Electric potential energy:

The electric potential energy of a system of point charges may be defined as the amount of work done in assembling the charges at their location by bringing them in, from infinity.

Expression for the potential energy of a system of two point charges:

Suppose a point charge q₁ is at rest at a point p₁ in space as shown in fig. it takes no work to bring their first charge q₁ because there is no field yet to work against therefore W₁ = 0

Electric potential due to charge q₁ at appoint P₂ at a distance r₁₂ from P₁ will be

If charge q₂ is moved in from infinity to point P₂. The work system (q₁ + q₂) so

 Potential energy of a system of three point charges:

AS shown in fig. now we bring in the charge q₃ from infinity to point P₃. Work has to be done against the forces exerted by q₁ and q₂ therefore.

W₃ = Potential at point P₃ due to q₁ and q₂ * charge q₃

Hence the electrostatic potential energy of the system q₁ + q₂ + q₃ is

U = Total word done to assemble the three charges

= W₁ + W₂ + W₃

Potential energy of a system of N point charges:

The expression for the potential of n point charges can be written as

As double summation counts every pair twice, to avoid this factor ½ has been introduced.

Potential energy of a single charge in an external field:

By definition, V at point is the amount of work done in bringing a unit positive charge from infinity to the point P. Thus the work done is

W = qV

This work done is stored as the potential energy of the charge q. if is the position vector of point P relative to some origin, then            

P.E. of a charge in an external field = charge * external electric potential

So we can define electric potential at a given point in an external field is the potential energy of a unit positive charge at that point.

Potential energy of a system of two charges in an external electric field:

Let V₁ and V₂ be the electric potentials of the field E at the points where q₁ and q₂ are located. Then potential energy of the two charges in the field E is

U = p.E. due to interaction between q₁ and  + P.E. due to interaction between q₂ and E + P.E. due to interaction between q₁ and q₂