Question

A small metal sphere carrying charge +Q is located at the centre of a spherical cavity in a large uncharged metallic spherical shell. Write the charges on the inner and outer surfaces of the shell. Write the expression for the electric field at the point $P_1$ is $E=\frac{xQ}{4\pi {ϵ}_0{r}_1^2}$. Then the value of x is :

Answer 1

The uniform charge $-Q$ will be induced on inner surface of the shell and $+Q$ will be induced on outer surface. This is follows from conservation of charge and no static charges resides in the interior of a metal in electrical equilibrium.
Using Gauss's law the field at $P_1$: $E.4\pi r_1^2=\frac{Q_{en}}{{ϵ}_0}$ where $Q_{en}=+Q$, charge inside gaussian surface of radius $r_1$.
Thus, $E=\frac{Q}{4\pi {ϵ}_0{r}_1^2}$