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

Explanation

Let q be the charge on a point charge placed at A. Let '+1C' of test charge be placed at infinity.
It is assumed that +1C of charge is brought from infinity to the point 'B' at a distance of 'r' from 'A'. The electric field is due to `'q_(A)'`
i.e., dV=-Edx for unit +ve test charge.
Now replacing '+1C' by `'q_(B)','dV'` by dW and 'E' by F, we get
`dW=-Fdx,dx`-displacement between C & B.

Integrating both sides we get,
`intdW=W=int_(x=0)^(x=r)-Fdx`
i.e., `W=-((1)/(4piepsi_(0)))(q_(A)q_(B))int_(0)^(r)(1)/(x^(2))dx`
i.e., `W=(-(1)/(4piepsi_(0)))(q_(A)q_(B))(-(1)/(x))_(0)^(r)`
or `W=((1)/(4piepsi_(0)))((q_(A)q_(B))/(r))`.