SJOERD STALLINGA PHYSICAL REVIEW E 73. 026606(2006) W=dP.E dod(o[(OF)2+0F2-FE] This energy balance equation follows directly from Max- wells equations [12] It appears that the rate of work can be written as the tin =中,E-3P=DE-3D,E,(35 derivative of a quantity that may be interpreted as the energy of the material subsystem been used, in a way similar to the derivation of Eq (27. b where the equation of motion naterial field f ha ap. E=Eo dod(a)oFE nondispersive to dispersive dissipation W is approximately zero in case dispersion is small. The total energy may then be approximated by the nondispersive contribution u. This doi(o)aF·(aF+a2F) leads to the default expression for the electromagnetic energy in a dielectric appearing in many textbooks [12, 29] doi(o)(aF)2+o3F].(27) B Momentum he transport and dissipation of electromagnetic (linear) Here, the equation of motion of the material field F, Eq (14), entum is described by the momentum balance equation is used to eliminate e in favor of F. The energy of the ma- terial subsystem thus follows as (36) uM=Eo dod(ol(a,F)+o'F].(28) where the momentum density Em, the momentum flux den- (28) sity (stress tensor)TEM and the density of the force on the material subsystem fa are given by Conservation of energy of the total system is expressed by ga(=ε0∈ eaByEBB (37) ,+v.S=0 where the total energy density is given by E0E2EB-=BaBB+。E2+ (38) E eo dod(o)l(a,F)2+0F] f (39) (30) These expressions correspond to the Abraham momentum density, the Maxwell stress tensor, and the Lorentz force den The total energy is the sum of squares, and therefore always sity. The momentum balance equation for the electromag- positive, which guarantees thermodynamic stability. The netic field can be derived from Maxwell,'s equations in a same energy conservation law has been found previously by straightforward manner [12] Tip [10], and by Glasgow, Ware, and Peatross by deduction The Lorentz force density can be written as the sum of from Maxwell's equations and the constitutive relation [18. temporal and spatial derivatives. This implies the existence The total energy may be split into parts in a variety of f a momentum balance equation without a source term, i.e., ways. A division between nondispersive and dispersive con- an equation that expresses the conservation of the total mo- mentum of the combined field-matter system. This rewriting is done in a number of steps. First, using Faraday's law it E.D+一HB follows that fa=fa+a(EaB,PpB,dp(-EaPB+EPSa p=50|do(o(a2+aF2-FE],(32) which gives rise to energy balance equations for the two where du D+VS=-w, (33) fa=3PBdEB-EBo PB=DB0EB-EBdoDE A second of the equation of motion for the where the dissipation from the nondispersive to the disper- material field F, Eq (14), in order to eliminate E in favor of sive energy density is given by F in the expression for f, 026606-4W =
t P · E.
26 This energy balance equation follows directly from Max￾well’s equations 12. It appears that the rate of work can be written as the time derivative of a quantity that may be interpreted as the energy of the material subsystem
t P · E = 0  0  d
ˆ


t F · E = 0  0  d
ˆ


t F ·

t 2 F +
2 F =
t 0  0  d
ˆ



t F 2 +
2 F2  .
27 Here, the equation of motion of the material field F, Eq.
14, is used to eliminate E in favor of F. The energy of the ma￾terial subsystem thus follows as uMT = 0  0  d
ˆ



t F 2 +
2 F2 .
28 Conservation of energy of the total system is expressed by
t u +
· S = 0,
29 where the total energy density is given by u = uEM + uMT = 0 2 E2 + 1 2 0 B2 + 0  0  d
ˆ



t F 2 +
2 F2 .
30 The total energy is the sum of squares, and therefore always positive, which guarantees thermodynamic stability. The same energy conservation law has been found previously by Tip 10, and by Glasgow, Ware, and Peatross by deduction from Maxwell’s equations and the constitutive relation 18. The total energy may be split into parts in a variety of ways. A division between nondispersive and dispersive con￾tributions uND and uDS may be defined by uND = 1 2 E · D + 1 2 H · B,
31 uDS = 0  0  d
ˆ



t F 2 +
2 F2 − F · E,
32 which gives rise to energy balance equations for the two parts
t uND +
· S = − W
,
33
t uDS = W
,
34 where the dissipation from the nondispersive to the disper￾sive energy density is given by W
=
t 0  0  d
ˆ



t F 2 +
2 F2 − F · E = 1 2
t P · E − 1 2 P ·
t E = 1 2
t D · E − 1 2 D ·
t E,
35 where the equation of motion for the material field F has been used, in a way similar to the derivation of Eq.
27. The nondispersive to dispersive dissipation W
is approximately zero in case dispersion is small. The total energy may then be approximated by the nondispersive contribution uND. This leads to the default expression for the electromagnetic energy in a dielectric appearing in many textbooks 12,29. B. Momentum The transport and dissipation of electromagnetic
linear momentum is described by the momentum balance equation
t g EM +
T EM = − f,
36 where the momentum density g EM, the momentum flux den￾sity
stress tensor T EM and the density of the force on the material subsystem f are given by g EM = 0 EB ,
37 T EM = − 0EE − 1 0 BB +  0 2 E2 + 1 2 0 B2 ,
38 f = −
PE + 
tPB .
39 These expressions correspond to the Abraham momentum density, the Maxwell stress tensor, and the Lorentz force den￾sity. The momentum balance equation for the electromag￾netic field can be derived from Maxwell’s equations in a straightforward manner 12. The Lorentz force density can be written as the sum of temporal and spatial derivatives. This implies the existence of a momentum balance equation without a source term, i.e., an equation that expresses the conservation of the total mo￾mentum of the combined field-matter system. This rewriting is done in a number of steps. First, using Faraday’s law it follows that f = f
+
t
 PB +
− EP + 1 2 E · P ,
40 where f
= 1 2 P
E − 1 2 E
P = 1 2 D
E − 1 2 E
D.
41 A second step is the use of the equation of motion for the material field F, Eq.
14, in order to eliminate E in favor of F in the expression for f
SJOERD STALLINGA PHYSICAL REVIEW E 73, 026606
2006 026606-4