Consider an electric field due to point charge Q placed at point O. Suppose VA is the electric potential at point A, which is at a distance of ra from charge Q.
It W∞ A is the work done in moving a small test charge Q0 from ∞ to A then potential is,
... (1)
Consider that at any time, the test charge is at point P. The electric field E due to the charge Q exerts force,
on the test charge placed at that point P.
If the test charge Q0 is moved a small distance PB = dl, then small work done is given by,
dW = F. dl
= - (QoE). dl
= -Qo E dl cos 180o
= QoEdl
dW = - QoEdr
Here we have taken dl as - dr, because the distance r decreases in the direction of dl.
Since, we know that
Using equation (2) in eqn. (1), we get