(a) An insulating sphere with radius a has a uniform charge density ρ . The sphere is not centered at the origin but at r → = b → . Show that the electric field inside the sphere is given by E → = ρ ( r → − b → ) / 3 ∈ 0 . (b) An insulating sphere of radius R has a spherical hole of radius a located within its volume and centered a distance b from the center of the sphere, where a < b < R (a cross section of the sphere is shown in Fig. P22.57 ). The solid part of the sphere has a uniform volume charge density ρ . Find the magnitude and direction of the electric field E → inside the hole, and show that E → is uniform over the entire hole. [ Hint: Use the principle of superposition and the result of part (a).] Figure P22.57
(a) An insulating sphere with radius a has a uniform charge density ρ . The sphere is not centered at the origin but at r → = b → . Show that the electric field inside the sphere is given by E → = ρ ( r → − b → ) / 3 ∈ 0 . (b) An insulating sphere of radius R has a spherical hole of radius a located within its volume and centered a distance b from the center of the sphere, where a < b < R (a cross section of the sphere is shown in Fig. P22.57 ). The solid part of the sphere has a uniform volume charge density ρ . Find the magnitude and direction of the electric field E → inside the hole, and show that E → is uniform over the entire hole. [ Hint: Use the principle of superposition and the result of part (a).] Figure P22.57
(a) An insulating sphere with radius a has a uniform charge density ρ. The sphere is not centered at the origin but at
r
→
=
b
→
. Show that the electric field inside the sphere is given by
E
→
=
ρ
(
r
→
−
b
→
)
/
3
∈
0
.
(b) An insulating sphere of radius R has a spherical hole of radius a located within its volume and centered a distance b from the center of the sphere, where a < b < R (a cross section of the sphere is shown in Fig. P22.57). The solid part of the sphere has a uniform volume charge density ρ. Find the magnitude and direction of the electric field
E
→
inside the hole, and show that
E
→
is uniform over the entire hole. [Hint: Use the principle of superposition and the result of part (a).]
A certain region of space bounded by an imaginary closed surface contains no charge. Is the electric field always zero everywhere on the surface? If not, under what circumstances is it zero on the surface?
A spherical ball of charged particles has a uniform charge density. In terms of the ball’s radius R, at what radial distances (a) inside and (b) outside the ball is the magnitude of the ball’s electric field equal to of the maximum magnitude of that field?
A solid insulating sphere of radius 0.07 m carries a total charge of 25 µC. Concentric with this sphere is a conducting spherical shell of inner radius 0.12 m and outer radius of 0.18 m and carrying a total charge of -54 µC. Find (a) the charge distribution for the insulating sphere and the conducting spherical shell, and the magnitude of the electric field at the following distances from the center of the two spheres and shell: (b) 0.05 m, (c) 0.10 m, (d) 0.15 m, and (e) 0.25 m.
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