**Potential energy of a single charge**

The external electric field E and the corresponding external
potential V may vary from point to point. By definition, V at a point P is the
work done in bringing a unit positive charge from infinity to the point P.

Work done in bringing a charge q from infinity to the point P in
the external field is qV. This work is stored in the form of potential energy
of q. If the point P has position vector r relative to some origin, we can
write:

**Potential energy of a system of two charges in an external field**

Work done in bringing the charge q

_{1}from infinity to r_{1}is q_{1}V(r_{1}). Consider the work done in bringing q_{2}to r_{2}. In this step, work is done not only against the external field E but also against the field due to q_{1}.
Work done on q

_{2}against the external field
Work done on q

_{2 }against the field due to q_{1}
By superposition principle for fields, add up the work done on q

_{2}against the two fields. Work done in bringing q_{2}to r_{2 }
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