 1 5 A Figure 7.29 For Problem 7.7. Figure 7.30 Current filament for Problem 7.10. *7.9 (a) Find H at (0, 0, 5) due to side 2 of the triangular loop in Figure 7.6(a). (b) Find H at (0, 0, 5) due to the entire loop. 7.10 An infinitely long conductor is bent into an L shape as shown in Figure 7.30. If a direct current of 5 A flows in the current, find the magnetic field intensity at (a) (2, 2, 0), (b) (0, -2, 0), and (C) (0, 0, 2)~ 7.11 Find H at the center C of an equilateral triangular loop of side 4 m carrying 5 A of current as in Figure 7.31. 7.12 A rectangular loop carrying 10 A of current is placed on z = 0 plane as shown in Fig- ure 7.32. Evaluate H at (a) (2, 2, 0) (c) (4, 8, 0) (b) (4, 2, 0) (d) (0, 0, 2) 7.13 A square conducting loop of side 2a lies in the 2: = 0 plane and carries a current I in the counterclockwise direction. Show that at the center of the loop V21 H = a 7m Z Z _ —2 0 2 O 4 8 Figure 7.31 Equilateral triangular loop for Figure 7.32 Rectangular loop of Problem 7.11. Problem 7.12. CHAPTER '7 MAGNETDSTATIB FIELDS 312 y 10A Figure 7.33 Filamentary loop of Problem 7.16; not drawn to scale. 7.14 A square conducting loop 3 cm on each side carries a current of 10 A. Calculate the mag- netic field intensity at the center of the loop. *7.15 (a) A filamentary loop carrying current I is bent to assume the shape of a regular polygon of n sides. Show that at the center of the polygon nI . 7r H = —- sm — 27rr n where r is the radius of the circle circumscribed by the polygon. (b) Apply this for the cases of n = 3 and n = 4 and see if your results agree with those for the triangular loop of Problem 7.11 and the square loop of Problem 7.13, respectively. (c) As 11 becomes large, show that the result of part (a) becomes that of the circular loop of Example 7.3. 7.16 For the filamentary loop shown in Figure 7.33, find the magnetic field strength at 0. 7.17 Two identical current loops have their centers at (0, 0, 0) and (0, 0, 4) and their axes the same as the z—axis (so that the “Helmholtz coil” is formed). If each loop has a radius of 2 m and carries a current of 5 A in 305, calculate H at (a) (0, 0, 0) (b) (0, 0, 2) 7.18 A solenoid 3 cm in length carries a current of 400 mA. If the solenoid is to produce a mag- netic flux density of 5 me/mz, how many turns of wire are needed? 7.19 A solenoid of radius 4 mm and length 2 cm has 150 turns/m and carries a current Of 500 mA. Find (a) at the center, (b) at the ends of the solenoid. 7.20 Plane x = 10 carries a current of 100 mA/m along az, while line x = 1, y = —2 carrieS a filamentary current of 207r mA along 32. Determine H at (4, 3, 2). 7.21 A wire bent as shown in Figure 7.34 carries a current of 4 A. Find H at (0, 0, 2) due to (a) the segment at —4 < x < — 1, (b) the segment at 1 < x < 4, (c) the semicircular portion- F’RDBLEMS 313 Figure 7.34 For Problem 7.21. SEBTIDN 7.3 AMPERE’E CIRCUIT LAW 7.22 (a) State Ampere’s circuit law. (b) A hollow conducting cylinder has inner radius a and outer radius 19 and carries current I along the positive z—direction. Find H everywhere. 7.23 A current sheet K1 = Sax A/m flows on y = 10, while K2 = —lOax A/m flows on y = -—4. Find H at the origin. 7.24 The 2 = 0 plane carries current K = 103x A/m, while current filament situated at y = 0, z = 6 carries current I along ax. Find I such that H(O, 0, 3) = 0. 7.25 (a) An infinitely long solid conductor of radius a is placed along the z—axis. If the con— ductor carries current I in the +z direction, show that I H = p 395 21rd2 within the conductor. Find the corresponding current density. (b) HI = 3 A and a = 2 cm in part (a), find H at (0, 1 cm, 0) and (0, 4 cm, 0). 7.26 If H = yax — xay A/m on plane z = 0, (a) determine the current density and (b) verify Ampere’s law by taking the circulation of H around the edge of the rectangle z=0,0<x<3,-1<y<4. 7.27 Let H = ko<£>a¢, p < a, where k0 is a constant. (a) Find J for p < a. (b) Find H for a p > a. 7.28 An infinitely long filamentary wire carries a current of 2 A along the z-axis in the +z—direction. Calculate the following: (a) B at (—3, 4, 7) (b) The flux through the square loop described by 2 S p S 6, 0 S z S 4. ¢ = 90° 314 CHAPTER l7 MAGNETDSTATIE FIELDS Figure 7.35 Electric motor pole of Problem 7.29. Pole Armature 7.29 The electric motor shown in Figure 7.35 has field 106 H = Tsin 2(1) ap A/m Calculate the flux per pole passing through the air gap if the axial length of the pole is 20 cm. 7.30 Consider the two-wire transmission line whose cross section is illustrated in Figure 7.36. Each wire is of radius 2 cm, and the wires are separated 10 cm. The wire centered at (0, 0) carries a current of S A while the other centered at (10 cm, 0) carries the return current. Find H at (a) (5 cm, 0) (b) (10 cm, 5 cm) BEETIDN “7.5 MAGNETIC F‘Lux DENSITY 7.31 Determine the magnetic flux through a rectangular loop (a X 19) due to an infinitely long conductor carrying current I as shown in Figure 7.37. The loop and the straight conductors are separated by distance d. Figure 7.36 Two-wire line of Problem 7.30. 