Tensors for Physics

8.5 Further Applications in Electrodynamics 149

one has

Eλ=ε− 01 ε−λκ^1 Dκ, Hλ=μ− 01 μ−λκ^1 Bκ,

and consequently

uel=

1

2

ε 0 −^1 Dλελκ−^1 Dκ=

1

2

DλEλ

umag=

1

2

μ− 01 Bλμ−λκ^1 Bκ=

1

2

BλHλ. (8.118)

For the special case of an isotropic linear medium, whereελκ=εδλκandμλκ=
μδλκ, with the scalar coefficientsεandμhold true, the equations for the electric and
magnetic energy density reduce to

uel=

1

2

(ε 0 ε)−^1 D^2 , umag=

1

2

(μ 0 μ)−^1 B^2. (8.119)

8.5.4 Momentum Balance for the Electromagnetic Field,

18.5.2 Maxwell Stress Tensor

The Lorentz force (3.47) describes the force, i.e. the time change of the linear momen-
tum, experienced by a charge in the presence ofEandBfields. When the “matter”
is characterized by the charge densityρand the current densityjν, the force density
exerted by the fields on the matter is

kμ=ρEμ+εμνλjνBλ.

With the help of the inhomogeneous Maxwell equations (7.56), this expression is
equal to

ρEμ+εμνλjνBλ=Eμ∇νDν+εμνλενκτ(∇κHτ)Bλ−εμνλ

(

∂

∂t

Dν

)

Bλ.

The term involving the time derivative can be rewritten as

−εμνλ

(

∂

∂t

Dν

)

Bλ=−εμνλ

∂

∂t

(DνBλ)+εμνλDν

∂

∂t

Bλ.

Due to the Faraday law, i.e. the homogeneous Maxwell equation (7.57) involving the
time derivative of theBfield, the last term is equal to

εμνλDν

∂

∂t

Bλ=−εμνλDνελκτ∇κEτ.