Michelle While
Jackson’s EM
November 3, 2014
Presentation Notes for November 10
Section 6.8 Poynting’s Theorem in Linear Dispersive Media with Losses
A thorough understanding of section 6.7 is necessary before approaching section 6.8. Therefore, we will follow
closely the arguments developed in section 6.7 and emphasize how these arguments are modified for dispersive
media with losses.
1. Both sections consider conservation of energy in various dielectric media. Section 6.6 goes into detail
how the internal motion of the atoms within a media gives rise to internal frequencies. One may be
deceived into thinking this motion from thermal agitation, zero point vibration and orbital motion
accounts for the dielectric’s electromagnetic properties. Interestingly, while these movements are
considerable and continuous; they average out so only EXTERNALLY applied oscillators
(Electromagnetic fields) contribute to the frequencies exhibited by the material.
⃗⃗⃗ 𝑎𝑛𝑑 𝐵
⃗ is:
2. The rate of doing work on a single charge by EXTERNAL electromagnetic fields 𝐸
𝑑𝑊
= 𝑞𝑣 ∙ 𝐸⃗
𝑑𝑡
Note that the magnetic field does NOT contribute to the work done because the magnetic field is
perpendicular to velocity.
3. The rate of doing work in a defined volume of medium with continuous distribution of charge and
current is:
Jackson Equation 6.103
𝑑𝑊
= ∫ 𝐽 ∙ 𝐸⃗ 𝑑3 𝑥
𝑑𝑡
𝑉
This is the electromagnetic energy converted into mechanical or thermal energy. Of course, that means
⃗⃗⃗ 𝑎𝑛𝑑 𝐵
⃗.
that energy has been removed from the electromagnetic fields 𝐸
Jackson Equation 6.105
⃗ ∙ (𝐸⃗ 𝑋 𝐻
⃗ ) + 𝐸⃗ ∙
∫ 𝑑 3 𝑥 𝐽 ∙ 𝐸⃗ = − ∫ 𝑑 3 𝑥 [∇
𝑉
𝑉
⃗
⃗
𝜕𝐷
𝜕𝐵
⃗ ∙ ]
+𝐻
𝜕𝑡
𝜕𝑡
This is where our section begins to diverge from section 6.7.
In section 6.7 the dielectric medium is linear in its electric and magnetic properties and isotropic. This
⃗ 𝑎𝑛𝑑 𝐸⃗ is linear as is the relationship between 𝐻
⃗ 𝑎𝑛𝑑 𝐵
⃗.
means the relationship between 𝐷
Jackson Equation 6.63
⃗ = 𝜖𝐸⃗ + 𝑃⃗
𝐷
1
⃗ −𝑀
⃗⃗
𝐵
𝜇
We are familiar with P as it describes the polarization of the dielectric material and M as it describes the
magnetization of the dielectric material.
⃗ =
𝐻
Page 1
Michelle While
Jackson’s EM
November 3, 2014
Presentation Notes for November 10
Our material is linear as well, however, in section 6.7 an assumption of negligible dispersion or losses
allows the use of Jackson Equation 4.89 and Equation 5.148, we are not afforded this simplification. Our
dielectric susceptibility (𝜖) and magnetic susceptibility (𝜇) are complex and frequency dependent.
4. We can no longer “ignore” the wave nature of our EM fields. The Fourier decompositions
⃗ 𝑎𝑛𝑑 𝐸⃗ and 𝐻
⃗ 𝑎𝑛𝑑 𝐵
⃗ exhibit this wave nature.
