CHAPTER 6
The "Curl Side":
Complementarity
We return to the model problem of Chapter 2, the magnetostatics version
of the "Bath cube" setup. This time, the two mirror symmetries with
respect to vertical planes are taken into account (Fig. 6.1): The field b is
obviously invariant with respect to both reflections, hence n. b = 0 on
these planes. 1
/'3
/ /
z ( ,c
FIGURE 6.1. Notations for solving the Bath-cube magnetostatics problem in a
quarter of the cavity. Symmetry planes bear the boundary condition n. b = 0,
hence contributin~to the S b boundary. The "link" c must go fromhthe upperh
pole (here, part S0 of the boundary S h) to the lower pole (part S 1 of S),
whereas the "cut" C should separate the two poles, while having its own
boundary 3C inside S b. (In the jargon of Fig. 4.6, c and C are "closed mod S h
[resp. mod S b] and non-bounding".)
Denoting by D the domain thus defined, and by S its boundary,
• h . • .
S b
with S the magnetic boundary and the part of S at the top of the
cavity (cf. Fig. 2.6) and in symmetry planes, we must have:
1We could reduce further to an eighth, by taking into account the invariance with
respect to a 90 ° rotation around the z-axis. But the symmetry group thus obtained is not
Abelian, a feature which considerably complicates the exploitation of symmetry. Cf. Refs.
[B1, B2] of Appendix A.
163
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