A hollow uncharged spherical conducting shell has an inner radius a and an outer radius b. A positive point charge q is in the cavity at the center of the sphere.

(a) Find the charge on each surface of the conductor
(surface a and surface b).

(b) Find the electric field everywhere.

(c) Find the potential everywhere, assuming that V = 0 at infinity.

Solution:

- Concepts:

Gauss' law, properties of conductors - Reasoning:

The field due to a spherically symmetric charge distribution can be found from Gauss' law.

The inside of a conductor is field free in electrostatics. - Details of the calculation:

(a) Charge on surface a: -q

Charge on surface b: q

(b) 0 < r < a:**E**= kq/r^{2}(**r**/r)

a < r < b:**E**= 0

r > b:**E**= kq/r^{2}(**r**/r)

(c) r > b: V = kq/r

a < r < b: V = kq/b

0 < r < a: V = kq/b - kq/a + kq/r = kq(1/r + 1/b - 1/a)

A spherical region of space of radius a contains a
charge Q which is uniformly distributed within the volume

(a) Use Gauss's law to determine the magnitude of the
electric field at any radius r from the center of the sphere.

(b) The total electrostatic energy of the sphere may be
calculated from the electric field, using

U = (ε_{0}/2)∫_{all_space}^{
}**E·E **dV
(SI units).

Evaluate this expression for the uniformly charged sphere.

(c) Calculate the work
required to bring a test charge +q from infinity to the center of the
sphere, using dW = **F· **d**r** = +qEdr.

Solution:

- Concepts:

Gauss' law, electrostatic energy - Reasoning:

A radial field is produced by a spherically symmetric charge distribution. - Details of the calculation:

(a) From Gauss' law we have E(r) = Q/(4πε_{0}r^{2}) for r > a, and E(r) = ρr/(3ε_{0}) for r < a. Here ρ = 3Q/4πa^{3}. Therefore E(r) = Qr/(4πa^{3}ε_{0}) for r < a. At r = 0 we have E(r) = 0.

(b) U = (4πε_{0}/2)[ ∫_{0}^{a}r^{2}dr Q^{2}r^{2}/(4πa^{3}ε_{0})^{2}+ ∫_{a}^{∞}r^{2}dr Q^{2}/(4πε_{0}r^{2})^{2}]

= (Q^{2}/(8πε_{0}))[ ∫_{0}^{a}r^{4}dr/a^{6}+ ∫_{a}^{∞}dr /r^{2}] = [Q^{2}/(8πε_{0})][1/(5a) + 1/a] = (3/5) Q^{2}/(4πε_{0}a).

(c) W = q[∫_{0}^{a}dr Qr/(4πa^{3}ε_{0}) + ∫_{a}^{∞}dr Q/(4πε_{0}r^{2})] = [qQ/(4πε_{0})][1/(2a) + 1/a]

= 3qQ/(8πε_{0}a).