38 CHAPTER 2 Magnetostatics: "Scalar Potential" Approach
same, because the sign of n intervenes twice, in the choice of direction,
and in the choice of which side one "jumps from". Hence the definition
of the jump of j as [n. j] = n 1 . Jl + n2" J2, where Jl and J2 are the values
on each side of S.
Such jumps often have interesting physical interpretations. For
instance, if j is a current density, the jump is equal to the intensity that
passes from the bulk of the conductor to the interface, and is thus instilled
in S, so to speak. As charge is conserved, the jump must vanish (and so
it goes at the interface between two conductors), unless some mechanism
exists to remove this charge flux. Which may happen: For instance, if S
corresponds to a highly conducting inclusion inside a normal conductor,
the current [n. j] entering S at some place will be conveyed along S, to
exit at some other place, where [n. j] will be negative. Note that such
considerations would apply to a M6bius band without any problem.
In the case of the electric induction d, the jump [n. d] s is the density
of electric charge present on surface S; hence the interface condition
[n .d] s = 0, unless there is a physical reason to have electric charge concen-
trated there. Same thing with b, and magnetic charge. Our proof above
that [n. b] = 0 across all interfaces made implicit use of the absence of
such charge. But there are problems in which the jump of a quantity
denoted n. b can be nonzero. This happens, for instance, when fictitious
surface magnetic charges are used as auxiliary quantities in integral
methods, and then [n. b] = q, the fictitious charge density.
A bit different is the case of vector quantities, such as the magnetic
field. The jump [h] s is simply the field, defined on S, obtained by
taking the jumps of the three coordinates. The subject of interest, however,
is more often the jump of the
tangential
part of h.
If h is smooth, we call
tangential part
and denote by h s the field of
vectors tangent to S obtained by projecting h(x), for all x in S, on the
tangent plane T x at x (Fig. 2.5, left). If h is smooth on both faces of S
but discontinuous there, there are two bilateral projections hsl and hs2,
and the jump of h s, according to the general definition, is [hs] S = hsl - hs2
in the case of Fig. 2.5. The sign of this of course depends on the crossing
direction. But the remark
(8)
[hs] s = - n x [n x hls = - n x (n I x h 1 + n 2 x h2)
points to the orientation-independent surface vector field [n x h]s. This
is equal to the jump [hs] s of the tangential part, up to a 90 ° rotation,
counterclockwise, around the normal. The cancellation of the tangential
jump is thus conveniently expressed by In x h]s = 0.
2.2 HONING OUR TOOLS
39
This field [n x h] is interesting for another reason. As suggested by
Fig. 2.5, right, [n x h] is always equal to minus the current density J s (a
surface vector field, thus modelling a "current sheet") supported by the
interface. For instance, if the crossing direction is from region I to region
2, and thus n = n 1 =- n2, then In x h] = n a x h I + n 2 x ha, which is -Js by
Amp6re's theorem. We find the same result with the other choice. Again,
if there is no way to carry along the excess current (such as, for instance,
a thin sheet of high conductivity borne by S), then Js = - [n x h] = 0,
which is the standard transmission condition about h we derived earlier.
nA h
x ~
~J -~Y/ ~~ -~n
JSSj ~ ~ D
, 2
FIGURE 2.5. Left: Definition of h s. Right: Relation between Js and the jump of
h s. (Take the circulation of h along the circuit indicated.)
In a quite similar way, [n x e] is equal, irrespective to the choice of
normal, to the time-derivative of the induction flux vector, 3tb s, along
the surface. This is most often 0mhence the transmission condition
[n x e] = 0rebut not always so. By way of analogy with the previous
example, the case of a thin highly permeable sheet will come to mind.
But there are other circumstances, when modelling a thin air gap, for
instance, or a crack within a conductor in eddy-current testing simulations,
when it may be necessary to take account of the induction flux in a
direction tangential to such a surface.
2.2.3 Alternatives to the standard formalism
Back to our critical evaluation of the ill-specified equation div b = 0:
What can be done about it? A simple course would be to explicitly
acknowledge the exceptions, and say, "We want div b = 0 wherever b
is effectively differentiable, and [n. b] = 0 across all material interfaces
and surfaces where a singularity might occur." Indeed, many textbooks
list transmission conditions as equations to be satisfied, and add them to
Maxwell equations, on almost the same footing.
40
CHAPTER 2 Magnetostatics: "Scalar Potential" Approach
It would be quite awkward, however, always to be forced to dot i's
and cross t's that way. Besides, and more importantly, the practice of
finite elements does not suggest that material interfaces should contribute
additional
equations. So there must be a way to stretch the meaning of
statements such as div b = 0 in order to
imply
the transmission conditions.
In fact there are two main ways. A radical one: differential forms; and a
moderate one: weak formulations, laid on top of the bedrock of classical
vector calculus.
The radical way will not be followed here, but must be mentioned,
because being aware of its existence helps a lot in understanding the
surprising analogies and formal symmetries that abound in the classical
approach. When looking for substitutes for the
differential, local
equations
div b = 0 and rot h = 0, we invoked
integral, global
relations: flux
conservation, Amp6re's theorem. All electromagnetic laws (apart from
constitutive laws) say things like "This circulation along that line is equal
to this flux across that surface, this volume integral equals that charge",
and so forth, with line, surface, and volume in a definite and simple
relationship, such as "is the boundary of". The laws thus appear as
relations between real quantities assigned to geometrical elements (points,
lines, surfaces, volumes), and the scalar or vector fields are there as a
way to compute these quantities.
Once we begin to see things in this light, some patterns appear.
Fields like e and h are definitely associated with
lines:
One takes their
circulations,
which are electromotive forces (emf) and magnetomotive
forces (mmf). The same can be said about the vector potential a. And it
can't be a coincidence either if when a curl is taken, one of these fields is
the operand. Fields like b and d, or j, in contrast, are
surface
oriented,
their fluxes
matter, and it's div, rarely rot, which is seen acting on them.
Even the scalar fields of the theory (charge density q, magnetic potential
q0, electric potential ~) have an associated dimensionality: Point values
of q0 and ~ matter, but only volume integrals of q are relevant, and
terms like grad q are never encountered, contrary to grad q0 or grad ~.
This forces us to shift attention from the fields to the linear mappings
of type
GEOMETRIC_ELEMENT ~ REAL_NUMBER
they help realize.
For instance, what matters about h, physically, is not its pointwise values,
but its circulations along lines (mmf). Thus, the status of h as a
LINE
REAL
linear map is more important than its status as a vector field. The
status of b as a
SURFACE --~ REAL
linear map is what matters (and in
this respect, b and h are different kinds of vector fields). The
(mathematical) fields thus begin to appear as mere props, auxiliaries in
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