First slide

Electric field

Question

Two concentric rings, one of radius R and total charge +Q and the second of radius 2R and total charge $- \sqrt{8} Q$ , lie in x-y plane (i.e., z = 0 plane). The common center of rings lies at origin and the common axis coincides with z-axis. The charge is uniformly distributed on both rings. At what distance from origin is the net electric field on z-axis zero

Moderate

A

$\frac{R}{2}$
B

$\frac{R}{\sqrt{2}}$
C

$\frac{R}{2 \sqrt{2}}$
D

$\sqrt{2} R$

Solution

Electric field at a point on z-axis distant r from origin is
$E = \frac{1}{4 π ε_{o}} (\frac{Q r}{{(r^{2} + R^{2})}^{3 / 2}} - \frac{\sqrt{8} Q r}{{(r^{2} + 4 R^{2})}^{3 / 2}}) = 0 \Rightarrow \frac{1}{{(r^{2} + R^{2})}^{3}} = \frac{8}{{(r^{2} + 4 R^{2})}^{3}} \Rightarrow \frac{1}{(r^{2} + R^{2})} = \frac{2}{(r^{2} + 4 R^{2})} \Rightarrow r^{2} + 4 R^{2} = 2 r^{2} + 2 R^{2}$
$\Rightarrow r = \sqrt{2} R$

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