Tensors for Physics

146 8 Integration of Fields

cf. Sects.2.6.4and5.3.4. Then the energy density is equal to

u=

1

2

EνDν=

1

2

ε 0 ενμEμEν. (8.104)

Clearly, the symmetric part ofενμonly contributes to the energy density.
The energy density for the more general case of a nonlinear relation between the
DandEfields is treated in Sect.8.5.3.

8.5.2 Force and Maxwell Stress in Electrostatics.

The forceFμacting on chargesqj, in vacuum, in the presence of a given electric
fieldEμis

Fμ=

∑

j

qjEμ(rj),

or, in terms of the charge densityρ,by

Fμ=

∫

ρ(r)Eμ(r)d^3 r=

∫

kelstatμ d^3 r. (8.105)

With the help of the Gauss lawρ=∇νDν,cf.(7.56), the electrostatic force density
kμelstatcan be rewritten as

kelstatμ =Eμ∇νDν=∇ν(DνEμ)−Dν∇νEμ. (8.106)

In electrostatics, one has∇×E=0, and consequently∇νEμ=∇μEν. Thus the last
term of (8.106) is equal toDν∇μEν. Provided that the interrelation betweenDandE
is linear, this term can also be written as the total spatial derivative( 1 / 2 )∇μ(EνDν).
Thus in vacuum, and the same applies for any linear medium, the force density is
given by

kelstatμ =∇ν(DνEμ)−

1

2

∇μ(EλDλ)=∇ν(DνEμ)−

1

2

∇μ(EλDλδμν)=∇νTνμelstat.
(8.107)
The electrostatic stress tensor for a linear medium is defined by

Tνμelstat≡DνEμ−

1

2

EλDλδμν. (8.108)

This tensor is symmetric, i.e.Tνμelstat=Tμνelstat, in vacuum and for an isotropic linear
medium, whereDν=εε 0 Eνapplies, with a scalar dielectric coefficientε. In general