Download Section 6.8 Poynting*s Theorem in Linear Dispersive Media with

November 10, 2014
Jackson’s Electrodynamics
Michelle While
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Summary
• Electrical and Magnetic Energy propagates
through vacuum and media via waves
• Media properties affect wave speed
(frequency) which make dielectric () and
magnetic () susceptibilities dependent
upon frequency of the EXTERNAL EM
energy
• Poynting’s Theorem utilizes conservation of
energy to determine how energy is lost
within a medium.
Background
• Atoms within substances move. They exhibit thermal
agitation, zeros point vibration and orbital motion
which gives rise to internal frequencies of the
substance, however, these motions average out so only
EXTERNAL applied oscillators contribute to the
frequencies exhibited by the material.
• Medium Characteristics
1. Linear or non-linear in nature
2. Isotropic or anisotropic
3. Non-dispersive with “no” energy losses or Dispersive
with losses
Energy Losses
1.
Rate of doing work on a single charge by EXTERAL EM fields
𝑑𝑊
= 𝑞𝑣 ∙ 𝐸
𝑑𝑡
Magnetic Field Does NOT Contribute to the Work Done Because it is
Perpendicular to Velocity
2.
Rate of doing work in a defined volume of medium with continuous
charge and current
𝑑𝑊
=
𝑑𝑡
𝐽 ∙ 𝐸 𝑑3𝑥
𝑉
Represents the EM energy converted into mechanical or thermal
energy. EM energy is being removed from the 𝐸 𝑎𝑛𝑑 𝐵 fields
3.
Energy Losses are described by
Jackson Equation 6.105
𝑑3𝑥 𝐽
𝑉
𝑑3𝑥
∙𝐸 =−
𝑉
𝜕𝐷
𝜕𝐵
𝛻∙ 𝐸𝑋𝐻 +𝐸∙
+𝐻∙
𝜕𝑡
𝜕𝑡
Linear and Isotropic Media
4.
Familiar Relationships
Jackson Equation 6.63
𝐷 = 𝜖𝐸 + 𝑃
𝐻=
5.
1
𝐵−𝑀
𝜇
Dielectric and Magnetic Susceptibilities become complex and frequency
dependent when the media is Dispersive
Fourier Transformations account for the wave nature of EM energy
∞
𝑑𝜔 𝐻 𝑥, 𝜔 𝑒 −𝑖𝜔𝑡
𝐻 𝑥, 𝑡 =
−∞
∞
𝑑𝜔 𝐵 𝑥, 𝜔 𝑒 −𝑖𝜔𝑡
𝐵 𝑥, 𝑡 =
−∞
Dispersive Media
6. Energy losses within the media affect the relationships between 𝐷 𝑎𝑛𝑑 𝐸 and
𝐻 𝑎𝑛𝑑 𝐵.
Jackson Equation 7.105 reveals the nonlocality in time condition that occurs with
dispersion.
∞
𝐷 𝑥, 𝑡 = 𝜀0 𝐸 𝑥, 𝑡 +
𝑑𝜏 𝐺 𝜏 𝐸 𝑥, 𝑡 − 𝜏
−∞
Basically, the value of 𝑫 at time t depends upon the value of the electric field
at times other than t.
Jackson Equation 7.106 the Temporal/Spatial Adjustment:
1
𝐺 𝜏 =
2𝜋
∞
−∞
𝜖 𝜔
𝑑𝜔
− 1 𝑒 −𝑖𝜔𝜏
𝜀0
Clearly when 𝜖 𝜔 is independent of 𝜔, 𝐺 𝜏 is directly proportional to the change
in time 𝛿𝜏 and the instantaneous connection between 𝐷 𝑎𝑛𝑑 𝐸 is re-acquired.
𝐷 𝑥, 𝑡 = 𝜀0 𝐸 𝑥, 𝑡
Once re-acquired, there is no dispersion.
Derivation of 𝐸 ∙
𝜕𝐷
𝜕𝑡
for Dispersive Media
Jackson Equation 6.105
𝑑3𝑥 𝐽 ∙ 𝐸 = −
𝑉
First we will write out 𝐸 ∙
dependence implicit.
𝑑3𝑥 𝛻 ∙ 𝐸 𝑋 𝐻 + 𝐸 ∙
𝑉
𝜕𝐷
𝜕𝑡
𝜕𝐷
𝜕𝐵
+𝐻∙
𝜕𝑡
𝜕𝑡
in terms of the Fourier integrals with spatial
Fourier integrals with spatial dependence:
𝐸 𝑡 =
𝑑𝜔 𝐸 𝜔 𝑒 −𝑖𝜔𝑡
𝐷 𝑡 =
𝑑𝜔 𝐷 𝜔 𝑒 −𝑖𝜔𝑡
𝜕𝐷
Take the partial derivative 𝜕𝑡
𝜕
𝐷 𝜔 𝑒 −𝑖𝜔𝑡 =
𝜕𝑡
𝑑𝜔 𝐷 𝜔
−𝑖𝜔 𝑒 −𝑖𝜔𝑡
Derivation of 𝐸 ∙
𝜕𝐷
𝜕𝑡
for Dispersive Media
Substitute 𝐷 𝑥, 𝜔 = 𝜖 𝜔 𝐸 𝜔′ .
