Chapter 24 Problems (b) the slanted surface, and (c) the entire
surface of the box.
1, 2, 3 = straightforward, intermediate,
Section 24.1 Electric Flux
1. An electric field with a magnitude of
3.50 kN/C is applied along the x axis. Calculate the electric flux through a
rectangular plane 0.350 m wide and 0.700 m Figure P24.4
long assuming that (a) the plane is parallel
to the yz plane; (b) the plane is parallel to 5. A uniform electric field aî + bĵ
the xy plane; (c) the plane contains the y intersects a surface of area A. What is the
axis, and its normal makes an angle of 40.0? flux through this area if the surface lies (a) with the x axis. in the yz plane? (b) in the xz plane? (c) in
the xy plane?
2. A vertical electric field of magnitude 42.00 × 10 N/C exists above the Earth’s 6. A point charge q is located at the
surface on a day when a thunderstorm is center of a uniform ring having linear brewing. A car with a rectangular size of charge density λ and radius a, as shown in
6.00 m by 3.00 m is traveling along a Figure P24.6. Determine the total electric roadway sloping downward at 10.0?. flux through a sphere centered at the point Determine the electric flux through the charge and having radius R, where R < a.
bottom of the car.
3. A 40.0-cm-diameter loop is rotated in
a uniform electric field until the position of
maximum electric flux is found. The flux in
5this position is measured to be 5.20 × 10
2N ? m/C. What is the magnitude of the electric field?
4. Consider a closed triangular box
resting within a horizontal electric field of 7. A pyramid with horizontal square 4magnitude E = 7.80 × 10 N/C as shown in base, 6.00 m on each side, and a height of Figure P24.4. Calculate the electric flux 4.00 m is placed in a vertical electric field of through (a) the vertical rectangular surface, 52.0 N/C. Calculate the total electric flux
through the pyramid’s four slanted surfaces.
11. Four closed surfaces, S1 through S4,
8. A cone with base radius R and together with the charges –2Q, Q, and –Q
height h is located on a horizontal table. A are sketched in Figure P24.11. (The colored horizontal uniform field E penetrates the lines are the intersections of the surfaces cone, as shown in Figure P24.8. Determine with the page.) Find the electric flux the electric flux that enters the left-hand through each surface.
side of the cone.
Section 24.2 Gauss’s Law Figure P24.11
9. The following charges are located 12. (a) A point charge q is located a
inside a submarine: 5.00 μC, –9.00 μC, 27.0 distance d from an infinite plane. Determine μC, and –84.0 μC. (a) Calculate the net the electric flux through the plane due to electric flux through the hull of the the point charge. (b) What If? A point
submarine. (b) Is the number of electric charge q is located a very small distance
field lines leaving the submarine greater from the center of a very large square on the
than, equal to, or less than the number line perpendicular to the square and going entering it? through its center. Determine the
approximate electric flux through the 10. The electric field everywhere on the square due to the point charge. (c) Explain surface of a thin spherical shell of radius why the answers to parts (a) and (b) are 0.750 m is measured to be 890 N/C and identical.
points radially toward the center of the
sphere. (a) What is the net charge within 13. Calculate the total electric flux the sphere’s surface? (b) What can you through the paraboloidal surface due to a conclude about the nature and distribution uniform electric field of magnitude E0 in the
of the charge inside the spherical shell? direction shown in Figure P24.13.
volume charge density in the layer of air
between these two elevations? Is it positive
17. A point charge Q = 5.00 μC is located at the center of a cube of edge L = 0.100 m. Figure P24.13 In addition, six other identical point charges having q = –1.00 μC are positioned 14. A point charge of 12.0 μC is placed at symmetrically around Q as shown in Figure the center of a spherical shell of radius 22.0 P24.17. Determine the electric flux through cm. What is the total electric flux through (a) one face of the cube. the surface of the shell and (b) any hemispherical surface of the shell? (c) Do the results depend on the radius? Explain.
