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Electric CHAPTER
23
The Field II: Continuous Charge Distributions
1* A uniform line charge of linear charge density = 3.5 nC/m extends from x = 0 to x = 5 m. (a) What is the total charge? Find the electric field on the x axis at (b) x = 6 m, (c) x = 9 m, and (d) x = 250 m. (e) Find the field at x = 250 m, using the approximation that the charge is a point charge at the origin, and compare your result with that for the exact calculation in part (d). 9 (a) Q = L Q = (3.510 5) C = 17.5 nC (b), (c), (d) Ex(x0) = kQ/[x0(x0 L)], Equ. 235 Ex(6) = 26.2 N/C; Ex(9) = 4.37 N/C; 3 Ex(250) = 2.5710 N/C 2 3 Ex(250) = 2.5210 N/C, within 2% of (d) (e) Ex kQ/x 2 Two infinite vertical planes of charge are parallel to each other and are separated by a distance d = 4 m. Find the electric field to the left of the planes, to the right of the planes, and between the planes (a) when each 2 plane has a uniform surface charge density = +3 C/m and (b) when the left plane has a uniform surface 2 2 charge density = +3 C/m and that of the right plane is = 3 C/m . Draw the electric field lines for each case. (a) E = 4k = 3.3910 N/C The field pattern is shown in the adjacent figure. The field between the plates is zero.
5
(b) Again, E = 3.3910 N/C. The field pattern is shown in the adjacent figure. The field is confined to the region between the two plates and is zero elsewhere.
5
3
A 2.75C charge is uniformly distributed on a ring of radius 8.5 cm. Find the electric field on the axis at (a) 1.2 cm, (b) 3.6 cm, and (c) 4.0 m from the center of the ring. (d) Find the field at 4.0 m using the approximation that the ring is a point charge at the origin, and compare your results with that for part (c).
Chapter 23
The Electric Field II: Continuous Charge Distributions
(a) Use Equ. 2310
Ex =
8.99_ 109 _2.75_ 106 _ 0.012 N/C= 4.69 _ 105 N/C 2 2 3/2 ((0.012) + (0.085) )
(b), (c) Proceed as in (a) 2 (d) Ex ~ kQ/x
Ex(0.036) = 1.1310 N/C; Ex(4) = 1.5410 N/C 3 Ex = 1.5510 N/C; this is slightly greater than (c) because the point charge is nearer x = 4 m than the ring.
2
6
3
4
A disk of radius 2.5 cm carries a uniform surface charge density of 3.6 C/m . Using reasonable approximations, find the electric field on the axis at distances of (a) 0.01 cm, (b) 0.04 cm, (c) 5 m, and (d) 5 cm. For x << r, the disk appears like an infinite plane. For x >> r, the ring charge may be approximated by a point charge. 5 (a), (b) Use Equ. 2312a Ex = 2.0310 N/C 2 2 2 (c) Ex = kQ/x = kr /x Ex = 2.54 N/C 2 2 4 Ex = 2.5410 N/C (d) Ex = kr /x ; this is not a good approximation since x = 2r is not much greater than r.
5* For the disk charge of Problem 4, calculate exactly the electric field on the axis at distances of (a) 0.04 cm and (b) 5 m, and compare your results with those for parts (b) and (c) of Problem 4. 2 5 (a) Use Equ. 2311; r = 2.5 cm, = 3.6 C/m Ex = 2.0010 N/C Ex = 2.54 N/C (b) Proceed as in (a) For x = 0.04 cm, the exact value of Ex is only 1.5% smaller than the approximate value obtained in the preceding problem. For x = 5 m, the exact and approximate values agree within less than 1%. 6 A uniform line charge extends from x = 2.5 cm to x = +2.5 cm and has a linear charge density of = 6.0 nC/m. (a) Find the total charge. Find the electric field on the y axis at (b) y = 4 cm, (c) y = 12 cm, and (d) y = 4.5 m. (e) Find the field at y = 4.5 m, assuming the charge to be a point charge, and compare your result with that for part (d). (a) Q = L Q = 0.3 nC kQ (b) Ey = 1.43 kN/C; (c) Ey = 184 N/C; , Equ. 238 (b), (c), (d) E y = 2 2 (d) Ey = 0.1332 N/C y (L/2 ) + y (e) Ey = kQ/y 7
2
Ey = 0.1332 N/C, in good agreement with (c)
A disk of radius a lies in the yz plane with its axis along the x axis and carries a uniform surface charge 1 density . Find the value of x for which Ex = 2 /20.
Chapter 23
The Electric Field II: Continuous Charge Distributions
From Equ. 2311, E x = gives x = a/ 3 . 8
1  20
. Ex = /40 when ( x2 + a2 ) x
x ( x + a2 )
2
= 1/2 . Solving for x
A ring of radius a with its center at the origin and its axis along the x axis carries a total charge Q. Find Ex at (a) x = 0.2a, (b) x = 0.5a, (c) x = 0.7a, (d) x = a, and (e) x = 2a. (f) Use your results to plot Ex versus x for both positive and negative values of x. a), (b), (c), (d), (e) Use Equ. 2310; the results a) Ex = 0.189 (b) Ex = 0.358 (c) Ex = 0.385 2 are given in units of kQ/a (d) Ex = 0.354 (e) Ex = 0.179 (f) The field along the x axis is plotted in the adjoining figure. The x coordinates are in units of 2 x/a and E is in units of kQ/a .
9* Repeat Problem 8 for a disk of uniform surface charge density . a), (b), (c), (d), (e) Use Equ. 2311; the results are (a) Ex = 0.804 (b) Ex = 0.553 (c) Ex = 0.427 given in units of /20. (d) Ex = 0.293 (e) Ex = 0.106 (f) The field along the x axis is plotted in the adjoining figure. The x coordinates are in units of x/a and E is in units of /20.
10 A disk of radius 30 cm carries a uniform charge density . (a) Compare the approximation E = 2k with the exact expression (Equation 2311) for the electric field on the axis of the disk by computing the fractional difference E/E x/ x 2 + R 2 for the distances x = 0.1, x = 0.2, and x = 3 cm. (b) At what distance is the neglected term 1% of 2k ?
Chapter 23
The Electric Field II: Continuous Charge Distributions x = 0.1 cm, E/E = 0.00333; x = 0.2 cm, E/E = 0.00667; x = 3 cm, E/E = 0.0995 x = R/100 = 0.3 cm
(a) Evaluate E/E = x/ x 2 + R 2 for x = 0.1, 0.2, and 3 cm with R = 30 cm (b) Set E/E = 0.01 and solve for x
11 Show that Ex on the axis of a ring charge of radius a has its maximum and minimum values at x = +a/ 2 and x = a/ 2 . Sketch Ex versus x for both positive and negative values of x. Take the derivative of Ex given in Equ. 2310 and set it equal to 0.
3 x2 kQ dE x = 1  2 = 0 . This gives x = a/ 2 as the values of x where Ex is a 2 dx ( x2 + a2 )3/2 x + a
maximum. A plot of E versus x/a is shown in units of kQ/a .
2
12 A line charge of uniform linear charge density lies along the x axis from x = 0 to x = a. (a) Show that the x component of the electric field at a point on the y axis is given by Ex = k + y k
2 y + a2
(b) Show that if the line charge extends from x = b to x = a, the x component of the electric field at a point on the y axis is given by Ex = k y + a2
2

k
2 y + b2
(a) The line charge and point (0, y) are shown in the drawing.
