Let A and B be two points in a non-uniform electric field. Let an external agent move a test charge q from A to B along any path. Let E be the electric field at any points P. The electric field exerts a force qE on the charge q. The external agent must apply a force F = -qE is order to move q without acceleration.

The work done for a small displacement dl along AB = F.dl

. : total work done in moving the charge from A to B is

W_AB = ∫ F.dl = - q ∫ E. dl (substituting F = - q E)

or    W_AB / q = - ∫ E. dl.

But, by definition, WAB / q is the potential difference VB – VA between the points A and B. Thus

    V_B – V_A = - ∫ E. dl

If the point A lies at infinity, VA = 0. Then the potential at the point B is
   
V_B = - ∫ E. dl