Four equal point charges $Q$ each are placed in the $x - y$ plane at $(0, 2), (4, 2), (4, - 2)$ and $(0, - 2)$ . The work required to put a fifth charge $Q$ at the origin of the coordinate system will be

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$\frac{Q^{2}}{4 {πε}_{0}} (1 + \frac{1}{\sqrt{5}})$

$\frac{Q^{2}}{2 \sqrt{2} {πε}_{0}}$

$\frac{Q^{2}}{4 {πε}_{0}} (1 + \frac{1}{\sqrt{3}})$

$\frac{Q^{2}}{4 {πε}_{0}}$

answer is A.

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Detailed Solution

Required work = change in potential energy of the particle

$= U_{f} - U_{i} = \frac{Q^{2}}{4 {πε}_{0}} (\frac{1}{2} + \frac{1}{2} + \frac{1}{\sqrt{20}} + \frac{1}{\sqrt{20}}) - 0 = \frac{Q^{2}}{4 {πε}_{0}} (1 + \frac{1}{\sqrt{5}})$

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