Tensors for Physics

7.5 Maxwell Equations in Differential Form 101

equations, the first one of which is referred to asOersted law,are

εμνλ∇νHλ=jμ, ∇μBμ= 0.

Applications of the stationary equations are given in Sects.8.2.7and8.3.5.
The first of the homogeneous Maxwell equations (7.57), viz.

εμνλ∇νEλ=−

∂

∂t

Bμ,

is referred to asFaraday law. It underlies the coupling between electric and magnetic
fieldsdiscoveredbyFaraday:atime-dependentB-fieldinducesanelectricfieldE.For
the application of this differential equation to theFaraday inductionsee Sect.8.2.8.

7.5.3 Electromagnetic Waves in Vacuum

In vacuum, whereD=ε 0 EandB=μ 0 H, and forρ=0,jμ=0, application of
εαβμ∇βon the first equation of (7.57), use of (4.10) for the double cross product,

and of∇νEν=0, yields−ΔEα=−μ (^0) ∂∂tεαβμHμ. The second equation of (7.56)
links the curl of theHfield with the time derivative ofε 0 E. This then leads to the
wave equation
E≡ΔE−

1

c^2

∂^2

∂t^2

E= 0 , (7.60)

with the speed of light, in vacuum, determined by

c^2 =(ε 0 μ 0 )−^1. (7.61)

The magnetic fieldHobeys the same type of wave equation. The symbol

≡Δ−

1

c^2

∂^2

∂t^2

(7.62)

is thed’Alembert operator. A solution of (7.60)is

Eμ=Eμ(^0 )f(ξ ), ξ=k̂νrν−ct, (7.63)

whereEμ(^0 )is a constant vector characterizing the polarization of the field,k̂νis a
unit vector parallel to the wave vector, pointing in the direction of propagation of
the radiation, andfis any function which can be differentiated twice. Notice that
̂kνE(ν^0 )=0, i.e. the electromagnetic radiation, in vacuum, is atransverse wave.