Question

A solid uniformly charged fixed non conducting sphere of total charge $Q$ and radius $R$ contains a tunnel of negligible diameter. If a point charge $-q$ of mass ${}^{\prime }{m}^{\prime }$ is released at rest from point $P$ as shown in figure then its velocity at the surface of the sphere is found to be $\sqrt{(\frac{nqQ}{12\pi {\epsilon }_{0}Rm})}$. Find the value of $n$ (Give integer value)

Answer 1

Let the potential at $P$ be $V_1$ and at the surface be $V_2$

$V_1=\frac{KQ}{r}=\frac{KQ}{2R}$ and $V_2=\frac{KQ}{R}$

Work done by electrostatic field, $W=q\Delta V$

$W=-q(\frac{KQ}{2R}-\frac{KQ}{R})=\frac{kqQ}{2R}$

Using work energy theorem
$W_{net}$= Change in kinetic energy

$\frac{KqQ}{2R}=\frac{1}{2}m{v}^2-0$

$\Rightarrow v=\sqrt{\frac{kqQ}{Rm}}=\sqrt{\frac{qQ}{4\pi {\epsilon }_{0}Rm}}$

$\therefore v=\sqrt{\frac{3qQ}{12\pi {\epsilon }_{0}Rm}}$

Comparing with the data given in the question we get,$n=3$

Accepted answer : $3$