Question

A charge $+Q$ is uniformly distributed over a thin ring of radius $R$. Find the velocity of a negative point charge $-Q$ at the moment it passes through the centre of the ring if it has mass $m$ and was initially at rest at infinity on the axis of the ring.

• A. $\sqrt{\frac{2kQ}{mR}}$
• B. $Q\sqrt{\frac{2k}{mR}}$
• C. $2Q\sqrt{\frac{k}{mR}}$
• D. $0$

Answer 1

The correct option is B $Q\sqrt{\frac{2k}{mR}}$

Given that,
Radius of the ring $=R$
Mass of charge $-Q=m$

Let the velocity of the charge $-Q$ is $v$ when it passes through the centre of the ring.

The potential energy at the centre of the ring, $U=\frac{k\left(Q\right)(-Q)}{R}=\frac{-k{Q}^2}{R}$

This is the final potential energy, so, $U_f=\frac{-k{Q}^2}{R}$

The initial potential energy when charge $-Q$ is at infinite distance away from the ring is zero. So, $U_i=0$.


Now, we know from conservation of Mechanical energy for the system,
Decrease in potential energy $=$ Increase in kinetic energy

$\Rightarrow 0-\left(\frac{-k{Q}^2}{R}\right)=\frac{1}{2}m{v}^2-0$

$\Rightarrow \frac{1}{2}m{v}^2=\frac{k{Q}^2}{R}$

$\Rightarrow v=Q\sqrt{\frac{2k}{mR}}$

Hence, option (b) is correct