2.15 Show that the electric susceptibility tensor is symmetric.
Background: Gauss's law for the electric field says
∇⋅E = Ï/ε₀
(2.111)
where Ï is the density of electric charge due to all sources, which includes unbound free charges.
Gauss's law becomes
∇⋅D = Ï_f
(2.112)
where D = εE. The quantity ε is the dielectric constant K. For anisotropic dielectrics, the dielectric constant gets replaced with the dielectric tensor
D_i = ε_iE_i
(2.113)
where ε_i = ε. The energy density stored in the electrostatic field, i.e. the work necessary to separate the charges and create the field, equals 1/2DE. Show from these considerations that ε_i = ε and thus ε_i = ε. See Panofsky and Phillips, p.99.