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)/(4pi epsilon_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_1 xx q_2` `[:. V=(w)/(q_0) implies w=q_0V]`
`w=(1)/(4pi epsilon_0)(q_1)/(r)xxq_2`
This work done is stored in the charges as potential energy and is given by
`PE=(1)/(4 pi epsilon_0)(q_1q_2)/(r)`