Tensors for Physics

152 8 Integration of Fields

Notice that

εκνμTνμ=εκνμDνEμ+εκνμBνHμ=(D×E)κ+(B×H)κ. (8.128)

Thus integration of the local balance equation over a volumeVleads to

d

dt

∫

V

r×(D×B)d^3 r=−

∫

V

(D×E+B×H)d^3 r−

∫

V

r×(ρE+j×B)d^3 r,(8.129)

provided that the surface integral vanishes. The term on the left hand side is the
time change of the angular momentum of the electromagnetic field, compensated by
torques exerted by matter. So the torque of the field on the matter is equal to the right
hand side, just with the opposite sign.
For electrically neutral matter withρ=0 andj=0, the torque acting on matter is
∫

V

(D×E+B×H)d^3 r.

Due to

D=ε 0 E+P, B=μ 0 (H+M),

andD×E=P×E,aswellasB×H=μ 0 M×H=M×B, the corresponding torque
Tacting on matter characterized by the electric polarizationPand the magnetization
Mis the sum of the electric and magnetic torquesTelandTmagdetermined by

T=Tel+Tmag, Tel=

∫

V

P×Ed^3 r, Tmag=

∫

V

M×Bd^3 r. (8.130)

For a spatially constant electric field, the integration ofPover the volume yields
the electric dipole momentpel, cf. Sect.10.4.2. Similarly, for a constant magnetic
field, the integration ofMgives the magnetic momentm. The relation (8.130) then
reduces to the expressions (5.31) for the electric and magnetic torques.
The pertaining torque densitytacting on the electric polarizationPand the mag-
netizationMis

t=P×E+M×B. (8.131)