7.5 Maxwell Equations in Differential Form 99
as in the Gaussiancgs-system of physical units. Such a description is inconvenient
for electrodynamics applied to macroscopic matter. There,ρandjstand for the
density offree chargesandfree currents, whereasbound charges,internal currents
andmagnetic momentsassociated with the spin of particles are incorporated into
theelectric polarizationPand themagnetizationM, respectively. These quantities
occur in the relations
Dμ=ε 0 Eμ+Pμ, Bμ=μ 0 (Hμ+Mμ). (7.58)Vacuum corresponds toP=0 andM=0. In matter, constitutive relations are
needed forPandMin order to obtain a closed set of equations. These constitutive
relations are specific for the materials considered.
Remarks on Parity and Time ReversalTheE- andD-fields are polar vectors,Band
Hare axial vectors. Sinceρis a true scalar, and∇as well asjare polar vectors, the
Maxwell equations (7.56) and (7.57) conserve parity. The constitutive relations forP
andM, however, have to obey certain restrictions, when parity conservation should
not be violated.
Furthermore,ρand the fieldsE,aswellasD, do not change sign under the time
reversal operation, whereasjand the fieldsB,aswellasH, do change sign. Thus the
Maxwell equations (7.56) and (7.57) are invariant under time reversal. Time reversal
invariance, however, can be broken by constitutive relations. An example isOhm’s
law, in differential form,j=σE. The non-negative coefficientσis the electrical
conductivity.
Parity (P) and time reversal (T) invariance hold true for charges, currents and fields
in vacuum, where the relationsD=ε 0 EandB=μ 0 Happly. Parity conservation
implies: when
E(t,r),D(t,r),B(t,r),H(t,r)are solutions of the Maxwell equations for given charge density and current density
ρ(t,r),j(t,r),then
−E(t,−r),−D(t,−r),B(t,−r),H(t,−r)are solutions for given
ρ(t,−r),−j(t,−r).
Similarly, from T-invariance follows:
E(−t,r),D(−t,r),−B(−t,r),−H(−t,r)