Question

A silver coated sphere of radius 5 cm carries a total charge of 12 nC uniformly
distributed on its surface in free space. Calculate (a) |D| on the surface of the
sphere. (b) D external to the sphere and (c) total energy stored in the field.

Answer 1

(a) According to Gauss's law:

$$\int D\cdot d S=q_{en},\\\space\\ D=\frac{q_{en}}{4\pi r^2}=3.82\cdot10^{-7}\text{ C/m}^2.$$

(b) D external to the sphere can be found for a>r:

$$D=\frac{q_{en}}{4\pi a^2}=9.55\cdot10^{-10}/a^2\text{ C/m}^2.$$

(c) The energy is

$$E=\frac{1}{2}\int D\cdot EdV=\frac{1}{2}\int\epsilon_0E^2dV=\\\space\\ =13\space\mu\text{J}.$$