(transformations) given in the text for 𝐷
∞
𝐸⃗ (𝑥, 𝑡) = ∫ 𝑑𝜔 𝐸⃗ (𝑥, 𝜔)𝑒 −𝑖𝜔𝑡
−∞
∞
⃗ (𝑥, 𝑡) = ∫ 𝑑𝜔 𝐷
⃗ (𝑥, 𝜔)𝑒 −𝑖𝜔𝑡
𝐷
−∞
∞
⃗ (𝑥, 𝑡) = ∫ 𝑑𝜔 𝐻
⃗ (𝑥, 𝜔)𝑒 −𝑖𝜔𝑡
𝐻
−∞
∞
⃗ (𝑥, 𝑡) = ∫ 𝑑𝜔 𝐵
⃗ (𝑥, 𝜔)𝑒 −𝑖𝜔𝑡
𝐵
−∞
⃗ 𝑎𝑛𝑑 𝐸⃗ and 𝐻
⃗ 𝑎𝑛𝑑 𝐵
⃗.
Additionally, dispersion affects the relationship between 𝐷
Jackson Equation 7.105 reveals the nonlocality in time condition that occurs with dispersion.
⃗ at time t depends upon the value of the electric field at times other than t.
Basically, the value of 𝐷
This temporal/spatial adjustment is given in
Jackson Equation 7.106.
∞
1
𝜖(𝜔)
𝐺(𝜏) =
∫ 𝑑𝜔 [
− 1] 𝑒 −𝑖𝜔𝜏
2𝜋
𝜀0
−∞
Clearly when 𝜖(𝜔) is independent of 𝜔 𝐺(𝜏) is directly proportional to the change in time (𝛿𝜏) and the
⃗ 𝑎𝑛𝑑 𝐸⃗ is re-acquired.
instantaneous connection between 𝐷
⃗ (𝑥, 𝑡) = 𝜀0 𝐸⃗ (𝑥, 𝑡)
𝐷
Once re-acquired, there is no dispersion.
5. Linearity and Isotropism implies:
⃗ (𝑥, 𝜔) = 𝜖(𝜔)𝐸⃗ (𝑥, 𝜔)
𝐷
⃗ (𝑥, 𝜔) = 𝜇(𝜔)𝐻
⃗ (𝑥, 𝜔)
𝐵
Page 2
Michelle While
Jackson’s EM
November 3, 2014
Presentation Notes for November 10
Other useful relations are provided in our text:
𝐸⃗ (𝑥, −𝜔) = 𝐸⃗ ∗ (𝑥, 𝜔)
⃗ (𝑥, −𝜔) = 𝐷
⃗ ∗ (𝑥, 𝜔)
𝐷
⃗ (𝑥, −𝜔) = 𝐵
⃗ ∗ (𝑥, 𝜔)
𝐵
⃗ (𝑥, −𝜔) = 𝐻
⃗ ∗ (𝑥, 𝜔)
𝐻
𝜖(−𝜔) = 𝜖 ∗ (𝜔)
6. Two terms in Equation 6.105 are affected by the frequency dependence of dielectric susceptibility and
magnetic susceptibility.
⃗
⃗
𝜕𝐷
𝜕𝐵
⃗ ∙
𝐸⃗ ∙
𝑎𝑛𝑑 𝐻
𝜕𝑡
𝜕𝑡
⃗
𝜕𝐷
First we will write out 𝐸⃗ ∙ in terms of the Fourier integrals with spatial dependence implicit.
𝜕𝑡
Fourier integrals with spatial dependence:
𝐸⃗ (𝑡) = ∫ 𝑑𝜔 𝐸⃗ (𝜔)𝑒 −𝑖𝜔𝑡
⃗ (𝑡) = ∫ 𝑑𝜔 𝐷
⃗ (𝜔)𝑒 −𝑖𝜔𝑡
𝐷
Take the partial derivative
⃗
𝜕𝐷
𝜕𝑡
𝜕
⃗ (𝜔)𝑒 −𝑖𝜔𝑡 ] = ∫ 𝑑𝜔 𝐷
⃗ (𝜔)[(−𝑖𝜔)𝑒 −𝑖𝜔𝑡 ]
[𝐷
𝜕𝑡
⃗ (𝑥, 𝜔) = 𝜖(𝜔)𝐸⃗ (𝜔′).
