Relation Between Electric Field and Potential

Consider two points A and B in a uniform electric field E, separated by a small distance ∆r. By definition, potential difference ∆V between A and B is equal to the work done in moving a unit positive test charge from A to B.

∆V = (Force on unit positive charge) × (AB)

= E. ∆r = E(∆r) cos 180°

= – E ∆r

E = – ∆V/∆r

The negative sign indicates that work is done against the electric field.

At any point, the electric field is equal to negative rate of change of potential with distance (called potential gradient) at that point in the direction of field.