Electric Potential Energy

Definition. The electric potential energy of a system of fixed point charges is equal to the work that must be done by an external agent to assemble the system, bringing each charge in from an infinite distance.

Consider a system AB of two charges q1 and q2, separated by a distance r. Let us imagine that the charge q2 at B has been removed to infinity. Now,
 the potential at B due }    = V = ___1___ q₁
 to the charge + q₁      }              __  4πε₀     r

This is the work required to be done in order to bring a unit charge from infinity to the point B. Therefore, the work done in bringing the charge q2 from infinity to the point B is

    W = q2 V = ___1___ q₁ q₂
                          __4πε₀       r

This work doe represents the electric potential energy (U) of the system of charges q1 and q2.

. :    U = ___1___ q₁ q₂
              ___4πε₀       r
In case of like charges, the potential energy is positive. In case of unlike charges, potential energy is negative.

For systems containing more than two charges, the potential energy of the system is obtained by computing the potential energy for every pair of charges separately and adding the results algebraically.

If we have n different charges in nay arrangement in space.

    U = 1 (___1___) ∑    ∑   qiqj
           2  _  4πε₀   i = j  j≠I     rij