24 CHAPTER 1 Introduction: Maxwell Equations
magnetostatics
in this book, with only a few indications about the other
models in Chapters 8 and 9. This disproportion is to some extent mitigated
by the paradigmatic character of the magnetostatics model. As pointed
out in the Preface, the difficulties encountered in computational electro-
magnetism in the 1970s, when one tried to extend then well-established
finite element or boundary integral 2D methods to three-dimensional
situations, appear in retrospect to be due not to the increased dimension-
ality per se, but to the essential difference between the "curl-curl" operator
and the "div-grad" operator to which it reduces in two dimensions, and
fortunately, all essential points about the curl-curl operator can be
understood in the simple, limited, and well-defined framework of linear
magnetostatics.
EXERCISES
The text for Exer. 1.1 is on p. 5.
Exercise 1.2. Let X be an affine space and V the associated vector
space, f: IR x X x V --~ IR a
repartition function,
which one interprets as
the time-dependent density of some fluid in configuration space X x V.
Let ~(t, x) be the acceleration imparted at time t to particles passing at
x at this instant, by some given external force field. Show that
(34) 0tf + v. V× f + 7. Vvf = 0
expresses
mass conservation
of this fluid. What if y, instead of being a
data, depended on velocity?
Exercise 1.3. What is the divergence of the field x --~ a x x, of type
E 3 --~ V 3, where a is a fixed vector? Its curl? Same questions for x ~ x.
(Cf. Subsection A.1.2 for the notion of type, and the notational convention,
already evoked in Note 13, and Note 29.)
Exercise 1.4. In the context of Exer. 1.2, what is the divergence of the
field v --~ e + v x b ?
Exercise 1.5. Establish Vlasov's equation (11).
Exercise 1.6. Prove, using (11), that charge and current as given by (8) do
satisfy the charge conservation relation (6).
EXERCISES 25
See p. 12 for Exer. 1.7, pp. 18 and 19 for Exers. 1.8 to 1.10.
Exercise 1.11. Show that, in a region of a conductor where ~ is not
constant (due to variations in temperature, or in the composition of an
alloy, etc.), q -- div d may not be zero, and that this can happen in
stationary situations (continuous current). Thus, there can exist a
permanent charge imbalance at some places in the conductor. But Lorentz
force acts on this charge packet.
Why doesn't it move?
HINTS
1.1. Don't worry about differentiability issues: Assume all fields are
smooth.
1.2. Imitate the classical computation about the convective derivative in
fluid dynamics (which is very close to our treatment of charge conservation,
p. 4).
1.3. For x ~ a x x, divergence: 0, curl: 2a. For x ~ x, curl-free, the
divergence is the constant scalar field x ~ 3.
1.4. Mind the trap. Contrary to e and b, this field does
not
live in 3D
Euclidean space! The
type
of the map will tell you unambiguously what
"divergence" means.
1.5. Apply Exer. 1.2, acceleration being Qe(e + v × b). By Exer. 1.4, there
is no extra term.
1.6. Ostrogradskii on {t, x} x V. Ensure suitable boundary conditions by
assuming, for instance, an upper bound for the velocity of charges.
1
1.7. A careless attempt, like 39 q÷(x- u(x)/2)~(x) *= -~ Vq+ . u, would
seem to lead to -~ q+ div(~u), and hence to a different result than above
if div u ¢ 0. This is the key: Why does this divergence matter?
1.8. A bar magnet between the plates of a condenser.
1.9. This is an extension of the integration by parts formula (2.9) 40 of the
next chapter, ~D V. grad u i =
- ~D
ui div v + ~s n. v u, i = 1, 2, 3.
39The star in *= serves as a warning that the assertion should not be believed blindly.
4°As a
rule, we'll refer to "Eq. (n)" inside a chapter, and to "Eq. (X.n)" for the equation
labelled (n) in Chapter X.
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