2.2 HONING OUR TOOLS
33
2.2
HONING OUR TOOLS
At least two things disqualify (1-3) as a proper formulation. One is the
non-uniqueness of b and h, a mild problem which we'll address later.
The other is the implicit and unwarranted assumption of
regularity,
or
smoothness, of these fields. For instance, div b = 0 makes perfect sense
if the three components
b 1, b 2,
b 3, in Cartesian coordinates, are dif-
ferentiable. Then (div b)(x) = 31bl(x)
+
32b2(x)
+
33bB(x), a well-defined
function of position x, and the statement "div b = 0" means that this
function is identically 0. No ambiguity about that. But we can't assume
such differentiability. 2 As one knows, and we'll soon reconfirm this
knowledge, the components of b are
not
differentiable, not even continu-
ous, at some material interfaces. Still, conservation of the induction flux
implies a very definite "transmission condition" on S.
2.2.1 Regularity and discontinuity of fields
Since smoothness, or lack thereof, is the issue, let's be precise, while
introducing some shorthands. D being a space domainf the set of all
functions continuous at all points of D is denoted C°(D). A function is
continuously differentiable
in D if all its partial derivatives are in C°(D),
and one denotes by C
I(D)
the set of such functions (an infinite-dimensional
linear space). Similarly, ck(D) or C~(D) denote the spaces composed of
functions which have continuous partial derivatives of all orders up to k
or of all orders without restriction, inside D. In common parlance, one
says that a function "is C k'', or "is C ~'' in some region, implying that
there is a domain D such that ck(D), or C~°(D), includes the restriction
of this function to D as a set element. "Smooth" means by default C,
but is often used noncommittally to mean "as regular as required", that
is, C k for k high enough. (I'll say "k-smooth" in the rare cases when
2This is not mere nit-picking, not one of these gratuitous "rigor" or "purity" issues.
We have here a tool, differential operators, that fails to perform in some cases. So it's not
the right tool, and a better one, custom-made if necessary, should be proposed, one which
will work also in borderline cases. Far from coming from a position of arrogance, this
admission that a mismatch exists between some mathematical concepts and the physical
reality they are supposed to model, and the commitment to correct it, are a manifestation of
modesty: When the physicist says "this tool works well almost all the time, and the
exceptions are not really a concern, so let's not bother", the mathematician, rather than
hectoring, "But you have no
right
to do what you do with it", should hone the tool in order
to make it able
also
to handle the exceptions.
3Recall the dual use of "domain", here meaning "open connected set" (cf. Appendix
A, Subsection A.2.3).
34 CHAPTER 2 Magnetostatics: "Scalar Potential" Approach
definiteness on this point is important.) These notions extend to vector
fields by applying them coordinatewise.
In principle, the 4gradient of a function is only defined at
interior
points of its domain of definition, since the gradient is a record of
variation rates in all directions. Depending on the local shape of the
boundary, it may still be possible to define a gradient at a boundary
point, by taking directional derivatives. How to do that is clear in the
case of a smooth boundary (on each line through a boundary point, there
is a half-line going
inwards).
But it's more problematic at a corner, at the
tip of a cusp, etc. This is why the concept of smoothness
over
a region
(not only
inside
it), including the boundary, is delicate. To avoid ambiguities
about it, I'll say that a function f is
smooth over
a region R (which may
itself be very irregular, devoid of a smooth boundary) if there is a domain
D containing R
in
which some extension of f (cf. A.1.2) is smooth. (See
Fig. 2.2.)
]
I I I i I
I I I
iR IR IR :
[ ] /
a b --~ .... ~--
R R
i
I
I
S
i I]R
I
FIGURE 2.2. Notions of smoothness, for a function of a real variable. Left to
right, functions which are: smooth
in
]a, b[, smooth
over
[a, b],
piecewise
smooth
in region R,
not
piecewise smooth.
Piecewise smooth,
then, has a precise meaning: It refers to a function,
the domain of which can be partitioned into a mosaic of regions, in finite
number,
over
each of which the function is smooth. This does not exclude
discontinuities across inner boundaries, but allows only frank disconti-
nuities (of the "first kind"), or as we shall say below, "jumps".
Exercise 2.1. Check that a piecewise smooth function f has a definite
integral
YD
I fl
on
a
bounded
domain. Is this latter assumption necessary?
Now let's return to the case at hand and see where exceptions to
smoothness can occur. In free space (IX = ~ and j = 0), rot h = 0 and
div h = 0, and the same is true of b. We have this well-known formula
which says that, for a ca-vector field u,
(4) rot rot u = grad div u - Au,
4The other meaning of the word (Subsection A.1.2).
2.2 HONING OUR TOOLS 35
where Au is the field, the components of which are {Au 1,
AU 2, AU3}.
So
both h and b are
harmonic,
All = 0 and Ab = 0, in free space. A rather
deep result,
Weyl's lemma,
can then be invoked:
harmonic functions are
C ~. So both b and h are smooth. 5 The same argument holds unchanged
in a region with a uniform g, instead of go"
h 2
b ' b
,, 2 ~ .... 1
~1 ~ - ~d- - - ih-/7../'a¢
J/ , JJ]' !n __
~2 7//~ ~t2 << lt'tl
~J 0 2
FIGURE 2.3. Flux line deviation at a material interface. The pair {h i, bi} is the
field on side i, where i= 1 or 2.
