Question

In a hypothetical situation if a point charge $q$ is brought from infinity (slowly so that heat developed in the shell is negligible) and placed at the centre of a conducting neutral spherical shell of inner radius $a$ and outer radius $b$, then work done by external agent is calculated to be $\frac{k{a}^2}{na}-\frac{k{q}^2}{2b}$ .Find the value of $n$ (Give integer value)

Answer 1

When charge is at infinity, potential energy of the system is zero.

So, $U_i=0$

When charge is placed at centre, induced Charges will appear on the shell as shown


The final potential energy of the system will be:

$U_f$ = Self energy of $S_1$ and $S_2$ +Interaction energy of $S_1$ and $S_2$ +Interaction energy of $q$ with $S_1$ +Interaction energy of $q$ with $S_2$
$U_f=(\frac{K(-q{)}^2}{2a}+\frac{K(q{)}^2}{2b})+\left(\frac{K(-q)\left(q\right)}{b}\right)+\left(\frac{K(-q)q}{a}\right)+\frac{K{q}^2}{b}$

$\Rightarrow U_f=\frac{k{q}^2}{2a}+\frac{k{q}^2}{2b}-\frac{k{q}^2}{b}-\frac{k{q}^2}{a}+\frac{k{q}^2}{b}$

So, Work done $W=U_f-U_i$
$W=\frac{k{q}^2}{2b}-\frac{k{q}^2}{2a}$

Comparing with the data given in the question we get,$n=2$

Accepted answer : $2$