Substitute 𝐷
⃗
𝜕𝐷
= ∫ 𝑑𝜔′ 𝜖(𝜔)[(−𝑖𝜔)𝐸⃗ (𝜔′)𝑒 −𝑖𝜔𝑡 ]
𝜕𝑡
⃗
𝜕𝐷
= ∫ 𝑑𝜔′ 𝜖(𝜔)[(−𝑖𝜔)𝐸⃗ (−𝜔′)𝑒 𝑖𝜔𝑡 ]
𝜕𝑡
⃗
𝜕𝐷
= ∫ 𝑑𝜔′ 𝜖(𝜔)[(−𝑖𝜔)𝐸⃗ ∗ (𝜔′)𝑒 𝑖𝜔𝑡 ]
𝜕𝑡
𝐸⃗ ∙
⃗
𝜕𝐷
= ∫ 𝑑𝜔 𝐸⃗ (𝜔)𝑒 −𝑖𝜔𝑡 ∫ 𝑑𝜔′ 𝜖(𝜔)[(−𝑖𝜔)𝐸⃗ ∗ (𝜔′)𝑒 𝑖𝜔𝑡 ]
𝜕𝑡
Some Re-arrangement here
⃗
𝜕𝐷
𝐸⃗ ∙
= ∫ 𝑑𝜔 ∫ 𝑑𝜔′ 𝐸⃗ ∗ (𝜔′)[(−𝑖𝜔)𝜖(𝜔)] ∙ 𝐸⃗ (𝜔)𝑒 −𝑖(𝜔−𝜔′)𝑡
𝜕𝑡
Page 3
Michelle While
Jackson’s EM
November 3, 2014
Presentation Notes for November 10
Second, split the integral into two equal parts
𝐸⃗ ∙
⃗
𝜕𝐷
1
1
= ∫ 𝑑𝜔 ∫ 𝑑𝜔′ 𝐸⃗ ∗ (𝜔′)[−𝑖𝜔𝜖(𝜔)] ∙ 𝐸⃗ (𝜔)𝑒 −𝑖(𝜔−𝜔′)𝑡 + ∫ 𝑑𝜔 ∫ 𝑑𝜔′ 𝐸⃗ ∗ (𝜔′)[−𝑖𝜔𝜖(𝜔)] ∙ 𝐸⃗ (𝜔)𝑒 −𝑖(𝜔−𝜔′)𝑡
𝜕𝑡 2
2
In the second integral make the following substitutions:
𝜔 → −𝜔′ 𝑎𝑛𝑑 𝜔′ → −𝜔
𝐸⃗ ∙
⃗
𝜕𝐷
1
1
′
= ∫ 𝑑𝜔 ∫ 𝑑𝜔′ 𝐸⃗ ∗ (𝜔′)[−𝑖𝜔𝜖(𝜔)] ∙ 𝐸⃗ (𝜔)𝑒 −𝑖(𝜔−𝜔′)𝑡 + ∫ 𝑑(−𝜔′) ∫ 𝑑(−𝜔) 𝐸⃗ ∗ (−𝜔)[−𝑖(−𝜔′)𝜖(−𝜔′)] ∙ 𝐸⃗ (−𝜔′)𝑒 −𝑖(−𝜔 +𝜔)𝑡
𝜕𝑡 2
2
𝐸⃗ ∙
⃗
𝜕𝐷
1
1
= ∫ 𝑑𝜔 ∫ 𝑑𝜔′ 𝐸⃗ ∗ (𝜔′)[−𝑖𝜔𝜖(𝜔)] ∙ 𝐸⃗ (𝜔)𝑒 −𝑖(𝜔−𝜔′)𝑡 + ∫ 𝑑(𝜔′) ∫ 𝑑(𝜔) 𝐸⃗ (𝜔)[𝑖(𝜔′)𝜖 ∗ (𝜔′)] ∙ 𝐸⃗ ∗ (𝜔′)𝑒 −𝑖(𝜔−𝜔′)𝑡
𝜕𝑡 2
2
𝐸⃗ ∙
⃗
𝜕𝐷
1
1
= ∫ 𝑑𝜔 ∫ 𝑑𝜔′ 𝐸⃗ ∗ (𝜔′)[−𝑖𝜔𝜖(𝜔)] ∙ 𝐸⃗ (𝜔)𝑒 −𝑖(𝜔−𝜔′)𝑡 + ∫ 𝑑(𝜔′) ∫ 𝑑(𝜔) 𝐸⃗ (𝜔)[𝑖(𝜔′)𝜖 ∗ (𝜔′)] ∙ 𝐸⃗ ∗ (𝜔′)𝑒 −𝑖(𝜔−𝜔′)𝑡
𝜕𝑡 2
2
Grouping Terms Results in Jackson Equation 6.124
𝐸⃗ ∙
⃗
𝜕𝐷
1
= ∫ 𝑑𝜔 ∫ 𝑑𝜔′ 𝐸⃗ ∗ (𝜔′)[−𝑖𝜔𝜖(𝜔) + 𝑖𝜔′𝜖 ∗ (𝜔′)] ∙ 𝐸⃗ (𝜔)𝑒 −𝑖(𝜔−𝜔′)𝑡
𝜕𝑡 2
Third, the dielectric susceptibility changes wrt to frequency. If we play the law of averages, we know
