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

Explanation

Consider 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 .r. from A. Electric potential at B due to `q_1` is given by
`V_(1) = 1 / (4piepsilon_(0))q_(1)/r`
Next the charge `q_2` is brought from infinity to the point B. When the charge `q_2` 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 is given by
`w=V_1xxq_(2) " " [ :. V=(w)/q_(0)impliesw=q_(0)V]`
`w = 1/(4piepsilon_(0))(q_1)/(r)xxq_2`
This work done is stored in the charges as potential energy and is given by `PE = 1/(4piepsilon_(0))(q_1q_2)/r`