The maximum electric field intensity on the axis of a uniformly charged ring of charge q and radius R will be :

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$\frac{1}{4 π ε_{0}} \frac{q}{3 \sqrt{3} R^{2}}$

$\frac{1}{4 π ε_{0}} \frac{3 q}{2 \sqrt{2} R^{2}}$

$\frac{1}{4 π ε_{0}} \frac{2 q}{3 R^{2}}$

$\frac{1}{4 π ε_{0}} \frac{2 q}{3 \sqrt{3} R^{2}}$

answer is C.

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Detailed Solution

$E = \frac{kqx}{{(R^{2} + x^{2})}^{3 / 2}},$ for maximum $E, \frac{dE}{dx} = 0$

$\Rightarrow \frac{kq (1)}{{(R^{2} + x^{2})}^{3 / 2}} + (- \frac{3}{2}) \frac{kqx}{{(R^{2} + x^{2})}^{5 / 2}} (2 x) = 0$

$\Rightarrow R^{2} + x^{2} = 3 x^{2} \Rightarrow x^{2} = \frac{R^{2}}{2}$

$\Rightarrow E_{\max} = \frac{kq (R / \sqrt{2})}{{(R^{2} + \frac{R^{2}}{2})}^{3 / 2}} = \frac{2 kq}{3 \sqrt{3} R^{2}}$

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