Question

Let's consider a solid nonconducting sphere with radius a. It has a uniform +Q charge distribution in its volume. A gold layer (conducting) with negligible thickness covers the sphere. A total charge of -2Q is placed on this layer. a) What is the electric field inside the sphere?
b) What is the electric field outside the sphere?

Answer 1

a) Inside the sphere:

The charge density inside the sphere is

$\frac{Q }{4/3 \pi a^3}$

The charge contained in the sphere of radius r (where r < a) is charge density times volume of the sphere of radius r.

$q=\frac{Q r^3}{a^3}$

The electric field at r from the center of the sphere(inside the sphere) is therefore

$E=\frac{q}{4\pi\epsilon_o r^2}=\frac{Q\,r}{4\pi\epsilon_o a^3}$

b) The total charge contained on the solid sphere and the gold layer is Q -2Q = -Q

Therefore the electric field outside the sphere is

E -Q 2 TET

(negative sign means the field is directed inwards)