Fundamentals of Plasma Physics

9.2 Vacuum magnetic fields 267

is in fact the lowest energy field. Thus, the only way to ensure thatBminis indeed the true

minimum is to require ∫

v

δB·Bmind^3 r= 0 (9.8)

no matter howδBis chosen. UsingδB=∇×δAand the vector identity∇·(A×B) =

B·∇×A−A·∇×B,Eq. (9.8) can be integrated by parts to obtain

∫

V

[∇·(δA×Bmin) +δA·∇×Bmin]d^3 r= 0. (9.9)

The first term can be transformed into a surface integral overSusing Gauss’s theorem.

This surface integral vanishes becauseδAmust vanish on the bounding surface (recall that

the variation satisfies the same boundary condition as the minimum energy field). Because

δBis arbitrary withinV,δAmust also be arbitrary withinVand so the only way for the

second term in Eq. (9.9) to vanish is to have∇×Bmin= 0.Thus,Bminmust be a vacuum

field.

An important corollary is as follows: Suppose boundary conditions are specified on the

surface enclosing some volume. These boundary conditions can be considered as “rules”

which must be satisfied by any solution to the equations. All configurations satisfying the

imposed boundary condition and having finite current within the volume arenotin the

lowest energy state. Thus, non-vacuum fields can, in principle, have freeenergy available

for driving boundary-condition-preserving instabilities.

magnetic

energy

vacuum field

force-free field (typical)

configuration

Figure 9.2: Sketch of magnetic energy dependence on configuration of a system withfixed,
specified boundary conditions. The variation of the configuration would correspond to
different internal current profiles. The force-free configurations are local energy minima
while the vacuum configuration has the absolute lowest minimum.