Potential Energy in an External Field

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potential energy is the energy that an object has due to its position in a force field or that a system has due to the configuration of its parts.

Consider an external electric field 'E' produced by an external source, where it can specified or unspecified. But the potential 'V' due to external source has to be specified. Let us consider charge 'q' does not effect the external source producing the electric field, which implies q is very small. A strong source placed at infinity produces a finite electric field 'E'.

But we are concerned in finding the potential of a given charge and not of a source producing the external field.The electric field 'E' and the potential 'V' may vary from point to point.

If the potential at infinity is zero, then work done in bringing a charge 'q' from infinity to the point 'p' in the external field is q.V.

This work is stored as potential energy of q. If 'p' has a position vector, 'r' relative to some origin,

then potential energy of q at r in an external field is:

= qV(r)
where V(r) is the external potential at point 'r'.

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