Experiments with Supersonic Beams as a Source of Cold Atoms

macroscopic Maxwell’s equations [74]. Doing this, a new field H is defined as
H = 1
μ0
B − M (4.22)
where M is the magnetization of the material, allowing Ampère’s law to be rewritten
as
∇× H = J. (4.23)
For many materials, B and H are linearly related via the equation B = μ H,whereμ
is the magnetic permeability of the material. However, for ferromagnetic materials, B
is a non-linear multivalued (due to hysteresis) function of H, making the effects of the
material on the field much more difficult to calculate. Calculating the fields produced
by the coils numerically allows the full non-linear behavior of the real materials near
the coils to be taken into account.
4.2.1 Switching of Electromagnetic Coils
The final aspect of producing magnetic fields via pulsed coils that needs to be
addressed is the switching of the coils. Maxwell’s equations give the relationship
∇× E + ∂ B
∂t
=0, (4.24)
which says that a changing magnetic field will produce an electric field. When this is
applied to a coil, the result is Lenz’s law, which states that
E = −N dΦ
, (4.25)
dt
where E is the electromotive force, N is number of turns in the coil, and Φ is the
magnetic flux through the coil. Thus, switching a coil will produce a potential that
induces a current in the direction that opposes changes in the field. This makes it dif-
ficult to quickly switch the current in a coil, and doing so creates large voltage spikes.
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