Derive an expression for potential energy of a system of two charges in the absence of the external electric field.

Explanation

Imagine the charges `q_1` and `q_2` are initially at infinity. First, the charge `q_1` is brought from infinity to the point A and no work is done for this. This is because there is no electric field to oppose this charge. Consider another point B at a distance from A. Electric potential at B due to r is given by,
`V_1 = (1)/(4pi epsi_0 )(q_1)/(r )`
Next the charge `q_2` is brought from infinity to the point B. When the charge 92 is moved, the electric field due to `q_1` opposes it. Hence work has to be done. Work done in bringing the charge `q-2` from infinity to the point B.
`W = V_1 xx q_2`
[since `V = (W)/(q_0) rArr W = qV`]
`W = (1)/(4pi epsi_0) (q_1)/(r ) xx q_2`
This work done is stored in the charges as electric potential energy and given by
`PE = U = (1)/(4pi epsi_0) (q_1 q_2)/(r )`