Question

A hollow conducting spherical shell of inner radius $R_1$ and outer radius $R_2$ encloses a charge $q$ inside, which is located at a distance $d,(d<R_1$) from the centre of the sphere. (assume potential to be zero at infinity) :-

• A. The potential at the centre of the shell is $\frac{kq}{d}-\frac{kq}{R_1}$
• B. The potential of the shell is $\frac{kq}{R_2}$
• C. The potential outside the shell at a distance R from the centre is$\frac{kq}{R}+\frac{kq}{d}-\frac{kq}{R_1}$
• D. The potential at the centre of the shell is $\frac{kq}{d}-\frac{kq}{R_1}+\frac{kq}{R_2}$

Answer 1

Answer: (B), (D)

The potential at the centre of the shell is $\frac{kq}{d}-\frac{kq}{R_1}+\frac{kq}{R_2}$

Charges induced on the shell can be consideres as shown below,


Potential at centre =Potential due to charge $q$ + potential due to induced charge +Potential due to charge $q$ on outer surface.

$V_0=\frac{kq}{d}-\frac{kq}{R_1}+\frac{kq}{R_2}$

Electric outside the shell at a distance of $x$ from centre is
$\frac{kq}{x^2}$

$V_s-V_{\infty }={\int }_{\infty }^{R_2}\frac{kq}{x^2}dx$
$V_s=\frac{kq}{R_2}$

For the point outside the shell
$V_{out}=\frac{kq}{R}$
where $R$ is the distance from the centre.