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Example of the converse principle

To illustrate the power of this converse principle, we cite an example that has significant physical consequences. In classical electromagnetism, the electric force is a path-independent force; i.e. the work done on a particle that has returned to its original position within an electric field is zero (assuming that no changing magnetic fields are present).

Therefore, the above theorem implies that the electric force field F : S → R is conservative (here S is some open, path-connected subset of R that contains a charge distribution). Following the ideas of the above proof, we can set some reference point a in S, and define a function U: S → R by

U e (r):= − ∫ γ F e (u) ⋅ d u {\displaystyle U_{e}(\mathbf {r}):=-\int _{\gamma }\mathbf {F} _{e}(\mathbf {u})\cdot \mathrm {d} \mathbf {u} }

Using the above proof, we know U is well-defined and differentiable, and F = −∇ U (from this formula we can use the gradient theorem to easily derive the well-known formula for calculating work done by conservative forces: W = −Δ U). This function U is often referred to as the electrostatic potential energy of the system of charges in S (with reference to the zero-of-potential a). In many cases, the domain S is assumed to be unbounded and the reference point a is taken to be "infinity", which can be made rigorous using limiting techniques. This function U is an indispensable tool used in the analysis of many physical systems.