Question

Find the potential energy to a system of four particles placed at the vertices of a square of side $l$. Also obtain the potential at the centre of the square.

Answer 1

The system can be made by bringing each charge from reference $\left(\infty \right)$ and

placing them on the vertices of the square

Then net work done in arranging the system = Potentital energy of teh system

For bringing 1st charge : $W_1=0$

For bringing 2nd charge : $W_2\frac{kqq}{l^2}$ (assuming each charge = q and side = l)

For bringing 3rd charge : $W_3=\frac{kqq}{l^2}+\frac{kqq}{(\sqrt{2}l{)}^2}$

For bringing 4th charge: $W_4=\frac{kqq}{l^2}+\frac{kqq}{l^2}+\frac{kqq}{(\sqrt{2}l{)}^2}$

Hence total work done = $W_1+W_2+W_3+W_4=\frac{5k{q}^2}{l^2}$ i.e. potential energy of system

The potential at centra = $4\times \frac{kq}{{\left(\frac{1}{\sqrt{2}}\right)}^2}=\frac{8kq}{l^2}$