Chapter 23
The Electric Field II: Continuous Charge Distributions
Also shown is the line element dx and the corresponding field dE. The x component of dE is then dE x =  k x + y
2 2
sin dx = 
kx ( x + y 2 )3/2
2
dx . Integrating
dEx from x = 0 to x = a gives the result stated in the problem. (b) Proceed as in part (a) but integrate from x = b to x = a. One obtains the expression given in the problem. 13* (a) A finite line charge of uniform linear charge density lies on the x axis from x = 0 to x = a. Show that the y component of the electric field at a point on the y axis is given by k a k E= sin 1 = 2 y y y + a2 where 1 is the angle subtended by the line charge at the field point. (b) Show that if the line charge extends from x = b to x = a, the y component of the electric field at a point on the y axis is given by k E y = ( sin 1 + sin 2) y where sin 2 = b/ y 2 + b2 . (a) The line charge and the point (0, y) are shown in the drawing. Also shown is the line element dx and the corresponding field dE. The y component of dE is then k k y cos dx = dx . dE y = 2 2 2 ( x + y 2 )3/2 x + y Integrating dEy from x = 0 to x = a one obtains ka k = = sin 1 , 2 y y y + a2 where here 1 = shown in the drawing. (b) Proceed as in part (a) but integrate dEy from x = b to x = a. The result is
Ey
Ey =
k a + y y 2 + a2
= k ( sin + sin ) , where 2 is the angle subtended by 1 2 2 y y + b2 b
the line segment b at the point y. 14 A semicircular ring of radius R carries a uniform line charge of . Find the electric field at the center of the semicircle.
Chapter 23
The Electric Field II: Continuous Charge Distributions
The semicircular ring is shown in the drawing. From symmetry it is 2 evident that Ey = 0. The field dE = r d/r = d/r and dEx = [( cos )/r] d. Integrating from = /2 to /2 one obtains E = Ex = 2/r
15 A hemispherical thin shell of radius R carries a uniform surface charge . Find the electric field at the center of the hemispherical shell (r = 0). Consider a ring with its axis along the z direction of radius r sin and width r d. The field dE is given by 2 Equ. 2310, where the distance to the point of interest is z = r cos . The charge on the ring is 2r sin d and 2 2 2 2 2 3/2 the field dE = 2k(r cos )r sin d/(r sin + r cos ) = 2k cos sin d. Integrating from = 0 to = /2, one obtains E = k. 16 A line charge of linear charge density with the shape of a square of side L lies in the yz plane with its center at the origin. Find the electric field on the x axis at an arbitrary distance x, and compare your result to 1 that for the field on the axis of a charged ring of radius r = 2 L with its center at the origin and carrying the same total charge. (Hint: Use Equation 238 for the field due to each segment of the square.) Note that any point on the x axis is on the perpendicular bisector of each of the four sides of the square. From symmetry, the total field has no z or y components. The x component of the field due to one of the four sides is Ex = Ex/ 2 a 2 + x 2 , where a = L/2. The field due to one side is E = 2ka/(a + x ). There are four line charges,
2 2
so the total field in the x direction is E = 8kax/[(a + x ) 2 a 2 + x 2 ] = 4kLx/[(x + L /4) x 2 + L 2 /2 ]. 2 2 3/2 For a ring of radius r = L/2 = a the field is E = 2kax/(a + x ) .
2 2 2 2
17* True or false: (a) Gauss's law holds only for symmetric charge distributions. (b) The result that E = 0 inside a conductor can be derived from Gauss's law. (a) False (b) False 18 What information in addition to the total charge inside a surface is needed to use Gauss's law to find the electric field? To use Gauss's law the system must display some symmetry. 19 Is the electric field E in Gauss's law only that part of the electric field due to the charge inside a surface, or is it the total electric field due to all charges both inside and outside the surface? The electric field is that due to all the charges, inside and outside the surface. 20 Consider a uniform electric field E = 2 kN/C i. (a) What is the flux of this field through a square of side 10 cm in a plane parallel to the yz plane? (b) What is the flux through the same square if the normal to its plane makes a 30 angle with the x axis? = (2103 102) N.m2/C = 20 N.m2/C (a) Use Equ. 2314 o = 20 cos 30N.m2/C = 17.3 N.m2/C (b) Here i.n = cos 30
Chapter 23
The Electric Field II: Continuous Charge Distributions
21* A single point charge q = +2 C is at the origin. A spherical surface of radius 3.0 m has its center on the x axis at x = 5 m. (a) Sketch electric field lines for the point charge. Do any lines enter the spherical surface? (b) What is the net number of lines that cross the spherical surface, counting those that enter as negative? (c) What is the net flux of the electric field due to the point charge through the spherical surface? (a) A sketch of the field lines and of the sphere is shown. Three lines enter the sphere. (b) The net number of lines crossing the surface is zero. (c) The net flux is zero.
22 An electric field is E = 300 N/C i for x > 0 and E = 300 N/C i for x < 0. A cylinder of length 20 cm and radius 4 cm has its center at the origin and its axis along the x axis such that one end is at x = +10 cm and the other is at x = 10 cm. (a) What is the flux through each end? (b) What is the flux through the curved surface of the cylinder? (c) What is the net outward flux through the entire cylindrical surface? (d) What is the net charge inside the cylinder? This problem is identical to Example 234 except for a change in the data. Following the procedure of that 2 Example, one obtains the following results. (a) right = left = 1.51 N.m /C, (b) curved = 0, (c) net = 3.02 N. m /C, and (d) Q = 0 net = 2.6710 C. 23 A positive point charge q is at the center of a cube of side L. A large number N of electric field lines are drawn from the point charge. (a) How many of the field lines pass through the surface of the cube? (b) How many lines pass through each face, assuming that none pass through the edges or corners? (c) What is the net outward flux of the electric field through the cubic surface? (d) Use symmetry arguments to find the flux of the electric field through one face of the cube. (e) Which, if any, of your answers would change if the charge were inside the cube but not at its center? (a) All N lines pass through the surface of the cube. (b) By symmetry, N/6 lines pass through each face of the cube. (c) = q/0 (see Equ. 2320) (d) Through one face, = q/60 (e) Parts (b) and (d) would change if the charge is not centered. 24 Careful measurement of the electric field at the surface of a black box indicates that the net outward flux 2 through the surface of the box is 6.0 kNm /C. (a) What is the net charge inside the box? (b) If the net outward flux through the surface of the box were zero, could you conclude that there were no charges inside the box? Why or why not? 8 (a) Q = 0 Q = 5.3110 C (b) One can only conclude that the net charge is zero. There may be an equal number of positive and negative charges present inside the box. 25* A point charge q = +2 C is at the center of a sphere of radius 0.5 m. (a) Find the surface area of the sphere. (b) Find the magnitude of the electric field at points on the surface of the sphere. (c) What is the flux of
2 11
Chapter 23
The Electric Field II: Continuous Charge Distributions
the electric field due to the point charge through the surface of the sphere? (d) Would your answer to part (c) change if the point charge were moved so that it was inside the sphere but not at its center? (e) What is the net flux through a cube of side 1 m that encloses the sphere? 2 2 (a) A = 4r A = 3.14 m 4 (b) Use Equ. 2319 E = 7.1910 N/C 5 2 (c) E.n = E, so = EA = 2.2610 N.m /C (d) No change in if q is inside sphere 5 2 (e) Apply Gauss's law; unchanged = 2.2610 N.m /C 26 Since Newton's law of gravity and Coulomb's law have the same inversesquare dependence on distance, an expression analogous in form to Gauss's law can be found for gravity. The gravitational field g is the force per unit mass on a test mass m0. Then for a point mass m at the origin, the gravitational field g at some position r is
g=
Gm r
2
r
Compute the flux of the gravitational field through a spherical surface of radius r centered at the origin, and show that the gravitational analog of Gauss's law is net = 4Gminside. 2 2 0g.n dA. Then g = (Gm/r )(4r ) = 4Gm. Define the gravitational flux as g =
S
27 A charge of 2 C is 20 cm above the center of a square of side length 40 cm. Find the flux through the square. (Hint: Don't integrate.) Assume the square is one face of a cube of side length 40 cm. Then the charge is at the center of the cube and 4 2 total = q/0. So for the face, = q/60 = 3.7710 N.m /C. 28 In a particular region of the earth's atmosphere, the electric field above the earth's surface has been measured to be 150 N/C downward at an altitude of 250 m and 170 N/C downward at an altitude of 400 m. Calculate the volume charge density of the atmosphere assuming it to be uniform between 250 and 400 m. (You may neglect the curvature of the earth. Why?) 1. Calculate the charge inside a cylinder of base area Q = AH; we take our zero at 250 m above the earth 2. Use Gauss's law to find Q; take up as positive surface and can consider the earth as flat. 3. Evaluate Q = (EHA E0A)0 = AH; = (E0 EH)/0H 12 3 = 1.1810 C/m 29* Explain why the electric field increases with r rather than decreasing as 1/r as one moves out from the center inside a spherical charge distribution of constant volume charge density. 3 2 The charge inside a sphere of radius r is proportional to r . The area of the sphere is proportional to r . Using 3 2 Gauss's law, one sees that the field must be proportional to r /r = r. 30 A spherical shell of radius R1 carries a total charge q1 that is uniformly distributed on its surface. A second, larger spherical shell of radius R2 that is concentric with the first carries a charge q2 that is uniformly distributed on its surface. (a) Use Gauss's law to find the electric field in the regions r < R1, R1 < r < R2, and r > R2. (b) What should the ratio of the charges q1/q2 and their relative signs be for the electric field to be zero for r > R2? (c) Sketch the electric field lines for the situation in part (b) when q1 is positive. (a) r < R1, Q = 0; R1 < r < R2, Q = q1; r > R2, Q = q1 + q2
2
Chapter 23
The Electric Field II: Continuous Charge Distributions
2
(b) Set E = 0 for r > R2
E = 0 for r < R1; E = kq1/r for R1 < r < R2 2 E = k(q1 + q2)/r for r > R2 q1 = q2; q1/q2 = 1
c) The electric field lines for the case (b) are shown in the figure.