In case two regions with different permeabilities ~ and g2 are
separated by a smooth surface S (Fig. 2.3), b and h will therefore be
smooth on both sides, and thus have well-defined flux lines. 6 But the
latter will not go straight through S. They deviate there, according to
the following "law of tangents":
(5) ~.L 2
tan 01
= gl
tan 02,
where 01 and
0 2 are
the angles the flux half-lines make with the normal
n at the traversal point x. So if g~ ~ g2, neither b nor h can be
continuous at x. Formula (5) is an immediate consequence of the two
equalities, illustrated in Fig. 2.3,
(6)
n.bl=n.b 2 on
S, (7)
n×hl=nXh 2 on
S,
called
transmission conditions,
which assert that the normal part of b and
5A similar, stronger result by H6rmander [H6] implies that h and b are smooth if g
itself is C ~. Cf. [Pe]. All this has to do with one (number 19) of the famous Hilbert
problems [Br].
6A
flux line
of field b through point x 0 is a trajectory t ~ x(t) such that x(0) = x 0
and (3tx)(t) = b(x(t)). If b is smooth and b(x0) ~ 0, there is such a trajectory in some
interval ]- ~, o~[ including 0, by general theorems on ordinary differential equations. See,
e.g., [Ar], [CL], [Fr], [LS].
36 CHAPTER 2 Magnetostatics: "Scalar Potential" Approach
the tangential part of h are continuous across S, and which we now
proceed to prove.
The proof of (6) comes from an integral interpretation of Faraday's
law. By the latter, the induction flux through any
closed
surface vanishes.
Let's apply this to the surface of the "flat pillbox" of Fig. 2.4, built from
the patch f2 (lying in S) by extrusion. This surface is made of two
surfaces f~ and
~'~2
roughly parallel to S, joined by a thin lateral band.
Applying Ostrogradskii 7 and letting the box thickness d go to 0, one
finds that ~ (n.
b I - n. b2) = 0/because
the contribution of the lateral
band vanishes at the limit, whereas n. b on f~ and f~ respectively
tend to the values
n. b I
and
n. b 2
of n. b on both sides of f2. Hence
(6), since f2 is arbitrary.
ii
n
FIGURE
2.4. Setup and notations for the proof of (6) and (7).
As for (7), we rely on the integral interpretation of Amp6re's theorem:
the circulation of h along a closed curve is equal to the flux of j + ~)t d
through a surface bound by this curve. Here we apply this to the "thin
ribbon" of Fig. 2.4, built by extrusion from the curve y lying in S. Since
j + 3td = 0 in the present situation, the circulation of h along the
boundary of the ribbon is zero. Again, letting the ribbon's width d go
to 0, we obtain Y~ (~. h I - I;. ha) = 0, which implies, since y is arbitrary,
the equality of the projections (called "tangential parts") of h 1 and h 2
onto the plane tangent to S. This equality is conveniently expressed
by (7).
Fields therefore fail to be regular at all material interfaces where ~t
presents a discontinuity, and div or rot cease to make sense there.
Some regularity subsists, however, which is given by the interface
conditions (6) and (7). For easier manipulation of these, we shall write
them [n. b] s = 0 and [n x h]s = 0, and say that the
jumps
of the normal
part of b and of the tangential part of h vanish at all interfaces. Before
discussing the possibilities this offers to correctly reformulate (1-3), let's
explain the notation and digress a little about jumps.
7Flux, circulation, and relevant theorems are discussed in detail in A.4.2.
2.2 HONING OUR TOOLS 37
2.2.2 Jumps
This section is a partly independent development about the bracket
notation [ ] for jumps, which anticipates further uses of it.
Consider a field (scalar- or vector-valued) which is smooth on both
sides of a surface S, but may have a discontinuity across S, and suppose
S is provided with a crossing direction. The
jump
across S of this
quantity is by definition equal to its value just before reaching the surface,
minus its value just after. (The jump is thus counted downwards; rather
a "drop", in fact.) Giving a crossing direction through a surface is
equivalent to providing it with a continuous field of normals. One says
then that the surface has been
externally oriented.8
Not all surfaces can thus be oriented. For one-sided surfaces (as
happens with a M6bius band), defining a continuous normal field is not
possible, and the crossing direction can only be defined locally, not
consistently over the whole surface. For surfaces which enclose a volume,
and are therefore two-sided, the convention most often adopted consists
in having the normal field point outwards. This way, if a function qo is
defined inside a domain D, and equal to zero outside, its jump [qo] s
across the surface S of D is equal to the trace qo s of qo, that is, its
restriction to S if qo is smooth enough, or its limit value from inside
otherwise. The conventions about the jump and the normal thus go
together well.
For interfaces between two media, there may be no reason to favor
one external orientation over the other. Nonetheless,
D
some quantities can be defined as jumps in a way which 2
does not depend on the chosen crossing direction, n2
Consider for instance the flux of some vector field j
through an interface S between two regions D 1 and nl
D 2
(inset). Let n I and n 2 the normal fields defined
according to both possible conventions:
n I
points from
D 1
towards
D2,
and n 2 points the other way. Suppose
we choose n~ as the crossing direction, and thus set n = n 1. Then the
jump of the normal component n. j across S is by definition equal to its
value on the D 1 side, that is, n I . j, minus the value of n. j on the D 2
side, which is -n 2 . j. The jump is thus the
sum
n I .
j + n 2 . j.
This is
symmetrical with respect
to D 1
and
D2, so
we are entitled to speak of
"the jump of n . j across S" without specifying a crossing direction.
Whichever this direction, the decrease of n. j when crossing will be the
8Which suggests there is also a different concept of
internal
orientation (cf. Chapter 5).
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