𝜕𝐷
=
𝜕𝑡
𝑑𝜔′ 𝜖 𝜔
−𝑖𝜔 𝐸 𝜔′ 𝑒 −𝑖𝜔𝑡
𝜕𝐷
=
𝜕𝑡
𝑑𝜔′ 𝜖 𝜔
−𝑖𝜔 𝐸 −𝜔′ 𝑒 𝑖𝜔𝑡
Note that 𝐸 −𝜔′ = 𝐸 ∗ 𝜔′ and make substitution
𝜕𝐷
=
𝜕𝑡
𝑑𝜔′ 𝜖 𝜔
−𝑖𝜔 𝐸 ∗ 𝜔′ 𝑒 𝑖𝜔𝑡
Multiply through by 𝐸
𝐸∙
𝜕𝐷
=
𝜕𝑡
𝑑𝜔 𝐸 𝜔 𝑒 −𝑖𝜔𝑡
𝑑𝜔′ 𝜖 𝜔
−𝑖𝜔 𝐸 ∗ 𝜔′ 𝑒 𝑖𝜔𝑡
Derivation of 𝐸 ∙
𝜕𝐷
𝜕𝑡
for Dispersive Media
Some Re-arrangement here
𝜕𝐷
𝐸∙
=
𝜕𝑡
𝑑𝜔
𝑑𝜔′ 𝐸 ∗ 𝜔′
−𝑖𝜔 𝜖 𝜔
∙ 𝐸 𝜔 𝑒 −𝑖
𝜔−𝜔′ 𝑡
Second, split the integral into two equal parts
𝜕𝐷 1
=
𝜕𝑡
2
𝐸∙
𝑑𝜔
𝑑𝜔′ 𝐸 ∗ 𝜔′ −𝑖𝜔𝜖 𝜔
∙ 𝐸 𝜔 𝑒 −𝑖
𝜔−𝜔′ 𝑡
+
1
2
𝑑𝜔
𝑑𝜔′ 𝐸 ∗ 𝜔′ −𝑖𝜔𝜖 𝜔
∙ 𝐸 𝜔 𝑒 −𝑖
𝜔−𝜔′ 𝑡
In the second integral make the following substitutions:
𝜔 → −𝜔 ′ 𝑎𝑛𝑑 𝜔 ′ → −𝜔
𝜕𝐷 1
=
𝜕𝑡
2
𝑑𝜔
𝑑𝜔′ 𝐸 ∗ 𝜔′ −𝑖𝜔𝜖 𝜔
∙ 𝐸 𝜔 𝑒 −𝑖
𝐸∙
𝜕𝐷 1
=
𝜕𝑡
2
𝑑𝜔
𝑑𝜔′ 𝐸 ∗ 𝜔′ −𝑖𝜔𝜖 𝜔
∙ 𝐸 𝜔 𝑒 −𝑖
𝜔−𝜔′ 𝑡
𝐸∙
𝜕𝐷 1
=
𝜕𝑡
2
𝑑𝜔
𝑑𝜔′ 𝐸 ∗ 𝜔′ −𝑖𝜔𝜖 𝜔
∙ 𝐸 𝜔 𝑒 −𝑖
𝜔−𝜔′ 𝑡
𝐸∙
𝜔−𝜔′ 𝑡
1
2
𝑑 −𝜔′
+
1
2
𝑑 𝜔′
𝑑 𝜔 𝐸 𝜔 𝑖 𝜔′ 𝜖 ∗ 𝜔′
∙ 𝐸 ∗ 𝜔′ 𝑒 −𝑖
𝜔−𝜔′ 𝑡
+
1
2
𝑑 𝜔′
𝑑 𝜔 𝐸 𝜔 𝑖 𝜔′ 𝜖 ∗ 𝜔′
∙ 𝐸 ∗ 𝜔′ 𝑒 −𝑖
𝜔−𝜔′ 𝑡
+
𝑑 −𝜔 𝐸 ∗ −𝜔 −𝑖 −𝜔′ 𝜖 −𝜔′
∙ 𝐸 −𝜔′ 𝑒 −𝑖
−𝜔 ′ +𝜔 𝑡
Dispersive Media-Energy Losses
Jackson Equation 6.124
𝜕𝐷 1
𝐸∙
=
𝑑𝜔
𝜕𝑡
2
𝑑𝜔′ 𝐸 ∗ 𝜔′ −𝑖𝜔𝜖 𝜔 + 𝑖𝜔′𝜖 ∗ 𝜔′ ∙ 𝐸 𝜔 𝑒 −𝑖
𝜔−𝜔′ 𝑡
Recall that the  changes wrt to frequency so those terms must be expanded
Jackson Equation 6.125
𝜕𝐷
𝐸∙
𝜕𝑡
1
=
𝑑𝜔 𝑑𝜔 ′ 𝐸 ∗ 𝜔 ′ ∙ 𝐸 𝜔 𝜔 𝐼𝑚 𝜖 𝜔 𝑒 −𝑖
2
∙𝐸 𝜔
𝑑
𝜔𝜖 ∗ 𝜔 𝑒 −𝑖
𝑑𝜔
𝜔−𝜔′ 𝑡
𝜕 1
+
𝜕𝑡 2
𝑑𝜔
𝑑𝜔 ′ 𝐸 ∗ 𝜔 ′
𝜔−𝜔 ′ 𝑡
Electric fields have a wave nature and in dielectric materials the  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.
