1. Electrostatic Potential

Chapter 2: Electrostatic Potential and Capacitance

Electrostatic Potential 

We define potential energy of a test charge q in terms of the work done on the charge q. This work is obviously proportional to q, since the force at any point is qE, where E is the electric field at that point due to the given charge configuration. It is, therefore, convenient to divide the work by the amount of charge q, so that the resulting quantity is independent of q. In other words, work done per unit test charge is characteristic of the electric field associated with the charge configuration
POTENTIAL DUE TO A POINT CHARGE
Take Q to be positive. We wish to determine the potential at any point P with position vector r from the origin. For that we must calculate the work done in bringing a unit positive test charge from infinity to the point P. For Q > 0, the work done against the repulsive force on the test charge is positive.

Potential due to an electric dipole
What is the electric potential due to an electric dipole at an equatorial point? Zero, as potential on equatorial point, due to charges of electric dipole, are equal in magnitude but opposite in nature and hence their resultant is zero.

Potential due to system charges
derive an expression for the electric field at a point due to a system of n point charges. When there is a group of point charges say q₁, q₂, q₃,….q_n is kept at a distance r₁, r_2, r₃,……r_n, we can get the electrostatic potential at any particular point. 

Equipotential surface.

If the points in an electric field are all at the same electric potential, then they are known as the equipotential points. If these points are connected by a line or a curve, it is known as an equipotential line. If such points lie on a surface, it is called an equipotential surface.
Work Done in Equipotential Surface
The work done in moving a charge between two points in an equipotential surface is zero. Then the work done in moving the charge is given by
W = q₀(V_A –V_B)
As V_A – V_B is equal to zero, the total work done is W = 0.