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without changing the fact that they are not independent. Note also that we
have only implemented the fact that the equilibrium electric field vanishes.
Off equilibrium, it does not, and as we know may drive a current through a
wire. It remains dependent, however, and is in this case given by
c ∇×(H + H D )=σE. (58)
There are two ways to arrive at the hydrodynamic theory for conductors, one
may either set to zero all terms ∼ D 0 and E 0 in the previous theory, or one
may start from scratch with a reduced set of variables. Both methods lead to
the same results, which are presented below.
The basic thermodynamic identity for an arbitrary inertial system is,
where
dε =dɛ tot − c 2 dρ = T ds + µ dρ + µc dρc
+v · dg tot + H · dB (59)
g tot =(Ts+ µρ+ µc ρc + ρc 2 ) v
1 v
− × B × H
c2 c c
≈ ρ v − 1
v
× B × H . (60)
c c
In equilibrium, the magnetic field satisfies
∇·B =0. (61)
To linear order in v,
the fields
c
H 0 = H , B 0 = B (62)
are Lorentz invariant.
The change in the equilibrium conditions are in the Euler-Lagrange equations
for the field. Only
∂tH =0, ∇ 0 × H + 1
µ + c 2 H ×∇0 µ =0, (63)
remain, with E 0 ≡ 0 implied. The rotational identity Eq(26) reduces to
H × B =0. (64)
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