Question

A spherical conducting sphere of inner radius $R_1$ and outer radius $R_2$ has a charge $Q$.A charge $q$ is placed at the centre of the shell.What is the surface charged density on the

$\left(i\right)$inner surface

$\left(ii\right)$outer surface of the shell?

What is the expression for the electric field at a point to $x>R_2$ from the centre of the shell?

Answer 1

$+Q$ charge is on the outer surface of the shell.

Charge $q$ is placed at the centre, it induces a charge $-q$ on the inner surface and charge $+q$ on the outer surface of the shell.

Total charge on the inner surface$=-q$

${\sigma }_{inner}=\frac{-q}{4\pi r_1^2}$

Total charge on the outer surface$=Q+q$

${\sigma }_{outer}=\frac{Q+q}{4\pi r_2^2}$

Consider a Gaussian surface which is a sphere of radius $r>r_2$ concentric with the given sphere.

Applying Gauss's law,$\oint \overrightarrow{E}.d\overrightarrow{S}=\frac{A+q}{{ϵ}_0}$

$\overrightarrow{E}=\frac{Q+q}{4\pi r_2^2{ϵ}_0}$