The authors review the general question of the physical interpretation of Poynting's theorem and compare two typical derivations of it. One of these uses the work done on a charge by an external electromagnet field, while the other considers the work done by the total field, external plus self-field; though this derivation is mathematically consistent, it leads to difficult problems of interpretation.

We present a momentum balance equation derived directly from Maxwell's equations. This equation contains a force density, which we call Maxwell's force density, which generalizes the Lorentz force density, now including total fields rather than only external fields, and arbitrary charge and current distributions. As a test for this balance equation we derive for gases the electrostatic and magnetostatic Helmholtz force densities. This deduction will be useful for advanced undergraduates and graduate students, as well as for specialists interested in the conceptual aspects of electromagnetism.

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We solve exactly the problem of calculating the electromagnetic fields produced by a finite wire with a constant current, by using two methods: retarded potentials and Jefimenko's formalism. One result in this particular case is that the usual Biot-Savart law of magnetostatics gives the correct magnetic field of the problem. We also show graphically the corresponding Poynting vector.