Total kinetic energy of an electromagnetic body

Abstract

It is here proposed to extend the notion of the kinetic energy of a moving and deformable electromagnetic material. This extended form of kinetic energy, which we suggest naming total kinetic energy, is consistent with the classical expression of the kinetic energy, to which it reduces in the absence of electromagnetic fields. The proposed expression of the total kinetic energy follows from a Lagrangian formulation and is based on the canonical momenta, which, in turn, depend on the material electromagnetic potentials. The material potentials are the classical scalar and vector potentials, which have been properly convected to the referential frame of the body. As a result, the total kinetic energy density of the electromagnetic body turns out to be a quadratic, though non-homogeneous, form in the velocity of the point and in the electromagnetic potentials. This form preserves most of the fundamental properties that one would expect (positive definiteness, additivity, etc.). However, additional and specific peculiarities also emerge from the introduced total kinetic energy. These novel features are discussed herein. In view of a thermomechanical treatment, a kinetic energy theorem is also established and discussed.

† “In the theory of electricity and magnetism adopted in this treatise, two forms of energy are recognised, the electrostatic and the electrokinetic (see Arts. 630 and 636), and these are supposed to have their seat, not merely in the electrified or magnetized bodies, but in every part of the surrounding space, where electric or magnetic force is observed to act.” (Maxwell, Treatise, Vol. 2, art. 782)

Notes

† “In the theory of electricity and magnetism adopted in this treatise, two forms of energy are recognised, the electrostatic and the electrokinetic (see Arts. 630 and 636), and these are supposed to have their seat, not merely in the electrified or magnetized bodies, but in every part of the surrounding space, where electric or magnetic force is observed to act.” (Maxwell, Treatise, Vol. 2, art. 782)