31 A spherical shell of radius 6 cm carries a uniform surface charge density = 9 nC/m . (a) What is the total charge on the shell? Find the electric field at (b) r = 2 cm, (c) r = 5.9 cm, (d) r = 6.1 cm, and (e) r = 10 cm. 2 (a) Q = 4r Q = 0.407 nC E(2 cm) = E(5.9 cm) = 0 (b), (c) Qin = 0 for r < 6 cm 2 E(6.1 cm) = 983 N/C; E(10 cm) = 366 N/C (d), (e) Qin = Q; E = kQ/r
2
32 A sphere of radius 6 cm carries a uniform volume charge density = 450 nC/m . (a) What is the total charge of the sphere? Find the electric field at (b) r = 2 cm, (c) r = 5.9 cm, (d) r = 6.1 cm, and (e) r = 10 cm. Compare your answers with Problem 31. 3 (a) Q = (4/3)r Q = 0.407 nC 3 E(2 cm) = 339 N/C; E(5.9 cm) = 1000 N/C (b), (c) For r < R, E = kQr/R 2 E(6.1 cm) = 984 N/C; E(10 cm) = 366 N/C (d), (e) For r > R, E = kQ/r
3
33* Consider two concentric conducting spheres (Figure 2334). The outer sphere is hollow and initially has a charge 7Q deposited on it. The inner sphere is solid and has a charge +2Q on it. (a) How is the charge distributed on the outer sphere? That is, how much charge is on the outer surface and how much charge is on the inner surface? (b) Suppose a wire is connected between the inner and outer spheres. After electrostatic equilibrium is established, how much total charge is on the outside sphere? How much charge is on the outer surface of the outside sphere and how much is on the inner surface? Does the electric field at the surface of the inside sphere change when the wire is connected? If so, how? (c) Suppose we return to the original conditions in (a), with +2Q on the inner sphere and 7Q on the outer. We now connect the outer sphere to ground with a wire and then disconnect it. How much total charge will be on the outer sphere? How much charge will be on the inner surface of the outer sphere and how much will be on the outer surface? (a) Since the outer sphere is conducting, the field in the thin shell must vanish. Therefore, 2Q, uniformly distributed, resides on the inner surface, and 5Q, uniformly distributed, resides on the outer surface. (b) Now there is no charge on the inner surface and 5Q on the outer surface of the spherical shell. The electric field just outside the surface of the inner sphere changes from a finite value to zero.
Chapter 23
The Electric Field II: Continuous Charge Distributions
(c) In this case, the 5Q is drained off, leaving no charge on the outer surface and 2Q on the inner surface. The total charge on the outer sphere is then 2Q. 34 A nonconducting sphere of radius R = 0.1 m carries a uniform volume charge of charge density = 2.0 3 nC/m . The magnitude of the electric field at r = 2R is 1883 N/C. Find the magnitude of the electric field at r = 0.5R. 1. Write the expression for E(r < R) E(r < R) = (4/3)rk 3 2. Evaluate E at r = 0.5R = 0.05 m and = 2 C/m E = 3.77 kN/C 35 A nonconducting sphere of radius R carries a volume charge density that is proportional to the distance from the center: = Ar for r R, where A is a constant; = 0 for r > R. (a) Find the total charge on the sphere 2 by summing the charges on shells of thickness dr and volume 4r dr. (b) Find the electric field Er both inside and outside the charge distribution, and sketch Er versus r. (a) The charge in a shell of thickness dr is dq = 4r dr = 4Ar dr. Q = 4 A r 3 dr = A R 4 0.
2 3 0
R
(b) For r < R, Qin = Ar and by Gauss's law E(r) = Ar /40. 4 2 For r>R, E(r) = AR /40r . 4 4 2 For r > R, Qin = AR and E(r > R) = AR /40r A plot of E(r) versus r/R is shown. Here E(r) is in units of A/40.
4 2
36 Repeat Problem 35 for a sphere with volume charge density = B/r for r < R ; = 0 for r > R. 2 (a) Following the procedure of the preceding problem, one finds Q = 2BR (b) Following the procedure of the preceeding problem, one 2 2 finds E(r < R) = B/20, E(r > R) = BR /20r . A plot of E(r) versus r/R is shown. Here E(r) is in units of B/20.
37*
Repeat Problem 35 for a sphere with volume charge density = C/r for r < R; = 0 for r > R.
2 2
(a) The charge in a shell of thickness dr is dq = 4r dr = 4Cdr. Q = 4 C
dr
0
R
= 4 C R .
Chapter 23
The Electric Field II: Continuous Charge Distributions
(b) For r < R, Qin = 4Cr and by Gauss's law E(r < R) = C/0r. 2 For r > R, Qin = 4CR and E(r > R) = CR/0r A plot of E(r) versus r/R is shown. Here E(r) is in units of C/0R.
38 The charge density in a region of space is spherically symmetric and is given by (r) = Ce r/a when r < R and = 0 when r > R. Find the electric field as a function of r. 2 2 r/a The element of charge in a shell of thickness dr is dq = 4r dr = 4Cr e dr. The charge within a sphere of radius r < R is obtained by integrating dq. Q(r < R ) = 4C r 2 e  r / a dr = 4C (a 3 e  r / a )(r 2 / a 2 + 2r / a + 2)
0
r
0
r
= 4C 2a 3  a 3 e  r / a r 2 / a 2 + 2r / a + 2 .
