A ubiquitous, nearby reservoir of electromagnetic energy hidden in the fourth spatial dimension as a consequence of Kelvin’s rule (for constant electric currents) and of the Poynting vector

When applying what is called Kelvin’s principle to the elementary currents of two permanent magnets that attract each other, an apparent energy paradox appears. For Kelvin’s principle says that when constant electric currents are displaced with respect to one another, the mechanical work yielded as a result of the action of magnetic forces is equal in amount to the increase (not decrease) in the energy of the total magnetic field. The energy provided by the power supply in order to keep the currents constant is thus twice as large as the mechanical work yielded during the displacement of the current-carrying wires. But when dealing with permanent magnets and their polarization currents, there is still the yield of mechanical work and also the increase in energy of the total magnetic field, but no such thing as a visible power supply. In this article, things are analyzed by using the Poynting vector as an instrument. As a result, the topological assumption of a hidden reservoir of energy sitting in the direction of a fourth spatial dimension turns out to be indispensable in order to save the principle of local conservation of energy and of action by contact. A recognition of this kind was foreshadowed by Mie 100 years ago, who postulated that, in certain, but nevertheless common situations, energy flowed into ambient space out of the particles themselves both in the gravitational and the electromagnetic case.