4cm PROBLEMS 315 Brass ring Figure 7.38 Cross section of a brass ring enclosing a long Figure 7.37 For Problem 731 straight wire; for Problem 7.34. 7.32 Let B = 4 cos<-:—y)e_3z ax Wb/mz. Calculate the total magnetic flux crossing the surface x=0,0<y<1,z>0. 7.33 In free space, the magnetic flux density is B = yzax + zzay + xzaz Wb/m2 (a) Show that B is a magnetic field (b) Find the magnetic flux through x = l, 0 < y < l, l < z < 4. (c) Calculate J. *7.34 A brass ring with triangular cross section encircles a very long straight wire concentrically as in Figure 7.38. If the wire carries a current I, show that the total number of magnetic flux lines in the ring is +b [b—alna ] Calculate ‘Pifa = 30 cm, 19 = 10cm, h = 5 cm, and] = 10A. SECTIDN 7.6 MAXWELL’S EQUATIDNE 7.35 Consider the following arbitrary fields. Find out which of them can possibly represent an electrostatic or magnetostatic field in free space. (a) A = y cos axax + (y + e"‘)aZ 20 (b) B = 721‘, (c) C = r2 sin63¢ 316 CHAPTER '7 MAGNETDSTATIB FIELDS 7.36 Reconsider Problem 7.35 for the following fields. (a) D = yzzax + 2(x + 1)yzay — (x + 1)z2az + 1 sin (Z ) cos gb a9 + (is p p (b) E = az 1 (c) F=—2(20030a,+ sindag) r SECTION 7.7 MAGNETIC] SGALAR AND VEBTUR PDTENTIALE 7.37 Find the magnetic flux density B for each of these vector magnetic potentials. (a) A = e” sin yax + (1 + cos y)ay % bA= a () fi+1z r6 '0 (c) A: CO: ar—I— sm a9 r I” 7.38 For a current distribution in free space, A = (2x2); + yz)ax + (xy2 — xz3)ay — (6xyz - 2362y2)az Wb/m (a) Calculate B. (b) Find the magnetic flux through a loop described by x = 1, 0 < y, z < 2. (c) ShowthatV-A = OandV-B = 0. 7.39 The magnetic vector potential of a current distribution in free space is given by A = 156” sin (15 az Wb/m Find H at (3, 7r/4, — 10). Calculate the flux through p = 5, O S (j) S 7r/2, 0 S z S 10. 7.40 An infinitely long conductor of radius a carries a uniform current with J = JO az. Show that the magnetic vector potential for p < a is 1 A = _Z MoJopzaz 7.41 The current sheet y = 0 has a uniform current density K = ko ax A/m. Determine the vector potential. 7.42 Find the B- field corresponding to the magnetic vector potential , 7rx 7ry A = s1n-—-cos—az 2 2 7.43 The magnetic vector potential of two parallel infinite straight current filaments in fr65 _ space carrying equal current I in the opposite direction is M d-p A=—l 27l'n p az 7.44 7.45 7.46 *7.47 7.48 7.49 7.50 7.51 PROBLEMS 317 where d is the separation distance between the filaments (with one filament placed along the z—axis). Find the corresponding magnetic flux density B. The magnetic field intensity in a certain conducting medium is H = xyzax + xzzay - yzzaz A/m (a) Calculate the current density at point P(2, - 1, 3). a (b) What is p” a: at P? Find the current density J due to 0 2 a2 Wb/m in free space. Prove that the magnetic scalar potential at (0, O, z) due to a circular loop of radius a shown in Figure 7.8(a) is I z Vm = — 1— ——~———— 2 I: [Z2 + a2]1/2] A coaxial transmission line is constructed such that the radius of the inner conductor is a and the outer conductor has radii 3a and 4a. Find the vector magnetic potential within the outer conductor. Assume AZ = 0 for p = 3a. The z-axis carries a filamentary current 12 A along az. Calculate Vm at (4, 30°, —2) if V," = 0 at (10, 60°, 7). Plane z = * 2 carries a current of 50:1y A/m. If Vm = 0 at the origin, find Vm at (a) (—2, 0, 5) (b) (10,3,1) Prove in cylindrical coordinates that (a)V><(VV)=0 (b) V-(VXA)=0 HR = r — r’ andR = IR|,show that R _—1l3— where V and V’ are del operators with respect to (x, y, z) and (x’, y’, 2’), respectively. ...
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This note was uploaded on 02/04/2012 for the course ECE 311 taught by Professor Peroulis during the Fall '08 term at Purdue.

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sadiku_chap7_probs - 310 ; : L‘PRLOB‘LLEMS CHAPTER '7...

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