Fundamentals of Plasma Physics

272 Chapter 9. MHD equilibria

term involving∇⊥B^2 portrays a magnetic force due to pressure gradients perpendicular
to the magnetic field and is a more precise expression of the hoop force.
In our earlier discussion it was shown that the vacuum magnetic field is the lowest
energy state of all fields satisfying prescribed boundary conditions. A vacuum field might
be curved for certain boundary conditions (e.g., a permanent magnet with finite dimensions)
and, in such a case, the two terms in Eq.(9.13) are both finite but exactly cancel each
other. We can think of the minimum-energy state for given boundary conditions as being
analogous to the equilibrium state of a system of stiff rubber hoses which have been pre-
formed into shapes having the morphology of the vacuum magnetic field and which have
their ends fixed at the bounding surface. Currents will cause the morphology of this system
to deviate from the equilibrium state, but since any deformation requires work to be done
on the system, currents invariably cause the system to be in a higher energystate.

9.6 Flux preservation, energy minimization, and inductance

Another useful way of understanding magnetic field behavior relates to theconcept of
electric circuit inductance. The self-inductanceLof a circuit component is defined as
the magneticfluxΦlinking the component divided by the currentIflowing through the
component, i.e.

L=

Φ

I

. (9.15)

Consider an arbitrary short-circuited coil with currentIlocated in an infinite volumeV
and letCdenote the three-dimensional spatial contour traced out by the wire constituting
the coil. The total magneticflux linked by the turns of the coil can be expressed as

Φ=

∫

B·ds=

∫

∇×A·ds=

∮

C

A·dl (9.16)

where the surface integral is over the area elements linked by the coil turns. The energy
contained in the magnetic field produced by the coil is

W =

∫

V

B^2

2 μ 0

d^3 r

=

1

2 μ 0

∫

V

B·∇×Ad^3 r. (9.17)

However, using the vector identity∇·(A×B)=B·∇×A−A·∇×Bthis magnetic
energy can be expressed as

W=

1

2 μ 0

∫

V

A·∇×Bd^3 r+

1

2 μ 0

∫

S∞

ds·A×B (9.18)

where Gauss’s law has been invoked to obtain the second term, an integral over the surface
at infinityS∞. This surface integral vanishes because (i) at infinity the magnetic field must
fall off at least as fast as a dipole, i.e.,B∼R−^3 whereRis the distance to the origin,
(ii) the vector potential magnitudeAscales as the integral ofBsoA∼R−^2 , and (iii) the
surface at infinity scales asR^2.