Single-Particle Electrodynamics - Assassination Science

or, on reärranging the terms,
−(∂ x B x + ∂ y B y ) = ∂ z B z . (4.38)
But the left-hand side of (4.38) is simply the parenthesised expression in
(4.37); hence, we find
{ }
F = nqεv orb i ∂x B z + j∂ y B z + k∂ z B z + O(ε 2 ).
But the expression in braces is simply ∇B z ; hence,
F = nqεv orb ∇B z .
Again noting the relations (4.23) and (4.26), and taking the point limit, we
therefore find
F = ∇(µ·B). (4.39)
The expression (4.39) is the force on an electric-current magnetic dipole. It
is the controversial “textbook force” on a magnetic dipole. For example, we
find the following in Jackson [113, Sec. 5.7]:
This can be written vectorially as
F = (m×∇)×B = ∇(m·B) − m(∇·B).
Since ∇·B = 0 generally, the lowest order force on a localized
current distribution in an external magnetic field B is
F = ∇(m·B).
This result holds even for time-varying external fields.
These statements are absolutely correct. (By “lowest-order” Jackson is referring
to the keeping of only the first term in the Taylor expansion (4.29) of the
magnetic field; this is rigorously true in the point limit.) Equation (4.39) is
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