150 8 Integration of Fields
Putting terms together, one arrives at
ρEμ+εμνλjνBλ+εμνλ
∂
∂t
(DνBλ)
=Eμ∇νDν−εμνλDνελκτ∇κEτ+εμνλενκτ(∇κHτ)Bλ.
The right hand side of this equation can be written as a total spatial derivative. First
notice that, due toεμνλελκτ=δμκδντ−δμτδνκ,
−εμνλDνελκτ∇κEτ=−Dν∇μEν+Dν∇νEμ.
With
−Dν∇μEν=−∇μ(DνEν)+Eν∇μDν
and
Dν∇νEμ=∇ν(DνEμ)−Eμ∇νDν,
one obtains
−εμνλDνελκτ∇κEτ+Eμ∇νDν=∇ν(DνEμ−DκEκδμν)+Eν∇μDν.
The last term on the right hand side is the gradient of the electric energy densityuel.
By analogy to (8.114), one has
Eν∇μDν=∇μuel,
provided that the medium is hysteresis-free. Then the terms involving the electric
fields are equal to
−εμνλDνελκτ∇κEτ+Eμ∇νDν=∇νTνμel,
where
Tνμel=DνEμ−(DκEκ−uel)δμν (8.120)
is the electric part of the Maxwell stress tensor. By analogy, the term in the momentum
balance involving the magnetic fields is equal to
εμνλενκτ(∇κHτ)Bλ=∇νTνμmag,
with the magnetic part
T
mag
νμ =BνHμ−(BκHκ−umag)δμν (8.121)