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

Answer 1

Answer: (A)

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