20 CHAPTER 1 Introduction: Maxwell Equations
When studying the dynamics of moving conductors, one should take
into account the momentum of the moving bodies
and
the momentum of
the field in the expression of momentum conservation. 33 In an interaction
between two solids, for instance, momentum lost by one of them may
temporarily be stored in the field, before being restituted to the other
body. Thus, action and reaction may seem not to balance, in apparent
violation of Newton's third law [Ke]. See for instance [Co], [Ho], and the
abundant literature on Feynman's "disk paradox", a situation in which a
disk, initially at rest in a static field, can acquire angular momentum
without any mechanical action, just because of a change in the
electromagnetic environment [Lm].
Remark
1.3. So there are
static
configurations in which ~ d x b ~ 0"
Surprising as this may appear, a static electromagnetic field
can
possess
linear momentum. (Cf. R.H. Romer, Am. J. Phys., 62, 6 (1994), p. 489.
See also [PP].) 0
Remark
1.4. The cross product is an orientation-dependent operation:
its very definition requires a rule for orienting ambient space. Yet we see
it appear in expressions such as e x h or d x b, which account for
energy or momentum flux, physical quantities which obviously do
not
depend on orientation conventions. How come? It must be that some of
the vector fields e, h, d, b themselves depend on orientation. No
surprise in that: The
mathematical
entities by which the physical field is
represented may depend on the structures of Euclidean space, whereas
the objective phenomena do not. The question is further discussed in
Section A.3 of Appendix A. 0
1.4 DERIVED MODELS
Concrete problems in electromagnetism rarely require the solution of
Maxwell equations in full generality, because of various simplifications
due to the smallness of some terms. The displacement currents term
3re, for instance, is often negligible; hence an important submodel,
eddy-currents
theory, which we shall later study in its own right:
(28)
3tb+rote=0, roth=j, j=c~e+j g,
33Many papers in which this commonsense rule is neglected get published,
notwithstanding, in refereed Journals. It has been asserted, for example, that the operation
of a railgun cannot be explained in terms of classical electrodynamics. See a refutation of
this crankish claim in [AJ].
1.4 DERIVED MODELS 21
with in particular, in passive conductors (where one may eliminate e
from (28) after division by G), 3t(~h) + rot ((~-1 rot h) = 0.
Another frequent simplification is the passage to complex numbers
representations. If the source current jg is sinusoidal in time, 34 that is,
of the form jg(t, x) = Re[jg(x) exp(i~ t)], where
Jg
is a
complex-valued
vector field, and
if
all constitutive laws are linear, one may 35 look for the
electromagnetic field in similar form, h(x) = Re[H(X) exp(i~t)], etc., the
unknowns now being the complex fields H, E, etc., independent of time.
Maxwell's model with Ohm's law (15-18) then assumes the following
form:
(29)
--i¢,OD + rotH
=Jg -t-
(~Ez i03B + rotE=0, D =~E, B=~tH.
It is convenient there to
redefine
e by assigning to this symbol the
complex value ~ + (~/(i0~), which allows the incorporation of the term
(~ E into ic0 D, whence the model
(29') -i~D + rotH =Jg, i00B + rot E=0,
D=EE,
B=~tH,
which is, with appropriate boundary conditions, the
microwave oven
problem. In (29'), e is now complex, and one often writes it as ¢ = ¢'-
i¢", where the real coefficients ¢' and ~", of same physical dimension as
~0, are nonnegative. (They often depend on temperature, and are measured
and tabulated for a large array of products, foodstuffs in particular. Cf.
eg., [FS, St, Jo]. Figure 1.4 gives an idea of this dependence.)
Nothing forbids accepting complex ~t's as well, and not only for the
sake of symmetry. This really occurs with ferrites 36 [La, Li], and also in
some modellings, a bit simplistic 37 perhaps, of hysteresis.
34One often says "harmonic", but be wary of this use, not always free of ambiguity.