[
(
)]
For r > R, Q, the charge within the sphere of radius r is given by the above expression where r is replaced by R. The electric field as a function of r is obtained from Gauss's law. That is, 3 2 r/a 2 2 E(r < R) = (Ca /0r )[2 e (r /a + 2r/a + 2)], and E(r > R) is given by the same expression with r replaced by R inside the square brackets. Note that as in previous problems, E is continuous at r = R. 39 A thick, nonconducting spherical shell of inner radius a and outer radius b has a uniform volume charge density . Find (a) the total charge and (b) the electric field everywhere. 3 3 (a), (b) 1. For r < a, Qin = 0, and E(r < a) = 0. 2. For a < r < b, Qin = (4/3) (r a ); using Gauss's law, 3 3 2 3 3 3 3 2 E(a < r < b) = (r a )/30r . 3. For r > b, Qin = (4/3) (b a ) = Qtot and E(r > b) = (b a )/30r . 40 A point charge of +5 nC is located at the origin. This charge is surrounded by a spherically symmetric r/a negative charge distribution with volume density (r) = Ce . (a) Find the constant C in terms of a if the total charge of the system is zero. (b) What is the electric field at r = a? 3 3 (a) See Problem 38; set integration limits 0 to Qtot = 5 C + 8Ca = 0; C = 5/8a C 3 Qin = [5 + 4Ca (2 5/e)] C = 4.6 C (b) Use the expression in Problem 38 for r = a to 2 4 2 find Qin and E E = kQin/a = 4.1410 /a N/C 41* A nonconducting solid sphere of radius a with its center at the origin has a spherical cavity of radius b with its center at the point x = b, y = 0 as shown in Figure 2335. The sphere has a uniform volume charge density . Show that the electric field in the cavity is uniform and is given by Ey = 0, Ex = b/30. (Hint: Replace the cavity with spheres of equal positive and negative charge densities.) Using the Hint we shall find the x and y components of the field due to the uniform positive charge distribution of the solid sphere, and then the x and y components of the field due to a uniform negative charge distribution centered at x = b. We denote the field due to the solid positively charged sphere as E+ and that due to the negatively charged sphere at x = b by E. The field E+ is (4/3)k r and its x and y components are E+x =
Chapter 23
The Electric Field II: Continuous Charge Distributions
(4/3)k x and E+y = (4/3)k y. For the negatively charged sphere, E = (4/3)k r', where r' is the radial distance from x = b, y = 0. We can again find the x and y components of E at a point (x, y); they are Ex = (4/3)k (x b) and Ey = (4/3)k y. Thus Ex = E+x + Ex = (4/3)kb = b/30 and Ey = E+y + Ey = 0. Since the system is symmetric for rotation about the x axis, Ez = Ey = 0. 42 Show that the electric field due to an infinitely long, uniformly charged cylindrical shell of radius R carrying a surface charge density is given by E r = 0, 0 Er = r<R 2328a
R = , r>R r 2 0 r 0
2328b
where = 2R is the charge per unit length on the shell. From symmetry, the field in the tangential direction must vanish. Construct a Gaussian surface in the shape of a cylinder of radius r and length L. If r < R, Qin = 0 and Er = 0. If r > R, Qin = L. The area of the Gaussian surface surrounding Qin is 2RL, neglecting the end areas since no flux crosses those. The charge Qin = 2RL, and using Gauss's law, one obtains Er = R/0r. 43 A cylindrical shell of length 200 m and radius 6 cm carries a uniform surface charge density of = 9 2 nC/m . (a) What is the total charge on the shell? Find the electric field at (b) r = 2 cm, (c) r = 5.9 cm, (d) r = 6.1 cm, and (e) r = 10 cm. (Use the results of Problem 42.) (a) Q = 2RL Q = 679 nC E(2 cm) = E(5.9 cm) = 0 (b), (c) See Equ. 2328a E(6.1 cm) = 1 kN/C; E(10 cm) = 610 N/C (d), (e) Use Equ. 2328b 44 An infinitely long nonconducting cylinder of radius R carries a uniform volume charge density of (r) = 0. Show that the electric field is given by
Er =
R2 1 = ,0 2 0 r 2 0 r
r>R
2329a
Er =
r= r, 0 2 0 2 0 R 2
r<R
2329b
where = R 2 0 is the charge per unit length. Proceed as in Problem 42, constructing cylindrical Gaussian surfaces. For r < R, the charge within the Gaussian 2 surface is Qin = r L. The area of the Gaussian surface, neglecting the end areas, is 2rL. Using Gauss's law 2 one obtains Er = r/20 = /2r0. For r > R, the charge within the Gaussian surface is R L and the area of 2 2 the Gaussian surface is 2rL. From Gauss's law one obtains Er = R /2r0 = /2r0, since = R . 45* A cylinder of length 200 m and radius 6 cm carries a uniform volume charge density of = 300 nC/m . (a) What is the total charge of the cylinder? Use the formulas given in Problem 44 to calculate the electric field at a point equidistant from the ends at (b) r = 2 cm, (c) r = 5.9 cm, (d) r = 6.1 cm, and (e) r = 10 cm. Compare
3
Chapter 23
The Electric Field II: Continuous Charge Distributions
your results with those in Problem 43.
(a) Q = R L (b), (c) Use Equ. 2329b (d), (e) Use Equ. 2329a
2
Q = 679 nC Er(2 cm) = 339 N/C; Er(5.9 cm) = 1.00 kN/C Er(6.1 cm) = 1.00 kN/C; Er(10 cm) = 610 N/C
46 Consider two infinitely long, concentric cylindrical shells. The inner shell has a radius R1 and carries a uniform surface charge density of 1, and the outer shell has a radius R2 and carries a uniform surface charge density of 2. (a) Use Gauss's law to find the electric field in the regions r < R1, R1 < r < R2, and r > R2. (b) What is the ratio of the surface charge densities 2/1 and their relative signs if the electric field is zero at r > R2? What would the electric field between the shells be in this case? (c) Sketch the electric field lines for the situation in (b) if 1 is positive. From symmetry, the field must be radial. (a) Apply Equs. 2328a and 2328b E = 0 for r < R1; E = 1R1/0r for R1 < r < R2; E = E1 + E2 = (1R1 + 2R2)/0r for r > R2 (b) Set E for r > R2 = 0 1R1 + 2R2 = 0, 1/2 = R2/R1 (c) The field lines for case (b) are shown. Here we assume that 1 is positive.