15. A point charge Q is located just
above the center of the flat face of a hemisphere of radius R as shown in Figure
P24.15. What is the electric flux (a) through the curved surface and (b) through the flat face?
18. A positive point charge Q is located
at the center of a cube of edge L. In addition,
six other identical negative point charges q are positioned symmetrically around Q as
shown in Figure P24.17. Determine the Figure P24.15
electric flux through one face of the cube.
16. In the air over a particular region at
19. An infinitely long line charge having an altitude of 500 m above the ground the
a uniform charge per unit length λ lies a electric field is 120 N/C directed downward.
distance d from point O as shown in Figure At 600 m above the ground the electric field
P24.19. Determine the total electric flux is 100 N/C downward. What is the average
through the surface of a sphere of radius R
centered at O resulting from this line charge.
. Consider both cases, where R < d and R > d
Section 24.3 Application of Gauss’s Law
Figure P24.19 to Various Charge Distributions
20. An uncharged nonconducting 23. Determine the magnitude of the hollow sphere of radius 10.0 cm surrounds electric field at the surface of a lead-208 a 10.0-μC charge located at the origin of a nucleus, which contains 82 protons and 126 cartesian coordinate system. A drill with a neutrons. Assume the lead nucleus has a radius of 1.00 mm is aligned along the z volume 208 times that of one proton, and axis, and a hole is drilled in the sphere. consider a proton to be a sphere of radius
–15Calculate the electric flux through the hole. 1.20 × 10 m.
21. A charge of 170 μC is at the center of 24. A solid sphere of radius 40.0 cm has a cube of edge 80.0 cm. (a) Find the total a total positive charge of 26.0 μC uniformly
flux through each face of the cube. (b) Find distributed throughout its volume. the flux through the whole surface of the Calculate the magnitude of the electric field cube. (c) What If? Would your answers to (a) 0 cm, (b) 10.0 cm, (c) 40.0 cm, and (d) parts (a) or (b) change if the charge were 60.0 cm from the center of the sphere.
not at the center? Explain.
25. A 10.0-g piece of Styrofoam carries a 22. The line ag in Figure P24.22 is a net charge of –0.700 μC and floats above the diagonal of a cube. A point charge q is center of a large horizontal sheet of plastic located on the extension of line ag, very that has a uniform charge density on its close to vertex a of the cube. Determine the surface. What is the charge per unit area on electric flux through each of the sides of the the plastic sheet?
cube which meet at the point a.
26. A cylindrical shell of radius 7.00 cm 31. Consider a thin spherical shell of and length 240 cm has its charge uniformly radius 14.0 cm with a total charge of 32.0 distributed on its curved surface. The μC distributed uniformly on its surface. magnitude of the electric field at a point Find the electric field (a) 10.0 cm and (b) 19.0 cm radially outward from its axis 20.0 cm from the center of the charge (measured from the midpoint of the shell) distribution.
is 36.0 kN/C. Find (a) the net charge on the
shell and (b) the electric field at a point 4.00 32. In nuclear fission, a nucleus of cm from the axis, measured radially uranium-238, which contains 92 protons, outward from the midpoint of the shell. can divide into two smaller spheres, each
having 46 protons and a radius of 5.90 ×
-1527. A particle with a charge of -60.0 nC m. What is the magnitude of the 10
is placed at the center of a nonconducting repulsive electric force pushing the two spherical shell of inner radius 20.0 cm and spheres apart?
outer radius 25.0 cm. The spherical shell
carries charge with a uniform density 33. Fill two rubber balloons with air.
