Electric field intensity can be calculated by using either Coulomb’s law/Gauss’s law . when the charge distribution is symmetric another way of obtaining is from the electric scalar potential V
Assume a test charge at A in an Electric field, let points A and B are located at units from the origin O,from Coulomb’s law the force acting on a test charge is
The work done in moving a point charge along a differential length is is given by
so the total work done in moving a point charge from A to B is
the direction of work done is always opposite to the direction of displacement.
where A is the initial point and B is the final point. Dividing the work done by the charge gives the potential energy per unit charge denoted by ,this is also known as potential difference between the two points A and B.
if we take B as initial point and A as final point , then
To derive the expression for V in terms of charge Q and distance r , we can use the concept of Electric field intensity produced by a charge Q, which is placed at a distance r
where and are the scalar potentials at the points A and B respectively. If A is located at with respect to origin ,with zero potential and B is located at a distance r with respect to origin. then the work done in moving a charge from A (infinity) to B is given by
hence the potential at any point is the potential difference between that point and a chosen point at which the potential is zero. In other words assuming Zero potential at infinity .
The potential at a distance r from a point charge is the work done per unit charge by an external agent in transferring a test charge from infinity to that point.
So a point charge located at a point P with position vector then the potential at another point Q with a position vector is
As like superposition principle is applicable to V also that is for n point charges located at points with position vectors
then the potential at is