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. $\frac{Q^2}{2\sqrt{2}\pi {\epsilon }_0}$
• B. $\frac{Q^2}{4\pi {\epsilon }_0}\left(1+\frac{1}{\sqrt{5}}\right)$
• C. $\frac{Q^2}{4\pi {\epsilon }_0}\left(1+\frac{1}{\sqrt{3}}\right)$
• D. $\frac{Q^2}{4\pi {\epsilon }_0}$

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 $\frac{Q^2}{4\pi {\epsilon }_0}\left(1+\frac{1}{\sqrt{5}}\right)$