Field Energy and Momentum

G.R.Dixon, 5/3/2005

In this article the momentum and energy in the fields of a spherical shell of charge, q, are calculated. The shell’s radius is R, and it moves with a constant velocity whose magnitude is v<<c. __v__ is along the x-axis and it points in the positive x-direction.

The electric field is assumed to differ negligibly from q’s electrostatic field. But of course the __E__ field as a whole moves along with q. The fields are calculated when the center of q is at the origin. q is the angle subtended by the positive x-axis and a field evaluation point above the x-axis. Such points are uniquely defined by (r,q), where r>R is the distance from the origin to the field evaluation point.

The volume element dV is defined to be the volume of a spherical shell in space, centered on the origin and with inner radius r and outer radius r+dr. In brief,

. (1)

The volume element d^{2}V is defined to be the volume of a ring segment of dV, said ring being concentric to the x-axis and at angle q. That is, each ring constitutes a "slice" of dV, the plane of said slice being perpendicular to the x-axis. The ring’s radius is (r sin(q)).

If we consider the ring whose geometric center is at x=0, then a cross section of d^{2}V, say at (r,p/2), approaches a rectangle in the limit of infinitesimal dr and dq. In the limit the area of this cross section is

. (2)

We can use this cross section area increment for all "slices" or rings. By a theorem of Pappus, the volume of the element d^{2}V is then d^{2}A times the circumference of the circle with radius r sin(q):

. (3)

As expected, this expression integrates to dV over the range 0<q<p:

. (4)

The energy density in the electric field is

. (5)

Since u_{E} depends only upon r, the energy in the entire electric field is

. (6)

The energy density in the magnetic field is not spherically symmetric. At the point (r,q)

. (7)

The magnetic field energy density is

. (8)

Thus

. (9)

__|B|__ __is__ single-valued in the annulus d^{2}V. Thus the magnetic field energy in d^{2}V is

. (10)

The magnetic field energy in dV is thus

. (11)

The magnetic field energy in all of space is therefore

. (12)

The electromagnetic field momentum density is

. (13)

The y- and z-components cancel out in pairs, and the x-component has magnitude

. (14)

This too is single-valued in any volume element d^{2}V. That is, the momentum in volume element d^{2}V points in the positive x-direction and has magnitude

. (15)

p_{x} in dV is then

. (16)

And finally, the momentum over all of space is

. (17)

In Eq. 12 E_{B} is proportional to v^{2}/2, and in Eq. 17 p_{x} is proportional to v. The constant of proportionality in both cases is q^{2}/6pe_{o}Rc^{2}. For obvious reasons it is customary to refer to this as the electromagnetic mass:

. (18)

It is noteworthy that the "kinetic energy" resides solely in the __magnetic__ field.

Although m_{ElecMag} is an attribute of the fields, it is customary to think of it as the electromagnetic mass of the distribution of source charge. Some justification for this can be found in the fact that we can interact dynamically with the fields only via their source electric charges. Indeed electric charge has been likened to a "hook" whereby we locally interact with fields that fill all of space. Different distributions of charge have different expressions for their electromagnetic masses.

The result that all of a charge’s electromagnetic kinetic energy resides in the magnetic field can be useful in determining the electromagnetic mass per unit volume (or per unit area, or …) in selected cases where v (or w or …) is constant in time. In such cases

. (19)

An example of this technique is provided in another article, where the electromagnetic mass per unit length of an infinitely long solenoid is derived.