Contribution of electric self-forces to electromagnetic momentum in a moving system

TL;DR

The paper investigates how electromagnetic momentum in moving charged systems should be computed. It shows that momentum from the vector potential, while capturing bulk energy transport, is incomplete unless stress contributions from the Maxwell stress tensor are included; Lorentz-transforming the rest-frame stress-energy tensor reveals extra terms that account for energy flow within the system. These stress-induced terms explain the historically puzzling 4/3 factor in moving charges and do not require modifications to the standard EM energy-momentum formalism. When all electromagnetic and non-electromagnetic (stabilizing) contributions are accounted for, the total momentum transforms as ${\cal E} v / c^2$, consistent with a 4-vector, across different plate geometries and for both non-relativistic and relativistic motion.

Abstract

In moving electromagnetic systems, electromagnetic momentum calculated from the vector potential is shown to be proportional to the field energy of the system. The momentum thus obtained is shown actually to be the same as derived from a Lorentz transformation of the rest-frame electromagnetic energy of the system, assuming electromagnetic energy-momentum to be a 4-vector. The energy-momentum densities of electromagnetic fields form, however, components of the electromagnetic stress-energy tensor, and their transformations from rest frame to another frame involve additional contributions from stress terms in the Maxwell stress tensor which do not get represented in the momentum calculated from the vector potential. The genesis of these additional contributions, arising from stress in the electromagnetic fields, can be traced, from a physical perspective, to electric self-forces contributing to the electromagnetic momentum of moving systems that might not always be very obvious. Such subtle contributions to the electromagnetic momentum from stress in electromagnetic fields that could be significant even for non-relativistic motion of the system. Such contributions from stress in electromagnetic fields also provide a natural solution to some curious riddles in electromagnetic momentum like the famous, century-old, enigmatic factor of 4/3, encountered in the electromagnetic momentum of a moving charged sphere.

Figures (4)

• Figure 1: Calculation of the scalar potential, ${\phi}$ at a point $x$ between the charged capacitor plates, due to circular rings of radii $r$, centered on $O_1$ and $O_2$. The capacitor plates, circular discs of radii $L$ are assumed to be lying in the y--z planes, with a plate separation $d$ along the $x$ direction. The surface charge densities are $+\sigma$ and $-\sigma$, on plates 1 and 2 respectively. The capacitor is moving along the $x$-axis with a velocity $\mathbf v$, assumed to be non-relativistic.
• Figure 2: In the charged parallel-plate capacitor, plate 1 is experiencing an electric attractive force ${\bf F}_{1}$ along $x$-axis, while plate 2 is attracted with electric force ${\bf F}_{2}$ in the opposite direction. We assume the capacitor to contain an ideal gas comprising molecules having 1-dimensional motion with non-relativistic speeds $\pm V$ along the x-axis, the direction of plate separation, and that the resulting pressure keeps the capacitor plates separated at a fixed distance $d$, by cancelling the electric attractive forces, through mechanical forces, ${\bf F}_{\rm m1}=-{\bf F}_{1}$ on plate 1 and ${\bf F}_{\rm m2}=-{\bf F}_{2}$ on plate 2.
• Figure 3: Computation of the scalar potential, ${\phi}$, at a point $y$ between the charged capacitor plates, due to circular rings of radii $r$, centered on $O_1$ and $O_2$, from the charged capacitor plates 1 and 2, lying in the x--z planes. The surface charge densities are $+\sigma$ and $-\sigma$, on plates 1 and 2 respectively. The capacitor system moves with velocity ${\bf v}$ with respect to the lab-frame.
• Figure 4: A spherical shell of radius $\epsilon$ with a uniform surface charge density $\sigma$, moving with velocity $\bf v$ along the x-axis. Shown in gray are two symmetrically placed circular rings, each of radius $\epsilon \sin \theta$ and of angular width $d\theta$, lying on two opposite sides of the spherical shell separated by a distance $l= 2\epsilon\cos \theta$ along the $x$-axis.