47 Figure 2336 shows a portion of an infinitely long, concentric cable in cross section. The inner conductor carries a charge of 6 nC/m; the outer conductor is uncharged. (a) Find the electric field for all values of r, where r is the distance from the axis of the cylindrical system. (b) What are the surface charge densities on the inside and the outside surfaces of the outer conductor? (a) The cylinders are conductors; E = 0 inside them. E = 0 for r < 1.5 cm and for 4.5 cm < r < 6.5 cm For 1.5 cm< r < 4.5 cm, r > 6.5 cm use Equ. 2328b. E = 108/r N/C 2 2 (b) E = 0 inside the outer cylinder; see Problem 1 = (6/20.015) nC/m ; 2,in = 1R1/R2 = 21.2 nC/m 2 2 46(b). Q2,net = 0; therefore, 2,inR2,inner = 2,outR2,outer 2,out = (21.24.5/6.5) nC/m = 14.7 nC/m 48 Repeat Problem 44 for a cylinder with volume charge density (a) (r) = ar and (b) = Cr
2
Chapter 23
The Electric Field II: Continuous Charge Distributions
r
(a) Find in within a radius r for r < R Use Gauss's law to determine E (b) Find in for r > R Find E using Gauss's law
in = 2 r (a r 2 ) dr = 2 a r 3 /3
0
E = (2ar /3)/20r = ar /30 for < r R
3 2
in = 2 r (a r 2 ) dr = 2 a R3 /3
0
R
E = aR /30r
3
49* Repeat Problem 44 with = C/r. (a) Find in within a radius r for r < R Use Gauss's law to determine E (b) Find in for r > R Find E using Gauss's law
in = 2 r (C/r) dr = 2 C r
0
r
E = 2Cr/20r = C/0 for r < R
in = 2 r (C/r) dr = 2 C R
0
R
E = CR/0r 50 An infinitely long, thick, nonconducting cylindrical shell of inner radius a and outer radius b has a uniform volume charge density . Find the electric field everywhere. 1. For r < a, in = 0. Use Gauss's law E = 0 for r < a r 2. For a < r < b, find in(r) in(r) = 2 r dr = ( r 2  a 2 ) Use Gauss's law to find E
a
3. For r > b, in = (b a ). Use Gauss's law
2 2
E = (r a )/20r for a < r < b 2 2 E = (b a )/20r
2 2
51 Suppose that the inner cylinder of Figure 2336 is made of nonconducting material and carries a volume 2 charge distribution given by (r) = C/r, where C = 200 nC/m . The outer cylinder is metallic. (a) Find the charge per meter carried by the inner cylinder. (b) Calculate the electric field for all values of r. (a) See Problem 49(b); inner = 2CR inner = 18.85 nC/m (b) 1. For r < 1.5 cm, see Problem 49(a) E = C/0 = 22.6 kN/C 2. For 1.5 cm < r < 4.5 cm, see Problem 49(b) E = 339/r N/C 3. For 4.5 cm < r < 6.5 cm, conductor E=0 4. For r > 6.5 cm, see part 2 E = 339/r N/C 52 A penny is in an external electric field of magnitude 1.6 kN/C directed perpendicular to its faces. (a) Find the charge density on each face of the penny, assuming the faces are planes. (b) If the radius of the penny is 1 cm, find the total charge on one face. 2 (a) E = /0; = E0 = 14.2 nC/m 2 (b) Q = A = r Q = 4.45 pC 53* An uncharged metal slab has square faces with 12cm sides. It is placed in an external electric field that is perpendicular to its faces. The total charge induced on one of the faces is 1.2 nC. What is the magnitude of the
Chapter 23
The Electric Field II: Continuous Charge Distributions
electric field? 2 2 = Q/L ; E = Q/L 0
E = 9.41 kN/C
54 A charge of 6 nC is placed uniformly on a square sheet of nonconducting material of side 20 cm in the yz plane. (a) What is the surface charge density ? (b) What is the magnitude of the electric field just to the right and just to the left of the sheet? (c) The same charge is placed on a square conducting slab of side 20 cm and thickness 1 mm. What is the surface charge density ? (Assume that the charge distributes itself uniformly on the large square surfaces.) (d) What is the magnitude of the electric field just to the right and just to the left of each face of the slab? 2 (a) = Q/A = 150 nC/m (b) Use Equ. 2312; E = /20 E = 8.47 kN/C 2 (c) Now Q on each face is 3 nC = 75 nC/m (d) Use Equ. 2325 E = 8.47 kN/C 55 A spherical conducting shell with zero net charge has an inner radius a and an outer radius b. A point charge q is placed at the center of the shell. (a) Use Gauss's law and the properties of conductors in equilibrium to find the electric field in the regions r < a, a < r < b, and b < r. (b) Draw the electric field lines for this situation. (c) Find the charge density on the inner surface (r = a) and on the outer surface (r = b) of the shell. 2 (a) For r < a, E = kq/r . For a < r < b, E = 0. For r > 2 b, E = kq/r (b) See the figure 2 2 (c) At r = a, = q/4a ; at r = b, = q/4b
56 The electric field just above the surface of the earth has been measured to be 150 N/C downward. What total charge on the earth is implied by this measurement? 2 6 2 9 5 The earth is a sphere, so Q = ERE /k Q = 150 (6.3710 ) /8.9910 = 6.7710 C 57* A positive point charge of magnitude 2.5 C is at the center of an uncharged spherical conducting shell of inner radius 60 cm and outer radius 90 cm. (a) Find the charge densities on the inner and outer surfaces of the shell and the total charge on each surface. (b) Find the electric field everywhere. (c) Repeat (a) and (b) with a net charge of +3.5 C placed on the shell. 6 2 2 (a) For 60 cm < r < 90 cm, E = 0 qin = 2.5 C;in = 2.510 /40.6 = 0.553 C/m 6 2 2 qinner + qouter = 0 qout = 2.5 C; out = 2.510 /40.9 = 0.246 C/m 2 4 2 (b) For r < 0.6 m, E = kq/r E = 2.2510 /r N/C E=0 For 0.6 m < r < 0.9 m, conductor 2 4 2 E = 2.2510 /r N/C For r > 0.9 m, E = kq/r
Chapter 23
The Electric Field II: Continuous Charge Distributions
2
(c) Since E = 0 in the conductor, qinner is again 2.5 C and inner = 0.553 C/m . Now qinner + qouter = 3.5 C; 2 4 2 consequently, qouter = 6.0 C and outer = 0.59 C/m . The fields are 2.2510 /r N/C for r < 0.6 m, zero within 4 2 the shell, and 5.410 /r N/C for r > 0.9 m. 6 58 If the magnitude of an electric field in air is as great as 310 N/C, the air becomes ionized and begins to conduct electricity. This phenomenon is called dieletric breakdown. A charge of 18 C is to be placed on a conducting sphere. What is the minimum radius of a sphere that can hold this charge without breakdown? 2 9 6 6 Rmin = kq/Emax Rmin = (8.9910 1810 /310 )1/2 m = 23.2 cm 59 A square conducting slab with 5m sides carries a net charge of 80 C. (a) Find the charge density on each face of the slab and the electric field just outside one face of the slab. (b) The slab is placed to the right of an 2 infinite charged nonconducting plane with charge density 2.0 C/m so that the faces of the slab are parallel to the plane. Find the electric field on each side of the slab far from its edges and the charge density on each face. 2 2 5 (a) per face = q/2L ; E = /0 /face = 1.6 C/m ; E = 1.8110 N/C 5 Etotal(away from plane) = 2.9410 N/C (b) Eslab = slab/20; Etotal = Eface + Eslab on face away 5 from the plane. On face near the plane, fields oppose. Etotal(near plane) = 0.6810 N/C 2 (near plane) = 0E(near plane) (near plane) = 0.6 C/m ; 2 (away from plane) = 0E(away from plane) (away from plane) = 2.6 C/m 60 Imagine that a small hole has been punched through the wall of a thin, uniformly charged spherical shell whose surface charge density is . Find the electric field near the center of the hole. 2 We can view the system as a spherical shell of surface charge density = Q/4r with a small disk with a 2 surface charge density . From Coulomb's law, the field just outside the shell is kQ/r = 4k = /0 and is zero inside. The field due to the disk is /20 inside the shell and immediately adjacent to the disk, pointing radially outward; just outside the disk the field is /20 and points radially inward. Adding these fields we see 2 that just inside the hole the field is continuous and has the value /20 = kQ/2r . 61* True or false: (a) If there is no charge in a region of space, the electric field on a surface surrounding the region must be zero everywhere. (b) The electric field inside a uniformly charged spherical shell is zero. (c) In electrostatic equilibrium, the electric field inside a conductor is zero. (d) If the net charge on a conductor is zero, the charge density must be zero at every point on the surface of the conductor. (a) False (b) True (assuming there are no charges inside the shell) (c) True (d) False 62 If the electric field E is zero everywhere on a closed surface, is the net flux through the surface necessarily zero? What, then, is the net charge inside the surface? Yes, because = E dA . The net charge inside the surface must be zero by Gauss's law.