3of -1.33 μC/m. A proton moves in a circular Suspend both of them from the same point orbit just outside the spherical shell. and let them hang down on strings of equal Calculate the speed of the proton. length. Rub each with wool or on your hair,
so that they hang apart with a noticeable 28. A nonconducting wall carries a separation from each other. Make order-of-
2uniform charge density of 8.60 μC/cm. magnitude estimates of (a) the force on each, What is the electric field 7.00 cm in front of (b) the charge on each, (c) the field each the wall? Does your result change as the creates at the center of the other, and (d) the distance from the wall is varied? total flux of electric field created by each
balloon. In your solution state the 29. Consider a long cylindrical charge quantities you take as data and the values distribution of radius R with a uniform you measure or estimate for them.
charge density ρ. Find the electric field at
distance r from the axis where r < R. 34. An insulating solid sphere of radius
a has a uniform volume charge density and 30. A solid plastic sphere of radius 10.0 carries a total positive charge Q. A spherical
cm has charge with uniform density gaussian surface of radius r, which shares a
throughout its volume. The electric field common center with the insulating sphere, 5.00 cm from the center is 86.0 kN/C is inflated starting from r = 0. (a) Find an
radially inward. Find the magnitude of the expression for the electric flux passing electric field 15.0 cm from the center. through the surface of the gaussian sphere
as a function of r for r < a. (b) Find an
expression for the electric flux for r > a. (c) length of 30.0 nC/m. Find the electric field
Plot the flux versus r. (a) 3.00 cm, (b) 10.0 cm, and (c) 100 cm from
the axis of the rod, where distances are
measured perpendicular to the rod. 35. A uniformly charged, straight
filament 7.00 m in length has a total
40. On a clear, sunny day, a vertical positive charge of 2.00 μC. An uncharged
electric field of about 130 N/C points down cardboard cylinder 2.00 cm in length and
over flat ground. What is the surface charge 10.0 cm in radius surrounds the filament at density on the ground for these conditions? its center, with the filament as the axis of
the cylinder. Using reasonable
41. A very large, thin, flat plate of approximations, find (a) the electric field at aluminum of area A has a total charge Q the surface of the cylinder and (b) the total uniformly distributed over its surfaces. electric flux through the cylinder.
Assuming the same charge is spread
uniformly over the upper surface of an 36. An insulating sphere is 8.00 cm in
otherwise identical glass plate, compare the diameter and carries a 5.70-μC charge
electric fields just above the center of the uniformly distributed throughout its
upper surface of each plate. interior volume. Calculate the charge
enclosed by a concentric spherical surface 42. A solid copper sphere of radius 15.0 with radius (a) r = 2.00 cm and (b) r = 6.00
cm carries a charge of 40.0 nC. Find the cm.
electric field (a) 12.0 cm, (b) 17.0 cm, and (c)
75.0 cm from the center of the sphere. (d) 37. A large flat horizontal sheet of
What If? How would your answers change charge has a charge per unit area of 9.00
2if the sphere were hollow? μC/m. Find the electric field just above the
middle of the sheet.
43. A square plate of copper with 50.0-
cm sides has no net charge and is placed in 38. The charge per unit length on a long,
a region of uniform electric field of 80.0 straight filament is –90.0 μC/m. Find the
kN/C directed perpendicularly to the plate. electric field (a) 10.0 cm, (b) 20.0 cm, and (c) Find (a) the charge density of each face of 100 cm from the filament, where distances the plate and (b) the total charge on each are measured perpendicular to the length of face. the filament.
44. A solid conducting sphere of radius Section 24.4 Conductors in Electrostatic 2.00 cm has a charge of 8.00 μC. A Equilibrium
conducting spherical shell of inner radius
4.00 cm and outer radius 5.00 cm is 39. A long, straight metal rod has a
concentric with the solid sphere and has a radius of 5.00 cm and a charge per unit
total charge of –4.00 μC. Find the electric
–8field at (a) r = 1.00 cm, (b) r = 3.00 cm, (c) r = charge of 4.00 × 10 C is placed on the plate. 4.50 cm, and (d) r = 7.00 cm from the center Find (a) the charge density on the plate, (b) of this charge configuration. the electric field just above the plate, and (c)
the electric field just below the plate. You 45. Two identical conducting spheres may assume that the charge density is each having a radius of 0.500 cm are uniform.
