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Consider an electrical conductor. Suppose that a battery with an
electromotive field is feeding energy into this conductor.
The energy is either dissipated as heat, or is used to generate a
magnetic field. Ohm's law inside the conductor gives

(886) 
where is the true current density, is the
conductivity, and is the inductive electric field. Taking
the scalar product with , we obtain

(887) 
The lefthand side of this equation represents the rate at which the
battery does work on the conductor. The first term on the righthand
side is the rate of ohmic heating inside the conductor. We tentatively
identify the remaining term with the rate at which energy is fed into
the magnetic field. If all fields are quasistationary
(i.e., slowly varying) then the displacement current can be neglected,
and the differential form of Ampère's law reduces to
. Substituting this expression into Eq. (887),
and integrating over all space, we get

(888) 
The last term can be integrated by parts using the vector identity

(889) 
Gauss' theorem plus the differential form of Faraday's law yield

(890) 
Since
falls off at least as fast as in
electrostatic and quasistationary magnetic fields ( comes from electric
monopole fields, and from magnetic dipole fields), the surface
integral in the above expression can be neglected. Of course, this is
not the case for radiation fields, for which and
fall off like . Thus, the constraint of ``quasistationarity'' effectively
means that the fields vary sufficiently slowly that any radiation fields
can be neglected.
The total power expended by the battery can now be written

(891) 
The first term on the righthand side has already been identified as the
energy
loss rate due to ohmic heating. The last term is obviously the
rate at which energy is fed into the magnetic field. The variation
in the magnetic field energy can therefore be written

(892) 
This result is analogous to the result (851) for the variation in the
energy of an electrostatic field.
In order to make Eq. (892) integrable, we must assume a functional
relationship between and . For a medium which
magnetizes linearly, the integration can be carried out in much the
same manner as Eq. (854), to give

(893) 
Thus, the magnetostatic energy density inside a linear magnetic
material is given by

(894) 
Unfortunately, most interesting magnetic materials, such as ferromagnets,
exhibit a nonlinear relationship between and .
For such materials, Eq. (892) can only be integrated between
definite states, and the result, in general, depends on the past
history of the sample. For ferromagnets, the integral of Eq. (892) has
a finite, nonzero value when is integrated around a
complete magnetization cycle. This cyclic energy loss
is given by

(895) 
In other words, the energy expended per unit volume when a magnetic
material is carried through a magnetization cycle is equal to the area of
its hysteresis loop, as plotted in a graph of against . Thus,
it is particularly important to ensure that the magnetic
materials used to form
transformer cores possess hysteresis loops with comparatively small
areas, otherwise the transformers are likely to be extremely lossy.
Next: Magnetic induction
Up: Dielectric and magnetic media
Previous: Boundary value problems with
Richard Fitzpatrick
20060202