Tensors for Physics

8.5 Further Applications in Electrodynamics 151

of the Maxwell stress tensor. The total stress tensor is

Tνμ=Tνμel+T

mag

νμ. (8.122)

Finally, the momentum balance is found to be

∇νTνμ=

∂

∂t

(εμνλDνBλ)+ρEμ+εμνλjνBλ. (8.123)

The term on the left hand side is the divergence of the momentum flux density,
the first term on the right side is the time change of the momentum density of the
electromagnetic field. The other terms on the right hand side describe the time change
of the momentum density of matter.
Integration of (8.123) over a volumeVand application of the Gauss theorem
yields

∮

∂V

nνTνμd^2 s=

d

dt

∫

V

εμνλDνBλd^3 r+

∫

V

(ρEμ+εμνλjνBλ)d^3 r. (8.124)

The balance equations (8.123) and (8.124) show that

D×B (8.125)

is the density of the linear momentum of the electromagnetic field. For fields in
vacuum, this quantity is proportional to the energy flux densityS,cf.(8.110), viz.

D×B=

1

c^2

E×H=

1

c^2

S, (8.126)

wherecis the speed of light in vacuum.

8.5.5 Angular Momentum in Electrodynamics

Multiplication of the momentum balance equation (8.123)byεκτμrτand use of

εκτμrτ∇νTνμ=∇ν(εκτμrτTνμ)−εκτμTνμ∇νrτ,

together with∇νrτ=δντ, leads to the angular momentum balance equation

∇ν(εκτμrτTνμ)=εκνμTνμ+εκτμrτ

(

∂

∂t

(εμνλDνBλ)+ρEμ+εμνλjνBλ

)

.

(8.127)