CHAPTER 8
Eddy-current Problems
We now move beyond magnetostatics to tackle a non-stationary model.
The starting point is again Maxwell's system with Ohm's law:
(1) - 3 t d + rot h =
j,
(2)
0 tb
+ rot e = 0,
(3) d = ~ e, (4) j = jg + rse, (5) b = ~t h,
where jg is a given current density. In almost all of this chapter, we
suppose jg "harmonic", that is, of the form I
(6) jg(t) = Re[J g exp(i03t)],
where jg is a
complex-valued
vector field, and we'll look for all fields in
similar form: h(t) = Re[H exp(io)t)], e(t)= Re[E exp(ic0t)], etc. The functional
spaces where these fields roam will still be denoted by IL 2, ILaro¢ etc., but
it should be clearly understood that
complexified
vector spaces are meant
(see A.4.3). By convention, for two complex vectors u = u R + iu~ and v =
v R + iv I, one has u . v = u R . v R -i u I . v I + i(u I . v R + u R . vi), the Hermitian
scalar product being u. v*, where the star denotes complex conjugation,
and the norm being given by I uI 2 = u . u*, not by u.u. Note that an
expression such as (rot u) 2 should thus be understood as rot u. rot u,
not as I rot u I 2.
The form (6) of the given current is extremely common in electrotech-
nical applications, where one deals with alternating currents at a well-
defined frequency f. The constant 03 = 2~ f is called
angular frequency.
Under these conditions, system (1-5) becomes
(7) - i0)D + rot H = J1 (8) ico B + rot E
=
0,
(9)
D --- I~ E,
(10)
J-- Jg + (~EI
(11)
B = ~t H.
1There are other possibilities, such as Im[J g exp(i03t)], or Re[q-2 Jg exp(i03t)], etc. What
matters is consistency in such uses.
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