Chapter 4: Energy and Energy Density

ENERGY DENSITY IN THE ELECTROSTATIC FIELD
We have introduced the potential concept by considering the work done, or energy expended, in moving a point charge around in an electric field, and now we must tie up the loose ends of that discussion by tracing the energy flow one step
further.
Bringing a positive charge from infinity into the field of another positive charge requires work, the work being done by the external source moving the charge. Let us imagine that the external source carries the charge up to a point near the fixed
charge and then holds it there. Energy must be conserved, and the energy expended in bringing this charge into position now represents potential energy, for if the external source released its hold on the charge, it would accelerate away from the fixed charge, acquiring kinetic energy of its own and the capability of doing work.
In order to find the potential energy present in a system of charges, we must find the work done by an external source in positioning the charges. We may start by visualizing an empty universe. Bringing a charge Q1 from infinity to any position requires no work, for there is no field present.2 The positioning of Q2 at a point in the field of Q1 requires an amount of work given by the product of
the charge Q2 and the potential at that point due to Q1. We represent this potential as V2,1, where the first subscript indicates the location and the second subscript the source. That is, V2,1 is the potential at the location of Q2 due to Q1. Then
Work to position Q2 = Q2V2
Similarly, we may express the work required to position each additional charge in the field of all those already present.