Dispersive Media-Energy Losses
Jackson Equation 6.125 Magnetic Analog
𝜕𝐵
𝐻∙
𝜕𝑡
1
=
𝑑𝜔 𝑑𝜔 ′ 𝐻 ∗ 𝜔 ′ ∙ 𝐻 𝜔 𝜔 𝐼𝑚 𝜇 𝜔 𝑒 −𝑖
2
𝑑
∙𝐻 𝜔
𝜔𝜇 ∗ 𝜔 𝑒 −𝑖
𝑑𝜔
𝜔−𝜔′ 𝑡
+
𝜕 1
𝜕𝑡 2
𝑑𝜔
𝑑𝜔 ′ 𝐻 ∗ 𝜔 ′
𝜔−𝜔′ 𝑡
Now we can take the average of 𝐸 ∙
𝜕𝐷
𝜕𝑡
𝑎𝑛𝑑 𝐻 ∙
𝜕𝐵
𝜕𝑡
Jackson Equation 6.126a
𝜕𝐷
𝜕𝐵
𝐸∙
𝑎𝑛𝑑 𝐻 ∙
𝜕𝑡
𝜕𝑡
= 𝜔0 𝐼𝑚𝜖 𝜔0 𝐸 𝑥, 𝑡 ∙ 𝐸 𝑥, 𝑡
+ 𝜔0 𝐼𝑚𝜇 𝜔0 𝐻 𝑥, 𝑡 ∙ 𝐻 𝑥, 𝑡
Effective Electromagnetic Energy Density is:
1
𝑑 𝜔𝜖
1
𝑑 𝜔𝜇
𝑢𝑒𝑓𝑓 = 𝑅𝑒
𝜔0 𝐸 𝑥, 𝑡 ∙ 𝐸 𝑥, 𝑡 + 𝑅𝑒
2
𝑑𝜔
2
𝑑𝜔
𝜔0
𝜕𝑢𝑒𝑓𝑓
+
𝜕𝑡
𝐻 𝑥, 𝑡 ∙ 𝐻 𝑥, 𝑡
Poynting’s Theorem
Jackson Equation 6.127
𝜕𝑢𝑒𝑓𝑓
+ 𝛻 ∙ 𝑆 = −𝐽 ∙ 𝐸 − 𝜔0 𝐼𝑚𝜖 𝜔0 𝐸 𝑥, 𝑡 ∙ 𝐸 𝑥, 𝑡
𝜕𝑡
− 𝜔0 𝐼𝑚𝜇 𝜔0 𝐻 𝑥, 𝑡 ∙ 𝐻 𝑥, 𝑡
−𝐽 ∙ 𝐸 represent the ohmic (resistance) losses
−𝜔0 𝐼𝑚𝜖 𝜔0 𝐸 𝑥, 𝑡 ∙ 𝐸 𝑥, 𝑡 represents absorptive dissipation in the
medium excluding conductive losses.
In section 6.7 is the analog to our Conservation of Energy Equation.
Jackson Equation 6.108
𝜕𝑢
+ 𝛻 ∙ 𝑆 = −𝐽 ∙ 𝐸
𝜕𝑡
References
Jackson, John David, Classical Electrodynamics, 3 rd Ed. John Wiley
& Sons, Inc. (1999).
Griffiths, David J. Introduction to Electrodynamics, 4 th Ed.
Pearson, NY (2013)
Landau, L.D. and Liftshitz, E.M. Electrodynamics of Continuous
Media Vol 8. 2nd Ed. Pergamon Press, NY (1984).