pdf-file - Institut für Theoretische Physik
pdf-file - Institut für Theoretische Physik
pdf-file - Institut für Theoretische Physik
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0= ˙ D + j D el + ρel v − c ∇×(H + H D ) (52)<br />
is negligible with respect to j D el = σ E 0 , it may then be eliminated. This is the<br />
quasi-stationary approximation[5], always justified in the low-frequency limit:<br />
Consider (for v ≡ 0)<br />
| ˙ D|≈ωεE ≪ j D = σE, (53)<br />
(where ε is the dielectric constant,) and find<br />
ω ε/σ ≪ 1. (54)<br />
We shall in this work refer to a system as a conductor if the quasi-stationary<br />
approximation is valid. (If not, the displacement current ∂tD needs to be<br />
included and, despite some residual conductivity, the equations of the last<br />
section apply.)<br />
In the Heaviside-Lorentz system of units, the conductivity of metals is of the<br />
order of σ =10 18 /s, eight to ten orders of magnitude above the frequencies<br />
at which local equilibrium reigns and the hydrodynamic theory is valid. So<br />
the quasi-stationary approximations always apply in these systems, which can<br />
therefore be considered as conductors without any qualification. If the conductivity<br />
is below 10 10 /s, circumstances are more complicated and depend on<br />
the given frequency.<br />
The elimination of the displacement current ∂tD is a qualitatively important<br />
step, as this is equivalent to the statement that D is no longer an independent<br />
variable of the hydrodynamic theory. Indeed, the relaxation time of the electric<br />
field is τ = ε<br />
σ ,as<br />
| ˙ E|≈E/τ ≈ ˙ D/ε ≈ σ E/ε . (55)<br />
and the electric field has ample time to relax for frequencies ωτ ≪ 1. So,<br />
taking E 0 as zero in equilibrium, the rest frame thermodynamics is now,<br />
dɛ tot = T ds +(µ + c 2 )dρ + µc dρc + H · dB . (56)<br />
Note that the electric field need not vanish in the lab-frame, v = 0, though of<br />
course only via the Lorentz transformation<br />
E = −v × B/c , D = −v × H/c , (57)<br />
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