Question

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

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

Answer 1

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

$\text{Step 1: Potential at point O}$

Potential due to a point charge, $V$ = $\frac{Kq}{r}$

Here, $r=\left(a\sqrt{2}\right)/2$

As all the charges at corner are equal in magnitude and are equidistance from the center O of the square, So potential due to them at center will be equal.

Potential being scalar quantity will be added.

$\therefore V_O=4V_q=4\times \frac{Kq}{r}$

$=\frac{4q}{4\pi {ϵ}_0\left(a\sqrt{2}\right)/2}=\frac{\sqrt{2}q}{\pi {ϵ}_{0}a}$

Potential at Infinity is assumed to be zero(Reference point), $V_{\infty }=0$

$\text{Step 3: Work done}$

So, Work done in carrying a charge $-q$ from its centre to infinity is-

$W=q\Delta V$ $=(-q)(V\infty -V_O)=\frac{\sqrt{2}{q}^2}{\pi {ϵ}_{0}a}$

Hence, Option D is correct.

$\text{Note:}$ Since it is not mentioned that work done by which force, only Work done is mentioned.

In such cases, Work done by external agent is being referred. Therefore use formula accordingly.