The electric potential energy of two point charges is equal to the work done to assemble the charges or work done in bringing each charge or work done in bringing a charge from infinite distance.
Fig. Potential energy of a system of charges q1 and q2 is directly proportional to the product of charges and inversely to the distance between them.
Let us consider a point charge q1, placed at A (Fig.). The potential at a point B at a distance r from the charge q1 is
V = q/4πε0 r
Another point charge q2 is brought from infinity to the point B.
Now the work done on the charge q2 is stored as electrostatic potential energy (U) in the system of charges q1 and q2.
∴ Work done, W = Vq2
Potential energy (U) = q1q2/4πε0 r
Keeping q2 at B, if the charge q1 is imagined to be brought from infinity to the point A, the same amount of work is done.
Also, if both the charges q1 and q2 are brought from infinity, to points A and B respectively, separated by a distance r, then potential energy of the system is the same as the previous cases.
For a system containing more than two charges (Fig.), the potential energy (U) is given by
U = 1/ 4πε0 [q1q2/r12+q1q3/r13+q2q3/r23]
Note :
Because of the conservative nature of the electrostatic force (or equivalently, the path independence of work done), the final expression for U, is independent of the manner in which the configuration is assembled.
The potential energy is characteristic of the present state of configuration, and not the way the state is achieved.
