14.4 Solution of Tensor Equations 267
A simple application, for=1, is the computation of the electrical conductivity
in the presence of a magnetic field, as discussed next. The case of the fourth rank
viscosity tensor of a fluid in the presence of a magnetic field, is treated in Sect.16.3.2.
14.4.2 Effect of a Magnetic Field on the Electrical
Conductivity
In a stationary situation, the linear relation between the electric flux densityjand an
applied electric fieldEis described by
jμ=σμνEν, (14.48)whereσμνis the electrical conductivity tensor. For the isotropic case, whereσμν∼
δμν, this corresponds to the local formulation of Ohm’s law. The influence of a
magnetic fieldB=Bh, withh·h=1, on the conductivity is analyzed next for a
simple model. Consider the case of single carriers with massm, chargee, number
densitynand an average velocityv, then the flux density isj=nev. The velocity is
assumed to obey the damped equation of motion
mv ̇=e(E+v×B)−mτ−^1 v,whereτis a relaxation time. For a stationary situation, the time derivativev ̇vanishes
and the equation above forvreduces to an expression of the type (14.43), just for
=1, viz.
vμ+φHμνvν=c 0 Eμ, (14.49)
withφ=eBτ/mandc 0 =eτ/m. The solution of this equation forv,cf.(14.44), is
vμ=eτ
m∑^1
k=− 1( 1 +ikφ)−^1 Pμν(k)Eν. (14.50)Thus the dc-conductivity tensor is
σμν=
ne^2 τ
m∑^1
k=− 1(
1 +ik
eBτ
m)− 1
Pμν(k). (14.51)This result is equivalent to
σμν=σ‖hμhν+σ⊥(δμν−hμhν)+σtransεμλνhλ, (14.52)