398 Appendix: Exercises...
Differentiation of the second of these equations with respect tot, insertion of the
time change of the magnetic field tensor as given by the third equation and use of
the first one leads to the wave equation
∇i∇iEj=ε 0 μ 0∂^2
∂t^2Ej.Again, the speedcof the radiation is determined byc^2 =(ε 0 μ 0 )−^1. The electric
field is perpendicular to the direction of the propagation, even in 2D.
How about 1D? Obviously, the Maxwell equations loose their meaning in a true
one-dimensional world, thus electromagnetic waves do not exist in 1D. On the other
hand, longitudinal sound waves still can propagate in 1D.
7.4 Radial and Angular Parts of the Nabla Operator, Compare(7.81)with
(7.78) (p.106)
Due to the definition (7.80), the termεμνλr̂νLλin (7.81) is equal to
εμνλr̂νελαβr̂α∂
∂r̂β=(δμαδνβ−δμβδνα)r̂νr̂α∂
∂r̂β=r̂μr̂ν∂
∂r̂ν−
∂
∂r̂μ.
Here, the relation (4.10) was used for the product of the two epsilon-tensors. The
last term can be written as∂r∂̂μ=δμν∂∂̂rνand hence
εμνλr̂νLλ=εμνλ̂rνελαβr̂α∂
∂r̂β=(r̂μr̂ν−δμν)∂
∂r̂ν.
Now (7.81) is seen to be equal to (7.78).
7.5 Prove the Relations(7.82)and(7.83)for the Angular Nabla Operator
(p.106)
From the definition (7.80) of the differential operatorLfollows
LμLν=εμαβrα∇βενκτrκ∇τ=εμαβrαενκτδβκ∇τ+εμαβενκτrαrκ∇β∇τ
=εμαβενβτrα∇τ+εμαβενκτrαrκ∇β∇τ.Due toεμαβενβτ=εμαβετνβ=δμτδνα−δμνδατ,cf.(4.10), one obtains
LμLν=rν∇μ−δμνrα∇α+εμαβενκτrαrκ∇β∇τ.The second term on the right hand side is obviously symmetric under the exchange of
μandν. The same applies to the third term. To see this, notice that the simultaneous
interchangesα↔κandβ↔τcorresponds to an interchangeμ↔ν. Thus one
has
ελμνLμLν=ελμνrν∇μ=−Lλ,