Four equal charges Q are placed at the four corners of a square of each side is ‘a’. Work done in removing a c

Question

Four equal charges Q are placed at the four corners of a square of each side is ‘a’. Work done in removing a charge –Q from its centre to infinity is

Accepted Answer

1. **Identify Geometry:** The distance $r$ from any corner of the square (side $a$) to the center is half the diagonal. $r = \frac{a\sqrt{2}}{2} = \frac{a}{\sqrt{2}}$.
2. **Calculate Potential at Center ($V_c$):** The electric potential is a scalar quantity, so we sum the potentials due to the four charges $+Q$ at the corners [Source 157].
   $$V_c = 4 	imes \frac{1}{4\pi\epsilon_0} \frac{Q}{r} = \frac{1}{\pi\epsilon_0} \frac{Q}{(a/\sqrt{2})} = \frac{\sqrt{2}Q}{\pi\epsilon_0 a}$$
3. **Calculate Potential Energy ($U$):** The potential energy of the charge $-Q$ at the center is $U = (-Q)V_c$ [Source 36, 161].
   $$U_{center} = -Q \left( \frac{\sqrt{2}Q}{\pi\epsilon_0 a} ight) = -\frac{\sqrt{2}Q^2}{\pi\epsilon_0 a}$$
4. **Calculate Work Done ($W$):** The work done to remove the charge to infinity (where potential energy is zero) is the change in potential energy.
   $$W = U_{final} - U_{initial} = 0 - U_{center} = \frac{\sqrt{2}Q^2}{\pi\epsilon_0 a}$$

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