Tensors for Physics

Appendix: Exercises... 397

is made. The Maxwell equation−∂Bμ/∂t=εμνλ∇νEλleads to

cB(μ^0 )=εμνλ̂kνE(λ^0 ).

Thus the magnetic field is perpendicular to both the wave vector and the electric
field.
(ii)Plane Wave(7.64). From the plane wave ansatz

Eμ=E(μ^0 )exp[ikνrν−iωt]

follows, by analogy to the calculations above,

∇νEμ=ikνEμ.

Again∇νEν=0, corresponding tokνEν=0, implies that theE-field is perpendic-
ular to the wave vectork. The second spatial derivative leads to

ΔEμ=−k^2 Eμ.

The first and second time derivatives of the field are

∂

∂t

Eμ=−iωEμ,

∂^2

∂t^2

Eμ=−ω^2 Eμ.

Thus the wave equation (7.60) imposes the condition

k^2 =ω^2 /c^2 ,

which proofs the dispersion relation (7.65).
Here−∂Bμ/∂t=εμνλ∇νEλleads to

ωBμ=εμνλkνEλ.

As expected, also for plane waves, the magnetic field is perpendicular to both the
wave vector and the electric field.

7.3 Electromagnetic Waves in Flatland?(p.105)
In flatland, one has just 2 dimensions. Cartesian components are denoted by Latin
letters i,j,...,i= 1 ,2,j= 1 ,2,etc. The summation convention is used. In vacuum,
and for zero charges and currents, the adapted Maxwell equations are

∇iEi= 0 , −∇iHij=ε 0

∂

∂t

Ej, ∇iEj−∇jEi=−μ 0

∂

∂t

Hij.

In 2D, there is no equation corresponding to∇λBλ=0.It is not defined and not
needed in 2D. The magnetic field tensors have only one independent component.