Tensors for Physics

104 7 Fields, Spatial Differential Operators

acting on a particle with chargee, moving with velocityv, in the presence of an

electric fieldEand a magnetic fieldB, cf. Sect.3.4.6, is equivalent to

Fμ=eEμ+evνBνμ. (7.73)

The antisymmetric field tensorHμνpertaining to the magnetic fieldHis defined by

analogy to (7.69). Using the magnetic field tensors, the Maxwell equations read

∇μDμ=ρ, −∇νHνμ=jμ+

∂

∂t

Dμ, (7.74)

and

∇μEν−∇νEμ=−

∂

∂t

Bμν, ∇ 1 B 23 +∇ 2 B 31 +∇ 3 B 12 = 0. (7.75)

Notice that the last equation stems from

ελμν∇λBμν= 0 ,

andBνμ=−Bμνwas used.

Why should one bother to look at the alternative version of the Maxwell equations,

rather than stick to the vector equations (7.56) and (7.57)? There are two answers to

this question.

First, in the 4D formulation of electrodynamics, which reflects the Lorentz invari-

ance of the Maxwell equations, the 3×3 field tensor (7.70) is enlarged to a the

4 ×4 field tensor, which also comprises the 3 components of the electric field. In 4D,

an antisymmetric second rank tensor has 6 independent components, just like the

two vectorsBandEcombined. The 3D tensorial notation of the Maxwell equations

makes it easier to see their connection with the 4D version, discussed in Chap. 18.

Notice, the non-euklidian metric of special relativity is used for that 4D-space.

A second, more mathematical reason is: the first three of the equations (7.74) and

(7.75) can be adapted to any D-dimensional space with Euklidian metric, andD≥2.

Thus it is possible to answer the question: do electromagnetic waves exist for 2D?

7.3 Exercise: Electromagnetic Waves in Flatland?
In flatland, one has just 2 dimensions. Cartesian components are denoted by Latin
lettersi,j,...;i= 1 ,2;j= 1 ,2. 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.

Deriveawaveequationfortheelectricfieldtoproofthatonecanhaveelectromagnetic
waves in 2D. How about 1D?