A ring of radius $R$ has charge $- q$ distributed uniformly over it. Calculate the charge that should be placed at the centre of the ring such that the electric field becomes zero at a point on the axis of the ring at a distance $R$ from the centre of the ring.

Question

Verified by Toppr

Answer 1

Electric field point at point $P$ due to charge of ring is
$E = \frac{k Q x}{( R ^{2} + x ^{2} ) ^{3 / 2}}$
At $x = R; E = \frac{k Q}{2 2 R ^{2}}$ , directed towards the centre
Electric field at $P$ due to charge at centre: $\frac{k q}{R ^{2}}$
For net field to be zero at $P$ :
$\frac{k q}{R ^{2}} = \frac{k Q}{2 2 R ^{2}} \Rightarrow q = \frac{Q}{2 2}$

Answer 2

Electric field point at point

P

due to charge of ring is

E = \frac{k Q x}{( R ^{2} + x ^{2} ) ^{3 / 2}}

At

x = R; E = \frac{k Q}{2 2 R ^{2}}

, directed towards the centre
Electric field at

P

due to charge at centre:

\frac{k q}{R ^{2}}

For net field to be zero at

P

:

\frac{k q}{R ^{2}} = \frac{k Q}{2 2 R ^{2}} \Rightarrow q = \frac{Q}{2 2}