Tensors for Physics

148 8 Integration of Fields

Notice, no specific form of an interrelation betweenEandDor betweenBandH
has been used. So the energy balance (8.109) holds true in general.
Integration of the local balance equation over a volumeVand use of the Gauss
theorem yields

∫

V

jμEμd^3 r+

∫

V

(

Eμ

∂

∂t

Dμ+Hμ

∂

∂t

Bμ

)

d^3 r+

∮

∂V

nμSμd^2 s= 0. (8.111)

The first term is the power which the current extracts from the electromagnetic field
and the second term stands for the power which the field takes from the current, both
within the volumeV. The last term describes the power given to the surroundings,
e.g. by radiation.
The field energy can be defined for a nonlinear, hysteresis-free medium, where
theEandHfields are uniquely determined by theDandBfields according to

E=E(r,D), H=H(r,B). (8.112)

The electric and magnetic energy densities are defined by

uel(D)=

∫D

0

Eμ(D′)dD′μ, umag(B)=

∫B

0

Hμ(B′)dB′μ. (8.113)

The time derivatives of these energy densities are

∂

∂t

uel=Eμ

∂

∂t

Dμ,

∂

∂t

umag=Hμ

∂

∂t

Bμ. (8.114)

Thus, for the hysteresis-free medium, the local energy balance (8.109) is equivalent to

∂

∂t

(uel+umag)+∇μSμ+jμEμ= 0. (8.115)

The definition (8.113)impliesthatEandHare derivatives of the electric and of the
magnetic energy density with respect toDandB, respectively:

Eλ=

∂uel

∂Dλ

, uel=uel(D), Hλ=

∂umag

∂Bλ

, umag=umag(B). (8.116)

For a linear medium characterized by the relative dielectric tensorελκand the mag-
netic permeability tensorμλκaccording to

Dλ=ε 0 ελκEκ, Bλ=μ 0 μλκHκ, (8.117)