connected by a light 2.00-m-long
conducting wire. A charge of 60.0 μC is 50. A conducting spherical shell of inner placed on one of the conductors. Assume radius a and outer radius b carries a net
that the surface distribution of charge on charge Q. A point charge q is placed at the
each sphere is uniform. Determine the center of this shell. Determine the surface tension in the wire. charge density on (a) the inner surface of
the shell and (b) the outer surface of the 46. The electric field on the surface of an shell.
irregularly shaped conductor varies from
56.0 kN/C to 28.0 kN/C. Calculate the local 51. A hollow conducting sphere is surface charge density at the point on the surrounded by a larger concentric spherical surface where the radius of curvature of the conducting shell. The inner sphere has surface is (a) greatest and (b) smallest. charge –Q , and the outer shell has net
charge +3Q. The charges are in electrostatic
equilibrium. Using Gauss’s law, find the 47. A long, straight wire is surrounded
charges and the electric fields everywhere. by a hollow metal cylinder whose axis
coincides with that of the wire. The wire
52. A positive point charge is at a has a charge per unit length of λ, and the
distance R/2 from the center of an cylinder has a net charge per unit length of
uncharged thin conducting spherical shell 2λ. From this information, use Gauss’s law
of radius R. Sketch the electric field lines set to find (a) the charge per unit length on the
up by this arrangement both inside and inner and outer surfaces of the cylinder and
outside the shell. (b) the electric field outside the cylinder, a
distance r from the axis.
Section 24.5 Formal Derivation of Gauss’s
Law 48. A conducting spherical shell of
radius 15.0 cm carries a net charge of –6.40
53. A sphere of radius R surrounds a μC uniformly distributed on its surface.
Find the electric field at points (a) just point charge Q , located at its center. (a) outside the shell and (b) inside the shell. Show that the electric flux through a
circular cap of half-angle θ (Fig. P24.53) is
49. A thin square conducting plate 50.0
cm on a side lies in the xy plane. A total
Qsurface. (b) What is the direction of the ;； ，？1？cos(E2electric field at r > c? (c) Find the electric ！0
field at r > c. (d) Find the electric field in the
region with radius r where c > r > b. (e) What is the flux for (b) θ = 90? and (c) θ =
Construct a spherical gaussian surface of 180??
radius r, where c > r > b, and find the net
charge enclosed by this surface. (f)
Construct a spherical gaussian surface of
radius r, where b > r > a, and find the net
charge enclosed by this surface. (g) Find the
electric field in the region b > r > a. (h)
Construct a spherical gaussian surface of
radius r < a, and find an expression for the
net charge enclosed by this surface, as a
function of r. Note that the charge inside
this surface is less than 3Q. (i) Find the
electric field in the region r < a. ( j)
Determine the charge on the inner surface
of the conducting shell. (k) Determine the
charge on the outer surface of the Figure P24.53
conducting shell. (l) Make a plot of the
magnitude of the electric field versus r. Additional Problems
54. A nonuniform electric field is given
ikby the expression E = ay + bz + cx, j
where a, b, and c are constants. Determine
the electric flux through a rectangular
surface in the xy plane, extending from x = 0
to x = w and from y = 0 to y = h.
55. A solid insulating sphere of radius a
carries a net positive charge 3Q , uniformly
distributed throughout its volume.
Figure P24.55 Concentric with this sphere is a conducting
spherical shell with inner radius b and
56. Consider two identical conducting outer radius c, and having a net charge –Q ,
spheres whose surfaces are separated by a as shown in Figure P24.55. (a) Construct a
small distance. One sphere is given a large spherical gaussian surface of radius r > c
net positive charge while the other is given and find the net charge enclosed by this
a small net positive charge. It is found that insulating sphere, (b) the net charge on the
the force between them is attractive even hollow conducting sphere, and (c) the
though both spheres have net charges of the charges on the inner and outer surfaces of
same sign. Explain how this is possible. the hollow conducting sphere.