S
63 A point charge Q is at the center of a spherical conducting shell of inner radius R1 and outer radius R2 as shown in Figure 2337. The charge on the inner surface of the shell is (a) +Q. (b) zero. (c) Q. (d) dependent on the total charge carried by the shell.
Chapter 23
The Electric Field II: Continuous Charge Distributions
(a) 64 For the configuration of Figure 2337, the charge on the outer surface of the shell is (a) +Q. (b) zero. (c) Q. (d) dependent on the total charge carried by the shell. (d) 65* Suppose that the total charge on the conducting shell of Figure 2337 is zero. It follows that the electric field for r < R1 and r > R2 points (a) away from the center of the shell in both regions. (b) toward the center of the shell in both regions. (c) toward the center of the shell for r < R1 and is zero for r > R2. (d) away from the center of the shell for r < R1 and is zero for r > R2. (b) 66 If the conducting shell in Figure 2337 is grounded, which of the following statements is then correct? (a) The charge on the inner surface of the shell is +Q and that on the outer surface is Q. (b) The charge on the inner surface of the shell is +Q and that on the outer surface is zero. (c) The charge on both surfaces of the shell is +Q. (d) The charge on both surfaces of the shell is zero. (b) 67 For the configuration described in Problem 66, in which the conducting shell is grounded, the electric field for r < R1 and r > R2 points (a) away from the center of the shell in both regions. (b) toward the center of the shell in both regions. (c) toward the center of the shell for r < R1 and is zero for r > R2. (d) toward the center of the shell for r < R1 and is zero for r > R1. (c) 68 If the net flux through a closed surface is zero, does it follow that the electric field E is zero everywhere on the surface? Does it follow that the net charge inside the surface is zero? The electric field need not be zero everywhere on the surface, but the net charge inside the surface is zero. 69* Equation 238 for the electric field on the perpendicular bisector of a finite line charge is different from Equation 239 for the electric field near an infinite line charge, yet Gauss's law would seem to give the same result for these two cases. Explain. The two expressions agree if r << L, where L is the length of the line charge of finite length. For r of the same order of magnitude as L or greater, the electric field does not have cylindrical symmetry, and one cannot use Gauss's law to determine E. 70 True or false: The electric field is discontinuous at all points at which the charge density is discontinuous. False; see, for example, the field of a uniformly charged sphere. is discontinuous at the surface, E is not. 71 Consider the three concentric metal spheres shown in Figure 2338. Sphere I is solid, with radius R1. Sphere II is hollow, with inner radius R2 and outer radius R3. Sphere III is hollow, with inner radius R4 and outer radius R5. Initially, all three spheres have zero excess charge. Then a negative charge Q0 is placed on sphere I and a positive charge +Q0 is placed on sphere III. (a) After the charges have reached equilibrium, will the electric field in the space between spheres I and II point toward the center, away from the center, or
Chapter 23
The Electric Field II: Continuous Charge Distributions
neither? (b) How much charge will be on the inner surface of sphere II? Give the correct sign. (c) How much charge will be on the outer surface of sphere II? (d) How much charge will be on the inner surface of sphere III? (e) How much charge will be on the outer surface of sphere III? (f) Plot E versus r. (a) toward the center (b) +Q0 (c) Q0 (d) +Q0 (e) 0 (f) See the adjacent figure. Here E and r are in arbitrary units.
72 An early model of the hydrogen atom considered the atom to consist of a proton, which is a uniform charged sphere of radius R, with an electron in an orbit of radius r0 inside the proton as shown in Figure 2339. (a) Use Gauss's law to obtain the magnitude of E (the field due to the proton) at the position of the electron. Give your answer in terms of e (the charge on a proton), r0, and R. (b) Find the frequency of revolution f in terms of r0 and the velocity of the electron v. (c) What is the force on the electron in terms of m,v, and r0? (d) What is the frequency f in terms of m, e, R, 0 , and r0? (Each of your answers need not include all of the specified quantities.) 3 3 (a) 1. Find the charge within r0; q r q = er0 /R 3 2 3 3 2. Use Gauss's law E = er0 /4r0 0R = er0/40R (b) f = 1/T; T = 2r0/v f = v/2r0 2 (c) F = mac F = mv /r0 (d) eE = F; express f in terms of variables f = (e/2) 1/4 0 m R 3 73* A nonuniform surface charge lies in the yz plane. At the origin, the surface charge density is = 3.10 2 C/m . Other charged objects are present as well. Just to the right of the origin, the x component of the electric 5 field is Ex = 4.6510 N/C. What is Ex just to the left of the origin? 5 Use Equ. 2324 Ex,left = Ex,right /0 = 1.1510 N/C 74 An infinite line charge of uniform linear charge density = 1.5 C/m lies parallel to the y axis at x = 2 m. A point charge of 1.3 C is located at x = 1 m, y = 2 m. Find the electric field at x = 2 m, y = 1.5 m. E = (2k/r) i = 6.74103 N/C i 1. Find the field at (2, 1.5) due to line charge 2 2. Find E due to point charge Ep = kq/r = 9.35 kN/C 3. Determine the x and y components of Ep Epx = 9.351/1.251/2 = 8.36 kN/C; Epy = 4.18 kN/C 4. E = E + Ep E = 1.62 kN/C i 4.18 kN/C j
Chapter 23
The Electric Field II: Continuous Charge Distributions
75 Two infinite planes of charge lie parallel to each other and to the yz plane. One is at x = 2 m and has a 2 surface charge density of = 3.5 C/m . The other is at x = 2 m and has a surface charge density of = 2 6.0 C/m . Find the electric field for (a) x < 2 m, (b) 2 m < x < 2 m, and (c) x > 2 m. (a) Find E of each plane; use Equ. 2312 E2 = 1.98 kN/C, pointing toward this plane E2 = 3.39 kN/C, pointing away from this plane E = 1.41 kN/C i For x < 2 m, E = E2  E2 E = 5.37 kN/C i (b) For 2 m < x < 2 m, E = E2 + E2 E = 1.41 kN/C i (c) For x > 2 m, E = E2 E2 76 An infinitely long cylindrical shell is coaxial with the y axis and has a radius of 15 cm. It carries a uniform 2 surface charge density = 6 C/m . A spherical shell of radius 25 cm is centered on the x axis at x = 50 cm and 2 carries a uniform surface charge density = 12 C/m . Calculate the magnitude and direction of the electric field at (a) the origin; (b) x = 20 cm, y = 10 cm; and (c) x = 50 cm, y = 20 cm. (See Problem 42.) E(0, 0) = (4 0.252 12106 8.99109/0.52) N/C i (a) For x < 15 cm, Ecyl = 0 5 = 3.3910 N/C i; Ecyl = 5.08105 N/C i; Esph = 8.47105 N/C (b) Both cylinder and sphere contribute to E 5 5 Esph,x = 8.0410 N/C, Esph,y = 2.6810 N/C Find Esph,x, Esph,y and E E = Ecyl + Esph = 1.31106 N/C i  2.68105 N/C j (c) At (0.5, 0.2) Esph = 0 E = 2.03105 N/C I 77* An infinite plane in the xz plane carries a uniform surface charge density 1 = 65 nC/m . A second infinite 2 plane carrying a uniform charge density 2 = 45 nC/m intersects the xz plane at the z axis and makes an angle of 30 with the xz plane as shown in Figure 2340. Find the electric field in the xy plane at (a) x = 6 m, y = 2 m and (b) x = 6 m, y = 5 m. E1(6, 2) = E1(6, 5) = (1/20) j; Find E1 and E2 for (6, 2) and (6, 5) E2(6, 2) = (2/20)(sin 30o i cos 30o j); E2(6, 5) = E2(6, 2) E(6, 2) = 1.27 kN/C i + 1.47 kN/C j (a) E(6, 2) = E1(6, 2) + E2(6, 2) (b) E(6, 5) = E(6, 5) + E2(6, 5) E(6, 5) = 1.27 kN/C i + 5.87 kN/C j
2
78 A ring of radius R carries a uniform, positive, linear charge density . Figure 2341 shows a point P in the plane of the ring but not at the center. Consider the two elements of the ring of lengths s1 and s2 shown in the figure at distances r1 and r2, respectively, from point P. (a) What is the ratio of the charges of these elements? Which produces the greater field at point P? (b) What is the direction of the field at point P due to each element? What is the direction of the total electric field at point P? (c) Suppose that the electric field due to a 2 point charge varied as 1/r rather than 1/r . What would the electric field be at point P due to the elements shown? (d) How would your answers to parts (a), (b), and (c) differ if point P were inside a spherical shell of uniform charge and the elements were of areas s1 and s2? (a) The charges on the two segments are proportional to r1 and r2, respectively, so q1/q2 = r1/r2. The fields are 2 2 proportional to 1/r1 and 1/r2 , respectively. Consequently, E1/E2 = r2/r1 and E1 > E2. (b) The two fields point away from their segments of arc. E points toward s2 (c) In this case, E1 = E2, and the total field at P would be zero.
