Question

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.

Answer 1

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