Question

A charge Q is placed at the centre of uncharged, hollow metallic sphere of radius a. (a) Find 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 charge densities on the inner and the outer surfaces ? (c) Find the electric field inside the sphere at a distance x form the centre in the situations (a) and (b).

Answer 1

Given : A sphere is uncharged metallic sphere. Due to induction the charge induced at the inner surface =-q and that in outer surface =+q

(a) Hence the surface charge density at inner and outer surface

$=\frac{charge}{\text{total surface area}}$

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

(b) Again if another surface charge Q is added to the surface,

We have inner surface charge density

$=-\frac{q}{4\pi a^2}$.

Because the added charged does not affect it on the other hand the external surface

charge density $=\frac{q+Q}{4\pi a^2}$ as the Q gets added up.

(c) For electric field let us assume in imaginary surface x from centre this is same in both the cases as the q is ineffective.

Now , E.ds=$\frac{q}{{\in }_0}$

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