Tensors for Physics

8.5 Further Applications in Electrodynamics 147

however, and also in an anisotropic linear medium with ενμ =0, the Maxwell
stress tensor contains an antisymmetric part, which is associated with a torque. For
a discussion of this point see Sect.8.5.5.

8.5.3 Energy Balance for the Electromagnetic Field

Point of departure of the formulation of the energy balance for the electromagnetic
field is the expression

jμEμ

for the power, i.e. for the time change of the energy, delivered by the electric fieldE
on the electric current densityj.
Side remark: A plausible argument for the power being given byjμEμ.
In mechanics, the time change of the kinetic energyddt^12 mvμvμ=vμddtmvμof a
particle moving with velocityvin a force fieldFis given byvμFμ. For one type of
carriers with the electric chargee, the current density is equal tojμ=nevμ, where
nis the number density of the charges andvtheir average velocity. Then the relation
stated above follows due toFμ=eEμ.
Now, multiplication of the inhomogeneous Maxwell equation involving the cur-
rent density, cf. (7.56), byEμleads to

εμνλEμ∇νHλ=jμEμ+Eμ

∂

∂t

Dμ.

The term on the left hand side can be rewritten as

εμνλEμ∇νHλ=∇ν(εμνλEμHλ)−Hλεμνλ∇νEμ.

Due to the homogeneous Maxwell equation referred to as Faraday law, cf. (7.57),
the last term of the equation above is equal to

−Hλεμνλ∇νEμ=Hλελνμ∇νEμ=−Hλ

∂

∂t

Bλ.

Thus the energy balance equation reads

jμEμ+Eμ

∂

∂t

Dμ+Hμ

∂

∂t

Bμ+∇μSμ= 0 , (8.109)

with the energy flux density, also calledPoynting vector,

Sμ≡εμνλEμHλ. (8.110)