Tensors for Physics

100 7 Fields, Spatial Differential Operators

are solutions for

ρ(−t,r),−j(−t,r).

As a consequence of the combined PT-invariance

−E(−t,−r),−D(−t,−r),−B(−t,−r),−H(−t,−r)

are solutions for

ρ(−t,−r),j(−t,−r).

7.5.2 Special Cases

Application of∇μto the second equation of (7.56) and use of the first of these
equations and of∇μεμνλ∇νHλ =0 yields the continuity equation for the time
change of the charge density:

∂

∂t

ρ+∇μjμ= 0. (7.59)

Without Maxwell’s current density∂∂tDμin the second inhomogeneous equation of
(7.56), one would just have∇μjμ=0, which is true for stationary processes, but
not in general. More important, the existence of electromagnetic waves hinges on
the term∂∂tDμin (7.56), see the next section.
Special cases, for a stationary situation, where the time derivatives vanish in
the Maxwell equations, are the equations ofelectrostatics,magnetostatics, and the
equations determining the magnetic field caused a steady current. For electrostatics,
one has

∇μDμ=ρ, εμνλ∇νEλ= 0.

The first of these equations is referred to as theGauss law.
The equations for magnetostatics, applicable to fields of permanent magnets, are
mathematically equivalent to those of electrostatics withDandEreplaced byBand
H, respectively, andρ=0, because there are no magnetic monopoles and conse-
quently there is no magnetic charge density. In the equations ruling electrostatics and
magnetostatics, there is no coupling between electric and magnetic fields, unless the
constitutive relations for the electric polarization and the magnetization contain such
terms.
The equations for the magnetic field caused by a stationary electric current are
associated with the namesOerstedandAmpère, who discovered and studied an effect
which reveals a coupling between electric and magnetic phenomena. The relevant