. P OTENTIAL E NERGY OF A S YSTEM OF C HARGES Consider first the simple case of two charges q and q with position vector r and r relative to some origin. Let us calculate the work done (externally) in building up this configuration. This means that we consider the charges q and q initially at infinity and determine the work done by an external agency to bring the charges to the given locations.
Suppose, first the charge q is brought from infinity to the point r . There is no external field against which work needs to be done, so work done in bringing q from infinity to r is zero. This charge produces a potential in space given by 1P where r 1P is the distance of a point P in space from the location of q . From the definition of potential, work done in bringing charge q from infinity to the point r is q times the potential at r due to q : work done on q = q q FIGURE .
From the potential to the field. where r is the distance between points and . Since electrostatic force is conservative, this work gets stored in the form of potential energy of the system. Thus, the potential energy of a system of two charges q and q is q q U ( .
) Obviously, if q was brought first to its present location and q brought later, the potential energy U would be the same. More generally, the potential energy expression, Eq. ( . ), is unaltered whatever way the charges are brought to the specified locations, because of path-independence of work for electrostatic force.
Equation ( . ) is true for any sign of q and q . If q q > , potential energy is positive. This is as expected, since for like charges ( q q > ), electrostatic force is repulsive and