that most of the time the system resides in a narrow range of frequencies compared with the entire
spectrum of frequencies the susceptibility is capable of spanning. These frequencies dominate the
electric field and allow us to expand 𝑖𝜔′𝜖 ∗ (𝜔′).
Jackson Equation 6.125
𝐸⃗ ∙
⃗
𝜕𝐷
1
𝜕 1
𝑑
′
′
= ∫ 𝑑𝜔 ∫ 𝑑𝜔′ 𝐸⃗ ∗ (𝜔′ ) ∙ 𝐸⃗ (𝜔)𝜔 𝐼𝑚 𝜖(𝜔)𝑒 −𝑖(𝜔−𝜔 )𝑡 +
∫ 𝑑𝜔 ∫ 𝑑𝜔′ 𝐸⃗ ∗ (𝜔′ ) ∙ 𝐸⃗ (𝜔)
𝜔𝜖 ∗ (𝜔)𝑒 −𝑖(𝜔−𝜔 )𝑡
𝜕𝑡 2
𝜕𝑡 2
𝑑𝜔
Thus exposing this more complex expression as compared with non-dispersive media in Section 6.7
⃗
𝜕𝐸⃗
𝜕𝐷
⃗ =
∙𝐷
∙ 𝐸⃗
𝜕𝑡
𝜕𝑡
This reveals the fact that electric fields have a wave type nature and in dielectric materials the dielectric
susceptibility is affected by the propagation of those EM waves through the material.
The first term represents the conversion of electrical energy to heat while the second term represents
energy density.
7. For completeness I will go through the magnetic field counterpart before proceeding to 6.126
⃗ ∙
First we will write out 𝐻
⃗
𝜕𝐵
𝜕𝑡
in terms of the Fourier integrals with spatial dependence implicit.
Fourier integrals with spatial dependence:
⃗ (𝑡) = ∫ 𝑑𝜔 𝐻
⃗ (𝜔)𝑒 −𝑖𝜔𝑡
𝐻
⃗ (𝑡) = ∫ 𝑑𝜔 𝐵
⃗ (𝜔)𝑒 −𝑖𝜔𝑡
𝐵
Page 4
Michelle While
Jackson’s EM
November 3, 2014
Presentation Notes for November 10
Take the partial derivative
⃗
𝜕𝐵
𝜕𝑡
𝜕
⃗ (𝜔)𝑒 −𝑖𝜔𝑡 ] = ∫ 𝑑𝜔 𝐵
⃗ (𝜔)[(−𝑖𝜔)𝑒 −𝑖𝜔𝑡 ]
[𝐵
𝜕𝑡
⃗ (𝑥, 𝜔) = 𝜇(𝜔)𝐻
⃗ (𝜔′).