Chapter 23
2
The Electric Field II: Continuous Charge Distributions
2 2 2 2
(d) For a spherical shell, q1 r1 and q2 r2 , so q1/q2 = r1 /r2 . Since the fields are proportional to 1/r , E1/E2 = 1. The two fields are of equal magnitude and oppositely directed. Hence E = 0. If E 1/r, then s2 would produce the stronger field at P, and E would point toward s1. 79 A ring of radius R that lies in the horizontal (xy) plane carries a charge Q uniformly distributed over its length. A mass m carries a charge q whose sign is opposite that of Q. (a) What is the minimum value of q/m such that the mass will be in equilibrium under the action of gravity and the electrostatic force on the charge q? (b) If q/m is twice that calculated in (a), where will the mass be when it is in equilibrium? (a) The minimum value of q/m will be where the field due to the ring is greatest, i.e. at z =  R/ 2 (see Problem 11). Substituting this value into Equ. 2310 one obtains E = 2kQ/ R 2 27 . At equilibrium, mg = qE, so q/m = g/E = R 2 g 27 /2kQ . (b) If q/m is twice as great as in (a), the E should be half the value in (a), i.e., E = 2kQ/ R 2 27 . The condition to be satisfied is 2 1 2 2 z . Let a = z /R . = 27 R 4 ( z 2 + R 2 )3 The equation for a is a + 3a 24a + 1 = 0. This cubic equation is plotted in the adjacent figure. The solutions (by trial and error) are a = 0.042 and 3.6. The distances for equilibrium are z = 0.205R and z = 1.9R. However only z = 1.9R is a location of stable equilibrium.
3 2
80 A long, thin, nonconducting plastic rod is bent into a loop with radius R. Between the ends of the rod, a small gap of length ! ( ! << R) remains. A charge Q is equally distributed on the rod. (a) Indicate the direction of the electric field at the center of the loop. (b) Find the magnitude of the electric field at the center of the loop. (a) The loop with the small gap is equivalent to a closed loop and a charge of Q ! /2R at the gap. The field at the center of a closed loop of uniform line charge is zero. Thus the field is entirely due to the charge Q ! /2R. If Q is positive, the field at the origin points radially outward. 3 (b) E = Q ! /2R 81* A rod of length L lies perpendicular to an infinitely long uniform line charge of charge density C/m (Figure 2342). The near end of the rod is a distance d above the line charge. The rod carries a total charge Q uniformly distributed along its length. Find the force that the infinitely long line charge exerts on the rod. Let y be the distance from the infinite line charge. The element of charge on the finite rod is dq = (Q/L)dy, and the field at the charge dq is 2k/y. The force on the rod is
Chapter 23
The Electric Field II: Continuous Charge Distributions
2kQ F= L
d +L
d
dy 2kQ L + d 1n = 0 y L d
82 A nonconducting sphere 1.2 m in diameter with its center on the x axis at x = 4 m carries a uniform volume 3 charge of density = 5 C/m . Surrounding the sphere is a spherical shell with a diameter of 2.4 m and a 2 uniform surface charge density = 1.5 C/m . Calculate the magnitude and direction of the electric field at (a) x = 4.5 m, y = 0; (b) x = 4.0 m, y = 1.1 m; and (c) x = 2.0 m, y = 3.0 m. (a) Point (4.5, 0) is inside the shell and inside the sphere. Eshell = 0; Esph = (4/3)kr. (b) Point (4, 1.1) is inside the shell but outside 3 2 the sphere. E = (4/3)kr1 /r , r1 = 0.6 m, r = 1.1 m (c) Point (2, 3) is outside shell and sphere. Find the equivalent point charge at (4, 0). Find E(2, 3) and its components E = 9.41104 N/C i E = 3.36104 N/C j Q= 4 3 2 r 1 + 4 r 2 =  26.2 C 0 3
E = 18.1 kN/C; E = 10.0 kN/C i 15.1 kN/C j 2 83 An infinite plane of charge with surface charge density 1 = 3 C/m is parallel to the xz plane at y = 0.6 2 m. A second infinite plane of charge with surface charge density 2 = 2 C/m is parallel to the yz plane at x = 1 m. A sphere of radius 1 m with its center in the xy plane at the intersection of the two charged planes (x = 1 2 m, y = 0.6 m) has a surface charge density 3 = 3 C/m . Find the magnitude and direction of the electric field on the x axis at (a) x = 0.4 m and (b) x = 2.5 m. E1 = 2k1 j = 169 kN/C j; E2 = 113 kN/C i (a) Point (0.4, 0) is within the sphere's surface. Find E1 and E2 due to the two planes at (0.4, 0) and E = (113 i + 169 j) kN/C; E = 203 kN/C, = 56.3 add. Q = 4 = 37.7 C Esph = (kQ/r2)[(1.5 m)/r i + (0.6 m)/r j]; r = 1.62 m (b) Point (2.5, 0) is outside the sphere; find Q on the Esph = 120 kN/C i 47.8 kN/C j sphere; find Esph at (2.5, 0) E = 233 kN/C i + 121 kN/C j E = E1 + E2 + Esph (note: here E2 = 113 kN/C i) E = 262 kN/C, = 153 84 An infinite plane lies parallel to the yz plane at x = 2 m and carries a uniform surface charge density = 2 2 C/m . An infinite line charge of uniform linear charge density = 4 C/m passes through the origin at an 3 angle of 45 with the x axis in the xy plane. A sphere of volume charge density = 6 C/m and radius 0.8 m is centered on the x axis at x = 1 m. Calculate the magnitude and direction of the electric field in the xy plane at
Chapter 23
The Electric Field II: Continuous Charge Distributions
x = 1.5 m, y = 0.5 m. 1. Find Eplane at P, where P is at (1.5, 0.5) 2. Find Eline at P; Eline = 2k/r; r = 0.707 m 3. r from center of sphere to P = 0.707 m 4. E = Eplane + Eline + Esphere
Eplane = (2106/28.851012) N/C i = 113 kN/C i Eline = 102 kN/C (cos 45 i sin 45 j) Esphere = (4/3)kr = 160 kN/C, directed to center of sphere; Esphere = 113 kN/C i 113 kN/C j E = 154 kN/C i 185 kN/C j o E = 241 kN/C, = 230
85* An infinite line charge is located along the z axis. A mass m that carries a charge q whose sign is opposite to that of is in a circular orbit in the xy plane about the line charge. Obtain an expression for the period of the orbit in terms of m, q, R, and , where R is the radius of the orbit. m 2 For a circular orbit mR = 2kq/R; so T = 1/f = 2/ = 2 R 2kq 86 A ring of radius R that lies in the yz plane carries a positive charge Q uniformly distributed over its length. A particle of mass m that carries a negative charge of magnitude q is at the center of the ring. (a) Show that if x << R, the electric field along the axis of the ring is proportional to x. (b) Find the force on the mass m as a function of x. (c) Show that if m is given a small displacement in the x direction, it will perform simple harmonic motion. Calculate the period of that motion. 3 3 (a) For x << R, Ex kQx/R (see Equ. 2310). (b) F = qEx = kqQx/R . (c) Assume that q and Q are of opposite 3 2 2 sign. Then Fx = (kQq/R )x = m(d x/dt ). This is the SHO equation of motion; the period is given by T = 2 m R3 k qQ
