1.2 CONSTITUTIVE LAWS 11
- 0 in such regions 2° and to write a generalized Ohm's law, valid for
generators, conductors, and insulators alike (and thus, most often,
uniformly valid in all space):
(13) j = o e + jg.
It all goes then as if the charge dynamics problem had been solved in
advance, the result being given by (13). One often calls
passive
conductors
those ruled by (12), generators being then dubbed
active.
One may then append (13) to (1-4), with p = 0 and m = 0. The
system of equations thus obtained (or "Maxwell's model with linear
conductors") embodies the theory of
nonmoving
(cf. Note 19) active and
passive conductors which are neither polarizable nor magnetizable (cf.
next section). It deals with a two-compartment system, EM and CM
again, but the theory we have accepted for the latter is so simple, being
all contained in (13), that one may easily overlook the coupled nature of
the whole system. (One should not.)
1.2.3 Dynamics of bound charges: dielectric polarization
Now, another case of two-compartment system for which the same
approach leads to a specific constitutive law. It deals with polarizable
materials, in which charges are too strongly bound to separate from their
original sites, but loose enough to be pulled a little off their equilibrium
position by Coulomb forces, when the material is subject to a macroscopic
electric field. This
polarization
phenomenon is important for some
materials, dubbed
dielectric.
The simple reasoning (or myth... ) that
follows shows how to account for it, by a simple relation between e and
the p of (3).
Despite its electrical neutrality at a macroscopic scale, matter contains
positive and negative charges (+ and - for brevity) which we may
imagine as being attached by pairs at certain material sites. Suppose the
density of + charges is equal to q+, a function of position x. In the
absence of any macroscopic electric field, the density of -charges must
equal - q+, by electric neutrality. Now, a field e being applied, let's
represent by a vector field u the separation of charged pairs that results,
as follows: A + charge [resp. a - charge] that was at point x is now at
x + u(x)/2 [resp. at x - u(x)/2]. To easily compute the new charge
2°This amounts to neglecting the internal resistance of the generator. In some modellings,
having a nonzero r~ there can be useful. Note this wouldn't change the form of (13).
12 CHAPTER 1 Introduction: Maxwell Equations
densi. .ty .qp due21 to this. change in localization,, let us treat it as a mathematical
&strzbutzon, that is, as the mapping ~ --~ Y qp ~, where ~ denotes a
so-called "test function". Expanding ~ to first order and integrating by
parts, we have
qp ~/= ~ q+(x) [~(x + u(x) / 2) - ~(x - u(x) / 2)] dx
~ J q+ u. grad ~ = J-div(q÷ u) ~,
hence qp = -div p, where p = q+ u. This field p, soon to be identified
with the one in (3), is the polarization of the dielectric.
Exercise 1.7. Try to do the same computation "the other way around",
by starting from ~ qp ~ = ~ q+(x - u(x)/2) ~(x) - ~..., etc. Why does it go
wrong this way?
The macroscopic manifestation of this local charge splitting is thus
the appearance of a distribution of charges in what was initially an
electrically neutral medium. Moreover, if the polarization changes with
time, the motion of charges + and - in opposite directions amounts to
a current density jp = 3t(qu) = 3 tp. (Note that 3 t qp + div jp = 0, as it
should be.)
We might treat this current density on the same footing as j, and
replace the polarized matter by vacuum plus polarization current. Then
d = ¢0 e, and Eqs. (1) and (3) would combine to give
(14)
- 3t(£0 e) +
rot h = j
+ ~)tP"
Instead, we use our option (cf. Exer. 1.1) to charge 3 t p on the account of
Eq. (3), by setting d = g0 e + p, hence - 3td + rot h = -~)t(~0 e + p) + rot h =
j + 3tp - 3 t p = j, leaving e unchanged. This separates macroscopic
21In the theory of distributions [Sc], functions are not defined by their values at points
of their domain of definition, but via their effect on other functions, called test functions. So,
typically, a function f over some domain D is known if one is given all integrals JD f V,
for all smooth ~ supported in D. It is thus allowable to identify f with the linear
mapping ~ --~ ~D f V. (The arrowed notation for maps is discussed in A.1.9.) This has the
advantage of making functions appear as special cases of such linear TEST_FUNCTION --~
REAL_NUMBER mappings, hence a useful generalization of the notion of function: One
calls such maps distributions, provided they satisfy some reasonable continuity requirements.
For instance, the map V ~ ~(a), where a is some point inside D, is a distribution
("Dirac's mass" at point a, denoted 8a). The generalization is genuine, since there is no
function fa such that ~(a) = ~D fa~ for all ~. It is useful, because some theories, such as
Fourier transformation, work much better in this framework. The Fourier tranform of the
constant 1, for instance, is not defined as a function, but makes perfect sense as a distribution:
It's (2~) d/2 times a Dirac mass at the origin,
i.e., (2~) d/2 80,
in spatial dimension d.
