Question

Four equal point charges $Q$ each are placed in the $xy$ 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 :

• A. $\frac{Q^{2}}{2\sqrt{2}\pi\varepsilon_{0}} $
• B. $\frac{Q^{2}}{4\pi\varepsilon_{0}} \left(1+ \frac{1}{\sqrt{5}}\right)$
• C. $\frac{Q^{2}}{4\pi\varepsilon_{0}} \left(1+ \frac{1}{\sqrt{3}}\right)$
• D. $\frac{Q^{2}}{4\pi\varepsilon_{0}}$

Answer 1

Answer: (B)

Potential at origin = $\frac{KQ}{2} + \frac{KQ}{2} +\frac{KQ}{\sqrt{20}} + \frac{KQ}{\sqrt{20}}$
(Potential at $\infty$ = 0) $ =KQ \left(1+ \frac{1}{\sqrt{5}}\right) $
$\therefore $ Work required to put a fifth charge $Q$ at origin is equal to $ \frac{Q^{2}}{4\pi\varepsilon_{0}} \left(1+ \frac{1}{\sqrt{5}}\right) $