Concerning Classical Forces, Energies, and Potentials for Accelerated Point Charges

Although the expressions for energy densities involving electric and magnetic fields are exactly analogous, the connections to forces and electromagnetic potentials are vastly different. For electrostatic situations, the changes in the $\mathit{\text{electric}}$ energy can be related directly to $\mathit{\text{electric}}$ forces and to the electrostatic potential. The situation involving magnetic forces and energy changes involves two fundamentally different situations. For charged particles moving with constant velocities, the changes in both electric and magnetic field energies are provided by the external forces that keep the particles’ velocities constant; there are no Faraday acceleration electric fields in this situation. However, for particles that change speed, the changes in $\mathit{\text{magnetic}}$ energy density are related to acceleration-dependent Faraday $\mathit{\text{electric}}$ fields. Current undergraduate and graduate textbooks deal only with highly symmetric situations where the Faraday electric fields are easily calculated from the time-changing magnetic flux. However, in situations that lack high symmetry, such as the magnetic Aharonov-Bohm situation, the back (Faraday) acceleration electric fields of point charges seem unfamiliar. In this article, we present a simple unsymmetric example and analyze it using the Darwin Lagrangian. In $\mathit{\text{all}}$ cases involving changing velocities of the current carriers, it is the work done by the back (Faraday) acceleration $\mathit{\text{electric}}$ fields that balances the $\mathit{\text{magnetic}}$ energy changes.