Question

A ring of radius $R$ is uniformly charged with linear charge density $\lambda$. A unit test charge of opposite sign is released from rest at a height $h>2R$ from the centre of the ring. The test charge moves along the axis. Ignore the effect of gravity. Then:

• A. The test charge will execute simple harmonic motion about the centre of the ring
• B. The magnitude of the force on the test charge when it moves towards the centre of the ring, increases and then decreases
• C. The magnitude of the force on the test charge kept on the axis and near the centre of the ring decreases with increasing radius of the ring
• D. The speed of the test charge at the centre of the ring is proportional to $\lambda$

Answer 1

Answer: (B), (C)

The correct options are
B The magnitude of the force on the test charge when it moves towards the centre of the ring, increases and then decreases
C The magnitude of the force on the test charge kept on the axis and near the centre of the ring decreases with increasing radius of the ring

Electric field of uniform ring on charge kept on air :-

$E_P=\frac{KQh}{{(h^2+R^2)}^{3/2}}$

As $h>2R$

$\Rightarrow h>>>R$

So, $E_P$ reduces to $=\frac{KQh}{h^3}$

$E_P=\frac{KQ}{h^2}$

Now, As $h$ reduces, $E_P$ increases.

And if $E_P=\frac{KQh}{{(h^2+R^2)}^{3/2}}$

and $R$ increases, then $E_P$ decreases.