Conclusions concerning the relation between electric fields and potential.

1. By definition E = -dV/dr 

 

V = -E.dr

V = -E*dr *cos(theta), where theta is the angle between the local electric field vector and displacement vector (dr)

 

Note that  attains its most negative value when . In other words, the direction of the electric field at point  corresponds to the direction in which the electric potential  decreases most rapidly. 

 

2. Electric field is perpendicular to the equipotential surface. And using the above formula, 

E = -dV/dr

 

we can conclude, that the magnitude of the electric field is given by the change in potential per unit displament in the direction normal to the equipotential surface.