Question

A charged $Q$ is placed at the centre of an uncharged, hollow metallic sphere of radius $a$. (a) Find the surface charge density on the inner surface and on the outer surface. (b) If a charge $q$ is put on the sphere, what would be the surface charges densities on the inner and the outer surfaces? (c) Find the electric field inside the sphere at a distance $x$ from the centre in the situations (a) and (b).

Answer 1

Given : The sphere is uncharged metallic sphere.

Due to induction the charge induced at the inner surface $=-Q$, and that outer surface $=+Q$

(a) Hence the surface charge density at inner and outer surface $=\frac{\text{charge}}{\text{total surface area}}$

$=-\frac{Q}{4\pi a^2}$ and $\frac{Q}{4\pi a^2}$ respectively.

(b) Again if another charge '$q$' is added to the surface. We have inner surface charge density $=\frac{Q}{4\pi a^2}$, because the added charge does not affect it.

On the other hand external surface charge density $Q+\frac{q}{4\pi a^2}$ as the '$q$' gets added up.

(c) For electric field let us assume an imaginary surface area inside the sphere at a distance '$x$' from centre. This is same in both the cases as the '$q$' in ineffective.

Now, $\oint Eds=\frac{Q}{{\epsilon }_0}$

So, $E=\frac{Q}{{\epsilon }_0}\times \frac{1}{4\pi x^2}=\frac{Q}{4\pi {\epsilon }_0{x}^2}$