87 When the charges Q and q of Problem 86 are 5 C and 5 C, respectively, and the radius of the ring is 8.0 cm, the mass m oscillates about its equilibrium position with an angular frequency of 21 rad/s. Find the angular frequency of oscillation of the mass if the radius of the ring is doubled to 16 cm and all other parameters remain unchanged. 3/2 3/2 Note that T R (see Problem 86) and 1/T. Therefore doubling R must change by the factor 2 . 3/2 The new angular frequency is = 21/2 rad/s = 7.42 rad/s. 88 Given the initial conditions of Problem 87, find the angular frequency of oscillation of the mass if the radius of the ring is doubled to 16 cm while keeping the linear charge density on the ring constant. If the linear charge density remains constant, doubling R doubles Q. Consequently, since is now proportional to 1/R (see Problem 86), the new angular frequency is = 21/2 rad/s = 10.5 rad/s. 89* A nonconducting cylinder of radius 1.2 m and length 2.0 m carries a charge of 50 C uniformly distributed throughout the cylinder. Find the electric field on the cylinder axis at a distance of (a) 0.5 m, (b) 2.0 m, and (c) 20 m from the center of the cylinder. We shall first solve in general terms and then insert appropriate numerical values. Let the origin of coordinates be at the center of the cylinder. Now consider a disk of radius R, the radius of the cylinder, and thickness dx. The charge carried by that disk is dq = (Q/L)dx , where Q is the total charge of the cylinder and L its length.
Chapter 23
The Electric Field II: Continuous Charge Distributions
2
The disk has an effective surface charge density = Q/R L. We can now use Equ. 2311 to find the field due to this disk along its axis. If the point of interest, P, is within the cylinder, the charge to the left of P will result in a field to the right; the charge to the right of P will give a field to the left. Thus,
L / 2+ x E = 2k 1  0
E = 2k [2 x 
L / 2 x dx  1  2 2 x +R 0
x
dx . Performing the indicated integrations, one obtains x 2 + R 2 x
(L/2 + x )2 + R 2 +
(L/2  x )2 + R 2 ] .
If P is beyond the end of the cylinder, the field at that point is given by x+ L / 2 x 1  dx = 2k [L  (L/2 + x )2 + R 2 + (L/2  x )2 + R 2 ] as before. E = 2k 2 2 x +R xL / 2 We can now substitute numerical values. (a) For x = 0.5 m, E = 119 kN/C. (b) For x = 2 m, E = 103 kN/C. (c) For x = 20 m, E = 1.12 kN/C; note that 2 since the distance of 20 m is much greater than the length of the rod, we could have used Ex kQ/x = 1.12 kN/C. 90 A uniform line charge of density lies on the x axis between x = 0 and x = L. Its total charge is Q = 8 nC. The electric field at x = 2L is 600 N/C i. Find the electric field at x = 3L. 2 2 Note that for a uniform line charge Ex = kQ/[x0(x0  L)]. If x0 = 2L, Ex(2L) = kQ/2L ; if x0 = 3L, Ex(3L) = kQ/6L . Thus, Ex(3L) = Ex(2L)/3 = 200 N/C. 91 Find the linear charge density (in C/m) of the line charge of Problem 90. 2 1. Ex(2L) = kQ/2L (see Problem 90); solve for L L = k Q/2 E x 2. = Q/L = 2 E x Q/k ; evaluate = 32.7 nC/m
92 A uniformly charged sphere of radius R is centered at the origin with a charge of Q. Find the force on a uniformly charged line oriented radially having a total charge q with its ends at r = R and r = R + d. 2 E = kQ/r 1. Find the field of sphere at distance r R 2 2 2. Find force on line element dr dF = (kQ/r ) dr = (kQq/r d) dr 3. Integrate dF from r = R to r = R + d F = kQq/[R(R + d)] 93* Two equal uniform line charges of length L lie on the x axis a distance d apart as shown in Figure 2343. (a) What is the force that one line charge exerts on the other line charge? (b) Show that when d >> L, the force 2 2 tends toward the expected result of k(L) /d . (a) Take x = 0 to be at the left hand end of the left rod. Then the field at x > L is kQ/[x(x L)]. Now consider the right hand line charge. An element of charge in dx is dx and experiences a force Ex dx. The total force due to the left hand line charge on the right hand line charge is therefore given by 2 L+d (d + L )2 2 F = kL dx = k 1n x(x  L ) d (2 L + d ) L+ d 2 2 (b) For d >> L, the expression in the square brackets reduces to 1 + L /d to lowest order in L/d. We can now 2 2 2 2 2 2 use the expansion ln (1 + ) = /2 + . . ., and again keeping only the first term, obtain F = k L /d = kQ /d . 94 A dipole p is located at a distance r from an infinitely long line charge with a uniform linear charge density
Chapter 23
The Electric Field II: Continuous Charge Distributions
. Assume that the dipole is aligned with the field due to the line charge. Determine the force that acts on the dipole. 2 The force acting on a dipole is given by F = p(dE/dr). Here E = 2k/r, so F = 2kp/r , the minus sign indicating that the dipole is attracted to the line charge. 2 95 Suppose that the charge on the rod in Problem 81 is given by (y) = ay , where y is the distance from the midpoint of the rod, and that the total charge on the rod is Q. (a) Determine the constant a. (b) Find the force dF that acts on an element of charge (y) dy. (c) Integrate the force obtained in part (b) between L/2 and L/2 to obtain the total force that acts on the rod.
L/2
(a) The element of charge is dq = ay dy. So Q =
2
 L/2
ay
2
dy = a L3 /12 and a = 12Q/L .
3
(b) The field at y due to the infinite line charge is given by E(y) = 2k/(b + y), where b = L/2 + d. 2 2 The force dF is then E(y)ay dy = [2kay /(y + b)] dy. (c) F = 2ka y2 dy , where c = L/2. The integral between the stated limits is y+b c
c
L L L + d L b+c L L+d I = b 2 1n  L + d 0 = + d + d ln  2bc = + d 1n  L 2 2 bc 2 2 2 d
2
and F = 2 k a
L L L + d 3 + d + d ln  L 0 , where a = 12Q/L . 2 2 d
96 Repeat Problem 95 with the charge on the rod being (y) = by, where y is measured from the midpoint of the rod with the positive y direction up. Following the same procedure as in Problem 95, we find: (a) Q = 0 and the constant b cannot be related to the total charge as in Problem 95. (b) The field is given by the same expression as in Problem 95; i.e., E(y) = 2k/(p + y), where p = L/2 + d. (c) dF = [2kby/(y + p)] dy. (d) Let c = L/2. Then F = 2kb y + p dy = 2kb1n p  c = 2kb1n
c
y
p+c
c
L+d . d