3. Electromagnetic Momentum and Kinetic Energy.
The electric and magnetic fields are two of the many manifestations of energy in the physical world. The formulas for their energy densities are
(3_1)
and
. (3_2)
As referenced in Sect. 2, the Poynting vector (S=eoc2E X B) is a measure of the rate at which electromagnetic field energy fluxes through a surface.
When both fields are present there may also be field momentum. The formula for field momentum density is
. (3_3)
The volume integral of g is a measure of the momentum in the electromagnetic field. In Sect. 28-2 of The Feynman Lectures on Physics, Volume 2, the momentum in the field of a spherical shell of charge with constant non-relativistic velocity, v, is derived. If the charge is q and the shell’s radius is R, then the total momentum in its electromagnetic field is found to be
. (3_4)
Although this momentum exists in the field, it is customary to define the charge’s electromagnetic mass to be
. (3_5)
It is not difficult to extend Feynman’s derivation of pElecMagField and to derive the energy, TElecMagField in the same spherical shell’s magnetic field. It is found that
. (3_6)
Like "material" particles, then, electric charge can have momentum and kinetic energy. But whereas kinetic energy and momentum seem be localized in the case of material particles, these quantities are distributed in space in the case of charge. As a charge’s speed is increased, part of the energy that flows into space is electromagnetic kinetic energy. (There may also be radiant energy.) And as its speed is decreased, part of the energy flowing in from the field (to the charge/decelerating agent) is also kinetic energy. Similar remarks apply to momentum.
A point charge can be modeled as the limit of a spherical shell of charge as its radius shrinks to zero. Eq. 3_5 thus indicates that the electromagnetic mass of a point charge is infinite. Following Feynman’s derivation, it is readily shown that the field momentum of a moving point charge is infinite. Among other things, it would theoretically take an infinite force to change the velocity of a point charge. Despite this difficulty, the concept of point charges is useful in theoretical discussions. If it becomes an issue, the charge can usually be modeled as a spherical shell of charge (or solid sphere of charge or …) of very small but nonzero radius, with little or no loss of rigor.
Electrons appear to be true "particles" (with no constituent parts such as the quarks that comprise protons and neutrons). The fact that the electron’s measured mass is only 9.11E-31 kg has been a matter of much conjecture and discussion over the years. One suggestion has been that the electron is a true particle, but that its charge is smeared out over a small volume of space. In any case the point charge field solutions, introduced in Sect. 1, will be used extensively in sections to come.