Page 73 - Electromagnetics Handbook

and one axial vector (B) that we also call “electric” and “magnetic.” If we follow the
lead of some authors and choose to define E and B through measurements of the Lorentz
force, then we recognize that B must be axial since it is not measured directly, but as
part of the cross product v × B that changes its meaning if we switch from a right-hand
to a left-hand coordinate system. The other polar vector (D) and axial vector (H) arise
through the “secondary” constitutive relations. Following this reasoning we might claim
that E and B are “fundamental.”
Sommerfeld also associates E with B and D with H. The vectors E and B are
called entities of intensity, describing “how strong,” while D and H are called entities
of quantity, describing “how much.” This is in direct analogy with stress (intensity) and
strain (quantity) in materials. We might also say that the entities of intensity describe
a “cause” while the entities of quantity describe an “effect.” In this view E “induces”
(causes) a polarization P, and the field D =
0 E + P is the result. Similarly B creates
M, and H = B/µ 0 − M is the result. Interestingly, each of the terms describing energy
and momentum in the electromagnetic field (D · E, B · H, E × H, D × B) involves the
interaction of an entity of intensity with an entity of quantity.
Although there is a natural tendency to group things together based on conceptual
similarity, there appears to be little reason to believe that any of the four field vectors are
more “fundamental” than the rest. Perhaps we are fortunate that we can apply Maxwell’s
theory without worrying too much about such questions of underlying philosophy.

2.7 Maxwell’s equations with magnetic sources

Researchers have yet to discover the “magnetic monopole”: a magnetic source from
which magnetic field would diverge. This has not stopped speculation on the form that
Maxwell’s equations might take if such a discovery were made. Arguments based on
fundamental principles of physics (such as symmetry and conservation laws) indicate
that in the presence of magnetic sources Maxwell’s equations would assume the forms
∂B
∇× E =−J m − , (2.169)
∂t
∂D
∇× H = J + , (2.170)
∂t
∇· B = ρ m , (2.171)
∇· D = ρ, (2.172)
where J m is a volume magnetic current density describing the flow of magnetic charge in
exactly the same manner as J describes the flow of electric charge. The density of this
magnetic charge is given by ρ m and should, by analogy with electric charge density, obey
a conservation law
∂ρ m
∇· J m + = 0.
∂t
This is the magnetic source continuity equation.
It is interesting to inquire as to the units of J m and ρ m . From (2.169) we see that if B
2 2
has units of Wb/m , then J m has units of (Wb/s)/m . Similarly, (2.171) shows that ρ m
3
must have units of Wb/m . Hence magnetic charge is measured in Wb, magnetic current
in Wb/s. This gives a nice symmetry with electric sources where charge is measured in

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