5.3 Applications 65
enough, such that terms nonlinear in the applied electric field can be disregarded.
In this linear regime, one haspνind=ε 0 ανκEκ,cf.(5.25). Hereανκis the molecular
polarizability tensor andε 0 is the dielectric permeability of the vacuum. Theα-tensor
is symmetric, it can be decomposed into its isotropic and symmetric traceless parts:
ανκ=^13 αττδνκ+ανκ. It is only the anisotropic, i.e. irreducible part of theα-tensor,
which contributes to the torque
Tμel,ind=ε 0 εμλναλκEκEν. (5.32)This equation can also be written as
Tel,ind=ε 0(
α·E)
×E.
An expression analogous to (5.32) applies for the induced magnetic dipole moment.
Again, only the anisotropic part of the magnetic polarizability tensor gives rise to
the torque.
Similar relations follow from the angular momentum balance of the Maxwell
equations, see Sect.7.5. In particular, the torque density associated with the electro-
magnetic fields isE×D+H×B.DuetoD=ε 0 E+PandB=μ 0 (H+M),
the torque density is also equal toE×P+μ 0 H×M.Hereμ 0 is the magnetic
permeability, also called magnetic induction constant, of the vacuum. In SI-units, it
is given byμ 0 = 4 π 10 −^7 As/Vm, where As/Vm stands for “Ampere seconds/Volt
meter”. On the other hand, the torque density exerted by the fields on the matter is
P×E+M×B, which involves the electric polarizationPand the magnetizationM.
For the induced part, one has, in the linear regime,
Pλind=ε 0 χλκelEκ, Mλind=μ− 01 χλκmagBκ, (5.33)whereχλκelandχλκmagare the electric and magnetic susceptibility tensors. By analogy
to (5.32), the torque density is determined by
εμλν(
ε 0 χλκel EκEν+μ− 01 χ
mag
λκ BκBν)
, (5.34)
where just the anisotropic parts of the susceptibility tensors contribute to the torque.
5.4 Geometric Interpretation of Symmetric Tensors.
5.4.1 Bilinear Form.
Symmetric second rank tensors can be visualized through geometric interpretations.
In one, the tensor is used as thecoefficient matrix of a bilinear form,e.g.
xμSμνxν=X^2 , (5.35)