59. A particle of mass m and charge q 57. A solid, insulating sphere of radius a
moves at high speed along the x axis. It is has a uniform charge density ρ and a total
initially near x = –?, and it ends up near x = charge Q. Concentric with this sphere is an
+?. A second charge Q is fixed at the point uncharged, conducting hollow sphere
x = 0, y = –d. As the moving charge passes whose inner and outer radii are b and c, as
the stationary charge, its x component of shown in Figure P24.57. (a) Find the
velocity does not change appreciably, but it magnitude of the electric field in the
acquires a small velocity in the y direction. regions r < a, a < r < b, b < r < c, and r > c. (b)
Determine the angle through which the Determine the induced charge per unit area
moving charge is deflected. Suggestion: The on the inner and outer surfaces of the
y integral you encounter in determining vhollow sphere.
can be evaluated by applying Gauss’s law
to a long cylinder of radius d, centered on the stationary charge.
60. Review problem. An early (incorrect) model of the hydrogen atom, suggested by
J. J. Thomson, proposed that a positive
cloud of charge +e was uniformly
distributed throughout the volume of a
sphere of radius R, with the electron an equal-magnitude negative point charge –e
at the center. (a) Using Gauss’s law, show
that the electron would be in equilibrium at Figure P24.57
the center and, if displaced from the center
a distance r < R, would experience a 58. For the configuration shown in
restoring force of the form F = –Kr, where K Figure P24.57, suppose that a = 5.00 cm, b =
23is a constant. (b) Show that K = kee/R. (c) 20.0 cm, and c = 25.0 cm. Furthermore,
Find an expression for the frequency f of suppose that the electric field at a point 10.0
simple harmonic oscillations that an cm from the center is measured to be 3.60 ×
3electron of mass me would undergo if 10 N/C radially inward while the electric
displaced a small distance (< R) from the field at a point 50.0 cm from the center is
2center and released. (d) Calculate a 2.00 × 10 N/C radially outward. From this
numerical value for R that would result in a information, find (a) the charge on the
15frequency of 2.47 × 10 Hz, the frequency of
the light radiated in the most intense line in
the hydrogen spectrum. 64. A sphere of radius 2a is made of a
nonconducting material that has a uniform 61. An infinitely long cylindrical volume charge density ρ. (Assume that the
insulating shell of inner radius a and outer material does not affect the electric field.) A radius b has a uniform volume charge spherical cavity of radius a is now removed
density ρ. A line of uniform linear charge from the sphere, as shown in Figure P24.64. density λ is placed along the axis of the Show that the electric field within the cavity shell. Determine the electric field is uniform and is given by Ex = 0 and Ey =
everywhere. ρa/3ε0. (Suggestion: The field within the
cavity is the superposition of the field due 62. Two infinite, nonconducting sheets to the original uncut sphere, plus the field of charge are parallel to each other, as due to a sphere the size of the cavity with a shown in Figure P24.62. The sheet on the uniform negative charge density –ρ.)
left has a uniform surface charge density σ,
and the one on the right has a uniform
charge density –σ. Calculate the electric
field at points (a) to the left of, (b) in
between, and (c) to the right of the two
65. A uniformly charged spherical shell
with surface charge density σ contains a
circular hole in its surface. The radius of the
hole is small compared with the radius of
the sphere. What is the electric field at the
center of the hole? (Suggestion: This
problem, like Problem 64, can be solved by using the idea of superposition.) Figure P24.62 66. A closed surface with dimensions a = 63. What If? Repeat the calculations for b = 0.400 m and c = 0.600 m is located as in Problem 62 when both sheets have positive Figure P24.66. The left edge of the closed uniform surface charge densities of value σ. surface is located at position x = a. The