Ncert Exercises & Solutions

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. 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)	q at the centre of the ring is in an unstable equilibrium within the plane of the ring for \(q>0\).

Choose the correct statement(s):
1. (a), (b), (c)
2. (a), (c), (d)
3. (b), (c), (d)
4. (c), (d)

(1) Hint: Use the concept of stable and unstable equilibrium.

Step 1: Find the electric field at the centre of the ring.

The positive charge Q is uniformly distributed at the surface of the ring. Thus, the electric field at the centre of the ring is zero.

So, the effect of the electric field on charge q due to the positive charge Q is zero.

Now, the only governing factor is the attractive and repulsive forces between charges (Q and q). There are two cases that arise:

Step 2: Find the condition of stable equilibrium.

Case I: When charge q > 0 i.e., q is a positive charge, there creates a repulsive force between charge q and Q. The repulsive forces of charge Q from all around the charge q will push it towards the centre if it is displaced from the centre of the ring.

Step 3: Find the condition of unstable equilibrium.

Case II: When charge q < 0 i.e., q is a negative charge, then there is an attractive force between charge Q and q. If q is shifted from the centre, then the positive charges nearer to this charge will attract it towards itself and charge q will never return to the centre.

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