Question

Four equal charges q each are placed at four corners of a square of side a each. Work done in carrying a charge q from its centre to infinity is:

• A. zero
• B. $\frac{2q^2}{\pi {\epsilon }_{0}a}$
• C. $\frac{\sqrt{2}{q}^2}{\pi {\epsilon }_{0}a}$
• D. $\frac{q^2}{2\pi {\epsilon }_{0}a}$

Answer 1

The correct option is C $\frac{\sqrt{2}{q}^2}{\pi {\epsilon }_{0}a}$

Potential at center $,O=V_o=4(\frac{kq}{r})$

Here, $r=\frac{\text{(diagonal)}}{2}=(\frac{a}{\sqrt{2}})$

$\therefore V_o=4\times (\frac{\sqrt{2}q}{4\pi {\in }_{0}a})=(\frac{\sqrt{2}q}{\pi {ϵ}_{0}a})...............................\left(1\right)$

Work done in shifting q from centre to infinity $=W$

$W=–q(V_{\infty }–V_o)=qVo=q\times (\frac{\sqrt{2}q}{\pi {ϵ}_{0}a})$.................from (1)

$W=(\frac{\sqrt{2}{q}^2}{\pi {ϵ}_{0}a})$

So, correct option is $\left(C\right)$