Question

A positive charge $Q$ is uniformly distributed along a circular ring of radius $R$. A small test charge $q$ is placed at the centre of the ring as shown in figure. Then

• A. if $q>0$, and is displaced away from the centre in the plane of the ring, it will be pushed back towards the centre
• B. if $q<0$ and is displaced away from the centre in the plane of the ring, it will never return to the centre and will continue moving till it hits the ring
• C. if $q<0$ it will perform $SHM$ for small displacement along the axis
• D. all of the above

Answer 1

Answer: (D)

At the centre of the ring, $E=0$ when a positive charge $q>0$ is displaced away from the centre in the plane of the ring, say to the right, force of repulsion on $q$, due to charge on right half increases and due to charge on left half decreases. Therefore, charge $q$ is pushed back towards the centre. So option (a) is correct. When charge $q$ is negative $(q<0)$, force is of attraction.

Therefore, charge q displaced to the right continues moving to the right till it hits the ring. Along the axis of the ring, at a distance $r$ from the centre.
$E=\frac{Qr}{4\pi {\epsilon }_0{\left(r^2+a^2\right)}^{3/2}}$
If charge $q$ is negative $(q<0)$, it will perform $SHM$ for small displacement along the axis.