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 a system consisting of two point charges `q_1` and `q_2` separated by a distance `r_12` in free space.
Let the charge `q_1` be moved from infinity to the point A . No work is done during this process as the charge `q_1` is moved in a field free region.
With the charge `q_1` fixed at A, let the other charge `q_2` be moved from infinity to the point B against the field due to charge `q_1`.
Work done in this process is
`W=V_Bq_2`...(1 ) `V_B` is potential at B due to `q_1`
But `V_B=1/(4piepsilon_0)q_1/r_12`
So equation (1) becomes
`W=1/(4piepsilon_0)(q_1q_2)/r_12`
This work done is stored in the form of potential energy of the system of two charges `q_1` and `q_2`.
Hence `U=1/(4piepsilon_0)(q_1q_2)/r_12`.