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Multipole Theory in ElectromagnetismClassical, quantum, and symmetry aspects, with applications$

Roger E. Raab and Owen L. de Lange

Print publication date: 2004

Print ISBN-13: 9780198567271

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198567271.001.0001

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Multipole Theory in Electromagnetism
Oxford University Press

We express the total force on a steady current distribution in a slowly varying external magnetic field (equation (1.58)) in terms of the magnetic multipole moments (equation (1.59)). The leading term in the expansion of the force (1.58) is from (A.15)

ɛ ijk ) l B k ) v J j r l = ɛ ijk ɛ jlm ( l B K ) m m = ( i B j ) m j .
In the last step leading to (C.1) we have used (A.7) and ∇ · B = 0.

The next term in (1.58) becomes from (A.16)

1 2 ɛ ijk ( m l B k ) V J j r l r m = 1 4 ɛ ijk ( m l B k ) ( ɛ jln m nm + ɛ jmn m nl ) = 1 2 ɛ ijk ɛ jln ( m l B k ) m nm = 1 2 ( j i B k ) m kj .
Because the sources of the external field B are outside the current distribution and B is magnetostatic, x B = 0. Thus in (C.2)
i B j = j B i .
From (C.1)–(C.3) the force in (1.58) can be written
F i = m j j B i + 1 2 m jk k j B i + ,
which is (1.59).