First slide

Electric field

Question

A ring of radius R has charge -Q distributed uniformly over it. A charge q is placed at the centre of the ring such that the electric field becomes zero at a point on the axis of the ring distant R from the centre of the ring. The value of charge q is

Moderate

A

$\frac{Q}{2} \sqrt{3}$
B

$\frac{Q}{4} \sqrt{2}$
C

$\frac{Q}{3} \sqrt{2}$
D

$\frac{Q}{4} \sqrt{3}$

Solution

Electric field at point P due to charge of ring is
$E = \frac{kQx}{{(R^{2} + x^{2})}^{3 / 2}}$
$At x = R : E = \frac{kQ}{2 \sqrt{2} R^{2}} directed towards the centre .$
Electric field at P due to centre charge: $\frac{kg}{R^{2}}$
For net field to be zero at P:
$\frac{kq}{R^{2}} = \frac{kQ}{2 \sqrt{2} R^{2}} \Rightarrow q = \frac{Q}{2 \sqrt{2}} = \frac{Q}{4} \sqrt{2}$ .

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