1.2 CONSTITUTIVE LAWS
13
currents j, which continue to appear on the right-hand side of the expression
of Faraday's law, and microscopic (polarization) currents jp = ()t P, now
hidden from view in the constitutive law. Notice that div d = q, where
q is the macroscopic charge, and div(e0e) = q + qp.
All this shuffling, however, leaves the polarization current to be
determined. The "(coupled) problem of bound charges" would consist
in simultaneously computing p and the electromagnetic field, while
taking into account specific laws about the way charges are anchored to
material sites. Just as above about conduction, one makes do with a
simple--and empirically well confirmedmsolution to this problem, which
consists in pretending (by invoking a "myth", again) that p and e are
proportional: p = )~e, as would be the case if charges were elastically
bound, with a restoring force proportional to e, and without any inertia. 22
Now, let us set e =~0 + )¢. Then, Eqs. (1) and (3) become
(1')
-Ot d +
rot h = j, (3') d = ee.
The coefficient ~ in (3') (called
permittivity,
or
dielectric constant
of the
medium 23) thus appears as the simple summary of a complex, but
microscopic-scale interaction, which one doesn't wish to know about at
the macroscopic scale of interest.
Another, simpler solution of the coupled problem obtains when one
may consider the field p, then called
permanent polarization,
as independent
of e. The corresponding behavior law, d = ~0 e + p with fixed p, is well
obeyed by a class of media called
electrets.
Of course one may superpose
the two behaviors (one part of the polarization being permanent, the
other one proportional to e), whence the law d = ee + p instead of (3),
with a fixed p.
1.2.4 Magnetization
It is tempting to follow up with a similar presentation of magnetization,
where a proportionality between m and h would be made plausible by
a simple myth about the interaction of magnetic moments (due to the
electrons' spins, mainly) with the magnetic field. This would be a little
artificial, however, because too remote from the real physics of magnetism
22The latter hypothesis will be reconsidered in the case of high frequencies. Note that
)¢ can be a tensor, to account for anisotropy.
23Terminology wavers here. Many authors call "permittivity" the ratio between ~ and
~0, and speak of "dielectric constant" when it comes to ~, or even to its real part in the case
when ~ is complex (see below). Note again that ~ may be a tensor.
14 CHAPTER 1 Introduction: Maxwell Equations
(cf., e.g., [OZ]), and the point is already made anyway: Constitutive laws
substitute for a detailed analysis of the interaction, when such analysis is
either impossible or unproductive. So let us just review typical constitutive
laws about magnetization.
Apart from
amagnetic
materials (m = 0), a simple case is that of
paramagnetic
or
diamagnetic
materials, characterized by the linear law m
= xh (whence b = lxh, with ~t = (1 + X)~t0), where the
magnetic susceptibility
is of positive or negative sign, respectively. It can be a tensor, in the
case of anisotropic materials. For most bodies, X is too small to matter
in numerical simulations, the accuracy of which rarely exceeds 1%
(X ~10 -4 for A1 or Cu).
Ferromagnetic
metals (Fe, Co, Ni) and their alloys are the exception,
with susceptibilities up to 10 5, but also with a nonlinear (and hysteretic 24)
behavior beyond some threshold. In practice, one often accepts the linear
law b = ~h as valid as far as the modulus of b does not exceed 1
tesla. 25
For
permanent magnets
[La, Li] a convenient law is m = xh + hm,
where h is a vector field independent of h and of time, supported by
the magnet (that is, zero-valued outside it), with X roughly independent
of h, too, and on the order of 1 to 4, in general [La]. This law's validity,
however, is limited to the normal working conditions of magnets, which
means h and b of opposite signs, and not too large. The characteristic
b = ~h
+
~t0h m is then called the "first order reversal curve".
1.2.5 Summing up" linear materials
Hysteresis, and nonlinearity in general, are beyond our scope, and we
shall restrict to the "Maxwell model of memoryless linear materials with
Ohm's law":
24Hysteresis
occurs when the value of b at time t depends not only on h(t), but on
past values.
Linearity
does not preclude hysteresis, for it just means that if two field
histories are physically possible, their superposition is possible too. This does not forbid
behavior laws "with memory", but only allows "convolution laws" of the form b(t) =
j~ M (t - s) h(s) ds. As we shall see in Section 1.4, this amounts to
B = ~I--I,
in Fourier space,
with a complex and frequency-dependent ~t.
25The weber per square meter, that is, the unit for b, is called the
tesla
(T). (One tesla is
10 000 gauss, the cgs unit still in use, alas.) The field h is measured in amp6res per meter
(A/m). An ordinary magnet creates an induction on the order of .1 to 1 T. The Earth
field is about 0.4 x 10 --4 tesla.
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.