Potential Energy of A System of Charges

Consider the charges q₁ and q₂ initially at infinity and determine the work done by an external agency to bring the charges to the given locations.

Suppose, charge q₁ is brought from infinity to the point r₁. There is no external field against which work needs to be done, so work done in bringing q₁ from infinity to r₁ is zero. This charge produces a potential in space given by

where r_1P is the distance of a point P in space from the location of q₁.

From the definition of potential, work done in bringing charge q₂ from infinity to the point r₂ is q times the potential at r₂ due to q₁:

where r₁₂ is the distance between points 1 and 2.

If q₁q₂ > 0, Potential energy is positive. For unlike charges (q₁ q₂ < 0), the electrostatic force is attractive.

Potential energy of a system of three charges q₁, qand q located at r₁, r₂, r, respectively. To bring q first from infinity to r₁, no work is required. Next bring q₂ from infinity to r₂. As before, work done in this step is

The total work done in assembling the charges at the given locations is obtained by adding the work done in different steps,

 The potential energy is characteristic of the present state of configuration, and not the way the state is achieved.