35This procedure is valid, a priori, each time one is certain about the
uniqueness
of the
solution of the problem "in the time domain", for if one finds a solution, by whatever
method, it's bound to be the right one. But it's the
linearity
of constitutive laws (cf. Note 24)
that makes the procedure effective. Moreover, linearity allows one to extend the method to
non-periodic cases, thanks to Laplace transform (then one has p, complex-valued, in lieu of
i0~). The passage to complex numbers is
in principle
of no use in nonlinear cases (for
instance, when iron or steel is present), and the notion of "equivalent (complex) permeability",
often invoked in applications to induction heating, is not theoretically grounded. (Its
possible empirical value is another question, to be considered in each particular instance.)
36One
refers to
linear
behavior there, and this complex permeability is not of the same
nature as the one of the previous note.
37Because of their essentially
linear
nature. Law B = (~t'- i~t")H amounts to ~t" 3th =
~t'h - b) in the time domain.
22 CHAPTER 1 Introduction: Maxwell Equations
E'/£0
5O
J
-40
o (oc)
0 40 - 40
£"/E'
0.5
0
0
(oc)
40
FIGURE 1.4. Typical curves for ¢' and ¢" as functions of temperature, for a
stuff with high water content. The ratio ¢"/E', shown on the right, is often
denoted by tan 8.
An even more drastic simplification obtains when one may consider
the phenomena as independent of time (steady direct current at the
terminals, or current with slow enough variations). Let us review these
models, dubbed
stationary,
derived from Maxwell's model by assuming
that all fields are independent of time.
In this case, one has in particular 3tb = 0, and thus rot e = 0. So,
after (5) and (17),
(30) rot e = 0, d = Ce, div d = q,
and this is enough to determine e and d in all space, if the electric
charge q is known: Setting e =-grad ~, where ~ is the
electric
potential,
one has indeed - div(¢ grad ~) = q, a Poisson problem which is,
as one knows, well posed. In the case where ¢ = ~0 all over, the solution
is given by
1 f~ q(y)
~(x)- 4xs0
--3
lx-yl
dy,
as one will check (cf. Exers. 4.9 and 7.5) by differentiating under the
summation sign in order to compute A~. Mode] (30) is the core of linear
electrostatics.
In a similar way, one has rot h = j, after (1), whence, taking into
account div b = 0 and (18), the model of linear
magnetostatics:
(31) rot h = j, b = l.th, div b = 0,
and this determines b and h in all space when j is given. If l.t = !% all
1.4 DERIVED MODELS 23
over, the solution is obtained in closed form by introducing the vector
field
~t° fg J(Y)
a(x)-- -~ 3 Ix-yl dy,
called
magnetic vector potential,
and by setting b = rot a. (By differentiating
inside the integral, one will find
Biot and Savart's formula,
which directly
gives h in integral form:
1 YE j(y) x (x - y)
(32) h(x)
-- ~ 3
IX -- y
I 3
dy.)
When, as in the case of ferromagnetic materials, constitutive laws more
involved than b = ~t h occur, problem (31) appears as an intermediate in
calculations (one step in an iterative process, for instance), with then in
general a position-dependent ~ An important variant is the magneto-
statics problem for a given distribution of currents and magnets, the
latter being modelled by b = ~th + ~t0h m with known ~t and (vector-valued)
38 Setting h m 0 in the air, one gets
hIIl* ---
rot h = j, b = ~th + ~10hm, div b = 0.
An analogous situation may present itself in electrostatics: d = Ce + p,
with p given, as we saw earlier.
Still under the hypothesis of stationarity, one has 3tq = 0, and thus
div
j = 0,
after (6), hence
(33) rot e = 0, j = r~e, div j = 0,
in passive conductors. This is the
conduction
or
electrokinetics model.
In
contrast to the previous ones, it does not usually concern the whole
space, and thus requires boundary conditions, at the air-conductor
interfaces, in order to be properly posed.
The formal similarity between these static models is obvious, and we
need examine only one in detail to master the others. We'll focus on
38A
legitimate question, at this stage, would be, "How does one know h m, for a given
permanent magnet?". Giving a rigorous answer would require the knowledge of the
conditions under which the material has been magnetized, as well as the details of its
hysteretic response, and a feasible simulation method of this process. In practice, most
often, a uniform magnetization field parallel to one of the edges of the magnets is a fair
representation. However, as more and more complex magnetization patterns are created
nowadays, the problem may arise to find h m from measurements of b by a computation
(solving an inverse problem).
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