Electrodynamics of Free- And Bound-Charge Electricity Generators Using Impressed Sources

Electric generators convert external energy, such as mechanical, thermal, nuclear, chemical, and so forth, into electricity and are the foundation of power station and energy harvesting operations. Inevitably, the external source supplies a force per unit charge (commonly referred to as an impressed electric field) to free or bound charge, which produces ac electricity. In general, the external impressed force acts outside Maxwell's equations and supplies a nonconservative electric action generating an oscillating electrodynamic degree of freedom. In this work we analyze the electrodynamics of ideal free- and bound-charge electricity generators by introducing a time-dependent permanent polarization, which exists without any applied electric field, necessarily modifying the constitutive relations and essential to oscillate free or bound charge in a lossless way. For both cases, we show that Maxwell's equations and, in particular, Faraday's law are modified, along with the required boundary conditions through the addition of an effective impressed magnetic current boundary source and the impressed electric field, related via the left-hand rule. For the free-charge case, we highlight the example of an electromagnetic generator based on Lorentz force, where the impressed force per unit charge that polarizes the conductor comes from mechanical motion of free electrons due to the impressed velocity of the conductor relative to a stationary dc magnetic field. In contrast, the bound-charge generator is simply an idealized permanently polarized bar electret, where the general case of a time-dependent polarized electret is the underlying principle behind piezoelectric nanogenerators. In the open-circuit state, both bound- and free-charge electricity generators are equivalent to idealized Hertzian dipoles, with the open-circuit voltage equal to the induced electromotive force (emf). Analyzing the short-circuit responses, we show that the bound-charge electricity generator has a capacitive source impedance. In contrast, we show that, for the ideal free-charge ac electricity generator, the back emf from the inductance of the loop that defines the short circuit directly cancels the source emf, so the voltage across the inductor is solely determined by the magnetic current boundary source. Thus, we find that the magnetic current boundary source best describes the output voltage of an ac generator, rather than the electric field.