Question

A dipole is placed at the centre of a non-conducting spherical shell of radius $r$. If the shell is uniformly charged by a surface charge density $\sigma$, then the interaction energy of dipole and the shell is
(The dipole moment is $p$)

• A. $\frac{p\sigma }{4{ϵ}_0}$
• B. $\frac{p\sigma }{{ϵ}_{0}r}$
• C. $\frac{p\sigma }{{ϵ}_0{r}^2}$
• D. zero

Answer 1

The correct option is D zero

Electric filed inside a uniformly charged shell is zero and electric potential is constant at a point inside the shell.

So, $V_{in}=V=$ constant


The separation $a$ between the constituent charges is so small, then it can be considered to be at centre.

$\Rightarrow V_{+q}=V_{-q}=V$

The potential for both charged is equal. The P. E $\left(U\right)$ of dipole at centre is

$U_c=U_{q+}+U_{q-}$

$\Rightarrow U_c=(+q)V+(-q)V$

$\Rightarrow U_c=qV-qV=0$

Work done is bringing the dipole from infinity to the centre of shell is

$W_{ext}=U_f-U_i=U_c-U_{\infty }$

$(\text{at}\infty ;U_{\infty }=0)$

$\Rightarrow W_{net}=0-0=0$

Since no work has been done by external force, the interaction energy of the dipole and the shell is zero.

Hence, option $\left(d\right)$ is the correct answer.

Why this question? Tip : If a charge configuration is assembled, then $W_{net}$ in bringing them from $\infty$ to their position gives rise to interaction energy of system.