﻿ The Dynamics of Electric amd Magnetic Fields in the Absence of Charges and Currents
San José State University

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Thayer Watkins
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The Dynamics of Electric and Magnetic Fields
in the Absence of Charges and Currents

This material investigates the dynamics of electric and magnetic fields; i.e,, the manner in which they change over time. 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 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 only field configurations with ρ and J equal to zero everywhere will be considered; in effect free fields.

Such fields 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 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. In Cartesian coordinates the i-th compponent of the vector Laplacian of a vector is equal to the scalar (ordinary) Laplacian of the i-th component of that vector.

For the case being considered in which there is no charge distribution ∇·E is equal to zero. 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. The wave equation is then of the form

#### (∂²E/∂t²) = C²∇²E

Suppose the space is one dimensional. Let f(x)=E(x,0). Then the solution, known as the d'Alembert solution, is of the form

#### E(x, t) = ½[f(x+Ct) + f(x-Ct)] Thus half of the initial profile moves to the right and half moves to the left. This constitutes moving electric fields but not radiation. It is more in the nature of a whoosh rather than a wave.

If the above procedure was carried out with H rather than E the same wave equation would result and likewise for D and B.

The solutions can be construed to be an energy flow but that terminology is a bit misleading. It is the fields that flow and take their energy along with them.