Question

A circular ring of radius $R$ and uniform linear charge density $+\lambda$ $C/m$ are kept in $x-y$ plane with its centre at the origin. The electric field at a point $\left(0,0,\frac{R}{\sqrt{2}}\right)$ is

• A. $\frac{\lambda }{3\sqrt{3}{ϵ}_{0}R}$
• B. $\frac{2\lambda }{3\sqrt{3}{R}^2}$
• C. $\frac{2}{3\sqrt{3}}\frac{\lambda }{{ϵ}_{0}R}$
• D. none of these

Answer 1

Answer: (A)

$E_{\text{axis }}=\frac{1}{4\pi {\epsilon }_0}\frac{\text{ qz }}{{\left(R^2+z^2\right)}^{\frac{3}{2}}},$ at $z=\frac{R}{\sqrt{2}}$
$\Rightarrow E_{axis}=\frac{2}{3\sqrt{3}}\frac{1}{4\pi {ϵ}_0}\frac{q}{R^2}$
$\Rightarrow E_{axis}=\frac{2}{3\sqrt{3}}.\frac{1}{4\pi {ϵ}_0}.\frac{\lambda .2\pi R}{R^2}$
$\Rightarrow E_{axis}=\frac{\lambda }{3\sqrt{3}{ϵ}_{0}R}$