For a uniformly charged ring of radius R , the electric field on its axis has the largest magnitude at a…

1. $\frac{R}{\sqrt{2}}$
2. $\frac{R}{\sqrt{5}}$
3. $R\sqrt{2}$
4. $R$

Solution

Electric field at a distance $h$ from centre on axis

$E=\frac{1}{4\pi {\epsilon }_0}\frac{Qh}{{\left(R^2+h^2\right)}^{\frac{3}{2}}}$

To get maximum $E,\frac{dE}{dh}=0$

$0=\frac{1}{4\pi {\epsilon }_0}\frac{Q{\left(R^2+h^2\right)}^{\frac{3}{2}}-Qh\frac{3}{2}{\left(R^2+h^2\right)}^{\frac{1}{2}}2h}{{\left(R^2+h^2\right)}^3}$

$\Rightarrow h=\pm \frac{R}{\sqrt{2}}$

Asked in: JEE Main 2019 (09 Jan Shift 1)

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