Substitute 𝐵
⃗
𝜕𝐵
⃗ (𝜔)𝑒 −𝑖𝜔𝑡 ∫ 𝑑𝜔′ 𝜇(𝜔)[(−𝑖𝜔)𝐻
⃗ ∗ (𝜔′)𝑒 𝑖𝜔𝑡 ]
= ∫ 𝑑𝜔 𝐻
𝜕𝑡
After Re-arrangement:
⃗
𝜕𝐵
⃗ ∙
⃗ ∗ (𝜔′)[(−𝑖𝜔)𝜇(𝜔)] ∙ 𝐻
⃗ (𝜔)𝑒 −𝑖(𝜔−𝜔′)𝑡
𝐻
= ∫ 𝑑𝜔 ∫ 𝑑𝜔′ 𝐻
𝜕𝑡
⃗ ∙
𝐻
Second, split the integral into two equal parts
⃗ ∙
𝐻
⃗
𝜕𝐵
1
1
⃗ ∗ (𝜔′)[−𝑖𝜔𝜇(𝜔)] ∙ 𝐻
⃗ (𝜔)𝑒 −𝑖(𝜔−𝜔′)𝑡 + ∫ 𝑑𝜔 ∫ 𝑑𝜔′ 𝐻
⃗ ∗ (𝜔′)[−𝑖𝜔𝜇(𝜔)] ∙ 𝐻
⃗ (𝜔)𝑒 −𝑖(𝜔−𝜔′)𝑡
= ∫ 𝑑𝜔 ∫ 𝑑𝜔′ 𝐻
𝜕𝑡 2
2
In the second integral make the following substitutions:
𝜔 → −𝜔′ 𝑎𝑛𝑑 𝜔′ → −𝜔
⃗ ∙
𝐻
⃗
𝜕𝐵
1
1
⃗ ∗ (𝜔′)[−𝑖𝜔𝜇(𝜔)] ∙ 𝐻
⃗ (𝜔)𝑒 −𝑖(𝜔−𝜔′)𝑡 + ∫ 𝑑(𝜔′) ∫ 𝑑(𝜔) 𝐻
⃗ (𝜔)[𝑖(𝜔′)𝜇∗ (𝜔′)] ∙ 𝐻
⃗ ∗ (𝜔′)𝑒 −𝑖(𝜔−𝜔′)𝑡
= ∫ 𝑑𝜔 ∫ 𝑑𝜔′ 𝐻
𝜕𝑡 2
2
Grouping Terms Results in:
⃗ ∙
𝐻
⃗
𝜕𝐵
1
⃗ ∗ (𝜔′)[−𝑖𝜔𝜇(𝜔) + 𝑖𝜔′𝜇∗ (𝜔′)] ∙ 𝐻
⃗ (𝜔)𝑒 −𝑖(𝜔−𝜔′)𝑡
= ∫ 𝑑𝜔 ∫ 𝑑𝜔′ 𝐻
𝜕𝑡 2
Expand 𝑖𝜔′𝜇∗ (𝜔′)
= 2𝜔 𝐼𝑚 𝜇(𝜔) − 𝑖(𝜔 − 𝜔′ )
𝑑
(𝜔𝜖 ∗ (𝜔)) + ⋯
𝑑𝜔
Jackson Equation 6.125 Magnetic Analog
⃗ ∙
𝐻
⃗
𝜕𝐵
1
𝜕 1
𝑑
′
⃗ ∗ (𝜔′ ) ∙ 𝐻
⃗ (𝜔)𝜔 𝐼𝑚 𝜇(𝜔)𝑒 −𝑖(𝜔−𝜔′)𝑡 +
⃗ ∗ (𝜔′ ) ∙ 𝐻
⃗ (𝜔)
= ∫ 𝑑𝜔 ∫ 𝑑𝜔′ 𝐻
∫ 𝑑𝜔 ∫ 𝑑𝜔′ 𝐻
𝜔𝜇∗ (𝜔)𝑒 −𝑖(𝜔−𝜔 )𝑡
𝜕𝑡 2
𝜕𝑡 2
𝑑𝜔
8. Now we can take the average of 𝐸⃗ ∙
⃗
𝜕𝐷
𝜕𝑡
⃗
⃗ ∙ 𝜕𝐵
𝑎𝑛𝑑 𝐻
𝜕𝑡
Jackson Equation 6.126a
⃗
⃗
𝜕𝑢
𝜕𝐷
𝜕𝐵
⃗ ∙
⃗ (𝑥, 𝑡) ∙ 𝐻
⃗ (𝑥, 𝑡)⟩ + 𝑒𝑓𝑓
𝐸⃗ ∙
𝑎𝑛𝑑 𝐻
= 𝜔0 𝐼𝑚𝜖(𝜔0 )⟨𝐸⃗ (𝑥, 𝑡) ∙ 𝐸⃗ (𝑥, 𝑡)⟩ + 𝜔0 𝐼𝑚𝜇(𝜔0 )⟨𝐻
𝜕𝑡
𝜕𝑡
𝜕𝑡
9. The Effective Electromagnetic Energy Density is:
1
𝑑(𝜔𝜖)
1
𝑑(𝜔𝜇)
⃗ (𝑥, 𝑡) ∙ 𝐻
⃗ (𝑥, 𝑡)⟩
(𝜔0 )] ⟨𝐸⃗ (𝑥, 𝑡) ∙ 𝐸⃗ (𝑥, 𝑡)⟩ + 𝑅𝑒 [
(𝜔0 )] ⟨𝐻
𝑢𝑒𝑓𝑓 = 𝑅𝑒 [
2
𝑑𝜔
2
𝑑𝜔
Section 6.7 counterpart to EM Energy Density is Jackson Equation 6.106
Page 5
Michelle While
Jackson’s EM
November 3, 2014
Presentation Notes for November 10
10. Conservation of Energy in dispersive media with losses (Poynting’s Theorem) reads:
Jackson Equation 6.127
𝜕𝑢𝑒𝑓𝑓
⃗ (𝑥, 𝑡) ∙ 𝐻
⃗ (𝑥, 𝑡)⟩
+ ⃗∇ ∙ 𝑆 = −𝐽 ∙ 𝐸⃗ − 𝜔0 𝐼𝑚𝜖(𝜔0 )⟨𝐸⃗ (𝑥, 𝑡) ∙ 𝐸⃗ (𝑥, 𝑡)⟩ − 𝜔0 𝐼𝑚𝜇(𝜔0 )⟨𝐻
𝜕𝑡
−𝐽 ∙ 𝐸⃗ is apparently ohmic (resistance) losses
−𝜔0 𝐼𝑚𝜖(𝜔0 )⟨𝐸⃗ (𝑥, 𝑡) ∙ 𝐸⃗ (𝑥, 𝑡)⟩ represents absorptive dissipation in the medium excluding conductive
losses.
Jackson Equation 6.108 is the section 6.7 analog to our Conservation of Energy Equation.
This concludes Section 6.8 presentation. Jackson cites additional sources for the interested student.
Reference
1. Jackson, John David, Classical Electrodynamics, 3rd Ed. John Wiley & Sons, Inc. (1999).
2. Griffiths, David J. Introduction to Electrodynamics, 4th Ed. Pearson, NY (2013)
3. Landau, L.D. and Liftshitz, E.M. Electrodynamics of Continuous Media Vol 8. 2nd Ed. Pergamon Press,
NY (1984).
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