Page 31 - Electromagnetics Handbook

Next we substitute for I (φ, t) to get

I 0 ωa ωa
∂ρ(r, t)
− sin φ δ(ρ − a)δ(z) cos ωt =− .
ρ c c ∂t
Finally, integrating with respect to time and ignoring any constant term, we have

ωa
I 0
ρ(r, t) = δ(ρ − a)δ(z) sin φ sin ωt,
c c
where we have set ρ = a because of the presence of the factor δ(ρ − a).

1.3.5 Magnetic charge

We take for granted that electric fields are produced by electric charges, whether
stationary or in motion. The smallest element of electric charge is the electric monopole:
a single discretely charged particle from which the electric field diverges. In contrast,
experiments show that magnetic fields are created only by currents or by time changing
electric fields; hence, magnetic fields have moving electric charge as their source. The
elemental source of magnetic field is the magnetic dipole, representing a tiny loop of
electric current (or a spinning electric particle). The observation made in 1269 by Pierre
De Maricourt, that even the smallest magnet has two poles, still holds today.

In a world filled with symmetry at the fundamental level, we find it hard to understand
why there should not be a source from which the magnetic field diverges. We would call
such a source magnetic charge, and the most fundamental quantity of magnetic charge
would be exhibited by a magnetic monopole. In 1931 Paul Dirac invigorated the search for
magnetic monopoles by making the first strong theoretical argument for their existence.
Dirac showed that the existence of magnetic monopoles would imply the quantization
of electric charge, and would thus provide an explanation for one of the great puzzles
of science. Since that time magnetic monopoles have become important players in the
“Grand Unified Theories” of modern physics, and in cosmological theories of the origin
of the universe.

If magnetic monopoles are ever found to exist, there will be both positive and negatively
charged particles whose motions will constitute currents. We can define a macroscopic
magnetic charge density ρ m and current density J m exactly as we did with electric charge,
and use conservation of magnetic charge to provide a continuity equation:

∂ρ m (r, t)
∇· J m (r, t) + = 0. (1.18)
∂t
With these new sources Maxwell’s equations become appealingly symmetric. Despite
uncertainties about the existence and physical nature of magnetic monopoles, magnetic
charge and current have become an integral part of electromagnetic theory. We often use
the concept of fictitious magnetic sources to make Maxwell’s equations symmetric, and
then derive various equivalence theorems for use in the solution of important problems.
Thus we can put the idea of magnetic sources to use regardless of whether these sources
actually exist.

© 2001 by CRC Press LLC