Single-Particle Electrodynamics - Assassination Science

the force on the electric-current magnetic dipole. But there is again a twist;
we shall return to this shortly.
We now turn to the question of the torque on an electric-current magnetic
dipole.
Fortunately, this problem is not subject to any controversy
whatsoever. In this nonrelativistic analysis, we simply sum the “momentarm”
torques on all of the circulating constituent charges:
N = n ∫ 2π
dθ { N
2π
+
(θ) + N −
(θ) } . (4.40)
0
Since each torque is only dependent on the force at that position θ, we can
immediately make use of the sum of the torques on the positive and negative
charges at that position:
From (4.36), we find
N(θ) ≡ N +
(θ) + N −
(θ) ≡ z(θ)×F (θ).
N(θ) = 2qz(θ)× ( v +
(θ)×B(θ) ) ;
using the identity
and noting that
a×(b×c) ≡ (a·c)b − (a·b)c,
z(θ)·v +
(θ) ≡ 0
(since the charge is moving in circular motion, i.e., with velocity perpendicular
to its radius vector), we thus find
N(θ) = 2q ( z(θ)·B(θ) ) v +
(θ). (4.41)
Substituting (4.41) into (4.40), using (4.22), (4.27) and (4.29), and integrating,
we find
{ }
N = nqεv orb −i By + jB x + O(ε 2 ). (4.42)
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