Maxwell's Equations for Free Fields

The behavior of electromagnetic fields is described by Maxwell's equations. The precise form of these equations depends upon the system of units used. Here the Gaussian system of J.D. Jackson's Classical Electrodynamics is used.

The Maxwell equations for electromagnetic fields in that system are:

∇·D = 4πρ
∇×H = (4π/c)J + (1/c)(∂D/∂t)
∇·H = 0
∇×E + (1/c)(∂B/∂t) = 0

where c is the speed of light in a vacuum, E and D are vector fields describing the electric field and B and H are vector fields for the magnetic field. The quantities ρ and J are the charge density and current density, respectively.

The relationships between E and D and B and H are

D = εE
B = μH

where ε is the dielectric constant of the material the fields are located in and μ is the permeability of that material.

Here free fields means field configurations with ρ and J equal to zero everywhere.

If such fields exist they change according to the dynamical equations

ε(∂E/∂t) = c∇×H
μ(∂H/∂t) = −c∇×E

If the first equation is differentiated once with respect to time the result is

ε(∂²E/∂t²) = c∇×(∂H/∂t)

The interchangeability of ∇× and (∂/∂t) was utilized in deriving the above equation.

The second equation above can then be used to replace (∂H/∂t) which yields

ε(∂²E/∂t²) = c∇×[−(c/μ)(∇×E)]
which reduces to
(∂²E/∂t²) = −(c²/(εμ))[∇×(∇×E)]

The curl of the curl of E, ∇×(∇×E), can be expressed as

∇×(∇×E) = ∇(∇·E) − ∇²E

where ∇²E is the vector Laplacian of E.

For the case being considered in which there is no charge distribution ∇·E is equal to zero and ∇(∇·E) is everywhere equal to the zero vector. Thus

∇×(∇×E) = − ∇²E
and hence
(∂²E/∂t²) = (c²/(εμ))∇²E

This type of partial differential equation is known as a wave equation. It can have solution of a sinusoidal nature but the solution depends upon the initial conditions.

The speed of electromagnetic radiation in the material is equal to c/(εμ)^½. Let that speed be designated as C. It is the speed of light in the material in which the fields exist. The wave equation is then of the form

(∂²E/∂t²) = C²∇²E

This equation may be used to compute the field E at any subsequent time from the initial condition of the field at t=0. That is to say

(∂E/∂t)(t+h) = (∂E/∂t)(t) + h(∂²E/∂t²)(t)
and
E(t+2h) = E(t+h) + h(∂E/∂t)(t+h)

where h is the time step of the iteration scheme.

Care must be taken in choosing the initial field. If there is an abrupy change in the value of E over space it will lead to an infinite value for (∂E/∂t) and abrupt change in E over time.

Some Simple Cases

Here are the electric field distributions for charged particles.

The sign of E is arbitrary just as the sign of the charge is arbitrary.

A photon is a ripple (perturbation) in the electric field, as shown below. But the ripple is moving outward at the speed of light.

If a charge were to disappear on its own then the its electric field would start disappearing outward at the speed of light, as depicted below.

If there were a photon in the particle's electric field the disappearance of the electric field would never catch up with it since both are traveling at the speed of light.

It needs to noted that there is no physical process that allows for the disappearance of a single charged particle on its own. Particles disappear only in particle-antiparticle annihilations. But when the fields of the particle and the antiparticle are brought together their fields cancel each other out. If the particle and anti-particle are brought together slowly enough then at the instant of annihilation the combined fields of the two particles have an intensity of zero.

But the annihilation of a positron and an electron typically produces two gamma photons traveling in opposite directions. Photons are perturbations in an electromagnetic field, but the only field available has an intensity everywhere of zero. The production of the two gamma photons tells us that a field of everywhere zero is still a field.

Further Analysis of the
Dynamics of a Free Field

Suppose the initial field distribution is spherically symmetric. Then all derivatives with respect to the angle coordinates are zero. The Laplacian ∇²E in spherical coorinates for a spherically symmetric field then reduces to

(∂²E/∂r²) + (2/r)(∂E/∂r)

Hence the wave equation reduces to

(∂²E/∂t²) = C²[(∂²E/∂r²) + (2/r)(∂E/∂r)]

For a point charge of Q the electric field is given by

E(r, φ, θ) = −kQ/r

Thus

(∂²E/∂t²) = C²[−kQ/r³ + 2kQ/r³] = C²kQ/r³

Thus the most rapid decline in the intensity of the field would take place for small r.

(To be continued.)