Introduction

Henning F. Harmuth    Retired, The Catholic University of America Washington, DC, USA

Beate Meffert    Humboldt-Universität Berlin, Germany

1.1 MODIFIED MAXWELL EQUATIONS AND BASIC RELATIONS

During the last century an enormous number of books and journal articles has been published on solutions of Maxwell’s equations. Practically all of them were about sinusoidal waves. These solutions were outside the conservation law of energy, since periodic sinusoidal waves, infinitely extended, must have infinite energy unless their power is zero. We have found only two papers that used Gaussian pulses rather than periodic sinusoidal functions to keep the energy finite (King and Harrison 1968, King 1993). The Gaussian pulse still starts at minus infinity, which prevents its use as a signal. A signal has to be zero before a certain finite time. This need was recognized by mathematicians who coined the term causal functions for functions that are zero before a finite time. Since signals—like any producible or observable electromagnetic wave—must have a finite energy, they are mathematically represented by quadratically integrable causal functions or signal solutions.

The lack of signal solutions of Maxwell’s equations must always have been a problem for its serious students. The best equations of the theory of electromagnetism did generally not provide the only solutions that could be produced or observed experimentally. Stratton (1941) tried to overcome this problem but his mathematical derivations were not well received and we do not find them in the textbooks published later.

From 1986 on it was recognized that Maxwell’s equations could generally not have solutions that satisfied the causality law, which explained the absence of signal solutions (Harmuth 1986a,b,c; Hillion 1991, 1992a,b; 1993). The addition of a magnetic (dipole) current density term corrected this shortcoming1 (Harmuth 1986a,b,c; Anastasovski et al. 2001). Rotating magnetic dipoles produce magnetic dipole currents just as rotating electric dipoles—e.g, in a material like barium-titanate—produce electric dipole currents.

Maxwell’s equations modified by a magnetic dipole current density can be written in the following form with international units in a coordinate system at rest:

?H=∂D∂t+ge

?E=∂B∂t+gm

?D=ρe

?B=0ordiv?B=ρm

We use here an old-fashioned notation but the problem of Maxwell’s equations with the causality law was found with this notation. It may well be that the physical meaning of equations is easier to grasp with this old notation. The operators ∇ and will be used when mathematical compactness is more important than physical meaning. The symbols E and H stand for the electric and magnetic field strength, D and B for the electric and magnetic flux density, ge and gm for the electric and magnetic current density, ρe and ρm for the electric and a possible magnetic charge density. The existence of magnetic charges is not accepted as confirmed experimentally, but there are serious theoretical arguments for their existence. The magnetic dipole current density gm does not depend on the existence of magnetic charges. The existence of magnetic dipoles is not disputed and rotating dipoles create dipole currents. The electric current density ge has always stood for electric monopole current densities requiring charges, but also for electric dipole and higher order multipole current densities whose total charge is zero. Without dipole currents no electric current could flow through a capacitor whose dielectric is an insulator for electric monopole currents.

The use of the term signal solution should not mislead one to believe that the modified Maxwell equations are only of interest in low-energy effects typically associated with information transmission. The electromagnetic pulse of an atomic bomb explosion or the electromagnetic radiation produced by a supernova explosion fit the definition of a signal solution too. The electric and magnetic field strengths produced in these examples are generally not defined by Maxwell’s equation but they are defined by the modified Maxwell equations.

Equations (1) to (4) are augmented by constitutive equations that connect D with E, B with H, ge with E, and gm with H. In the simplest case this connection is provided by scalar constants called permittivity , permeability μ, electric conductivity σ, and magnetic conductivity s. The electric and magnetic conductivities may be monopole current conductivities as well as dipole or higher order multipole current conductivities:

=ϵE

=μH

e=σE

m=sH

In more complicated cases , μ, σ, and s may vary with location, time, and direction, which requires time-variable tensors for their representation. In still more complicated cases Eqs.(5) to (8) may be replaced by partial differential equations. For sinusoidal time variation of E, H, D, B, ge, and gm one may use functions of frequency (ω), μ(ω), σ(ω), and s(ω) but this takes one beyond the conservation law of energy and the causality law. The widespread use of (ω), μ(ω), and σ(ω) demonstrates our ability to derive useful results from wrong theories.

A number of basic relations derived from the modified Maxwell equations will be needed. They are listed here without derivation. References for their derivation are given.

The electric and magnetic field strength in Maxwell’s equations are related to a vector potential Am and a scalar potential ϕe:

=−∂Am∂t−grad?ϕe

=cZ?curl?Am

For the modified Maxwell equations we have to add a vector potential Ae and a scalar potential ϕm. Equations (9) and (10) are replaced by the following relations2:

=−Zc?curl?Ae−∂Am∂t−grad?ϕe

=cZ?curl?Am−∂Ae∂t−grad?ϕm

The vector potentials are not completely specified since Eqs.(11) and (12) define only curl Ae and curl Am. Two additional conditions can be chosen that we call the extended Lorentz convention:

?Am+1c2∂ϕe∂t=0

?Ae+1c2∂ϕm∂t=0

The potentials of Eq.(11) and (12) then satisfy the following inhomogeneous partial differential equations:

2Ae−1c2∂2Ae∂t2≡□Ae=−1Zcgm

2Am−1c2∂Am∂t2≡?□Am=−Zcge

2?ϕe−1c2∂2ϕe∂t2≡□ϕe=−Zcρe

2?ϕm−1c2∂2ϕm∂t2≡□ϕm=−cZρm

Particular solutions of these partial differential equations may be represented by integrals taken over the whole space. We note that the magnetic charge density ρm may be always zero, which implies ϕm =...