Tensors for Physics

8.5 Further Applications in Electrodynamics 145

8.5 Further Applications in Electrodynamics

8.5.1 Energy and Energy Density in Electrostatics

TheCoulomb energy Uof chargesqiandqj, located in vacuum, is

U=

1

4 πε 0

∑

i<j

∑

j

qiqj

|ri−rj|

=

1

4 πε 0

1

2

∑

i=j

∑

j

qiqj

|ri−rj|

. (8.99)

The last expression is equivalent to

U=

1

2

∑

i

qiφ(ri), φ(ri)=

1

4 πε 0

∑

j

qj

|ri−rj|

, (8.100)

whereφis the electrostatic potential. With the continuous charge density

ρ(r)=

∑

i

qiδ(r−ri),

the Coulomb energy reads

U=

1

2

∫

ρ(r)φ(r)d^3 r. (8.101)

Notice, the conditioni= joccurring in (8.99), is ignored in (8.100) and lost in
(8.101).
Now use of the Gauss lawρ =∇νDν,cf.(7.56), of the relationφ∇νDν=
∇ν(Dνφ)−Dν∇νφandEν=−∇νφ, and the application of the Gauss theorem
leads to

U=

1

2

∫

φ(r)∇νDνd^3 r=

1

2

∫

EνDνd^3 r+

∮

nνDνφd^2 s. (8.102)

The last term vanishes, when the integral

∮

...d^2 sis performed over a far away
surface where, at leastφ=0, ornνDν=0, holds true. Then the energy is given by

U=

1

2

∫

u(r)d^3 r, u=

1

2

EνDν, (8.103)

whereu(r)is the energy density. So far charges in vacuum were considered, thus
one hasDν=ε 0 Eνand consequentlyu=^12 ε 0 EνEν. The relations still apply to a
linear medium characterized by the dielectric tensorενμaccording to

Dν=ε 0 ενμEμ,