Transcript Slides
November 10, 2014 Jackson’s Electrodynamics Michelle While USD
• • •
Summary
Electrical and Magnetic Energy propagates through vacuum and media via waves Media properties affect wave speed (frequency) which make dielectric ( ) and magnetic ( energy ) susceptibilities dependent upon frequency of the EXTERNAL EM Poynting’s Theorem utilizes conservation of energy to determine how energy is lost within a medium.
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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.
• 1.
Medium Characteristics
Linear or non-linear in nature 2.
3.
Isotropic or anisotropic 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 𝑑𝑡 𝑣 ∙ 𝐸 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 3.
𝑑𝑊 = 𝐽 ∙ 𝐸 𝑑 3 𝑥 𝑑𝑡 𝑉 Represents the EM energy converted into mechanical or thermal energy. EM energy is being removed from the 𝐸 𝑎𝑛𝑑 𝐵 fields Energy Losses are described by
Jackson Equation 6.105
𝑑 3 𝑥 𝐽 ∙ 𝐸 = − 𝑑 3 𝑥 𝛻 ∙ 𝐸 𝑋 𝐻 + 𝐸 ∙ 𝜕𝐷 𝜕𝑡 + 𝐻 ∙ 𝜕𝐵 𝜕𝑡 𝑉 𝑉
4.
Linear and Isotropic Media
Familiar Relationships
Jackson Equation 6.63
𝐷 = 𝜖𝐸 + 𝑃 𝐻 = 1 𝜇 𝐵 − 𝑀 5.
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 −𝑖𝜔𝑡 −∞ ∞ 𝑑𝜔 𝐵 −𝑖𝜔𝑡 −∞
6.
Dispersive Media
Energy losses within the media affect the relationships between 𝐻 𝑎𝑛𝑑 𝐵 . 𝐷 𝑎𝑛𝑑 𝐸 and Jackson Equation 7.105 reveals the nonlocality in time condition that occurs with dispersion. ∞ 𝐷 0 −∞
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:
𝐺 𝜏 = ∞ 1 2𝜋 −∞ 𝑑𝜔 𝜖 𝜔 𝜀 0 − 1 𝑒 −𝑖𝜔𝜏 Clearly when 𝜖 𝜔 in time 𝛿𝜏 is independent of 𝜔, 𝐺 𝜏 is directly proportional to the change and the instantaneous connection between 𝐷 𝑎𝑛𝑑 𝐸 is re-acquired. 𝐷 0 𝑥, 𝑡 Once re-acquired, there is no dispersion .
Derivation of
𝐸 ∙ 𝜕𝐷 𝜕𝑡
Jackson Equation 6.105
for Dispersive Media
𝑉 𝑑 3 𝑥 𝐽 ∙ 𝐸 = − 𝑑 3 𝑥 𝛻 ∙ 𝐸 𝑋 𝐻 + 𝐸 ∙ 𝜕𝐷 𝜕𝑡 + 𝐻 ∙ 𝜕𝐵 𝜕𝑡 𝑉 First we will write out 𝐸 ∙ dependence implicit.
𝜕𝐷 𝜕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 Take the partial derivative 𝜕𝐷 𝜕𝑡 𝜕 𝐷 𝜔 𝑒 −𝑖𝜔𝑡 𝜕𝑡 = 𝑑𝜔 𝐷 𝜔 −𝑖𝜔 𝑒 −𝑖𝜔𝑡
Derivation of
𝐸 ∙ 𝜕𝐷 𝜕𝑡
for Dispersive Media
Substitute 𝐷 . 𝜕𝐷 = 𝑑𝜔′ 𝜖 𝜔 𝜕𝑡 −𝑖𝜔 𝐸 𝜔′ 𝑒 −𝑖𝜔𝑡 𝜕𝐷 = 𝑑𝜔′ 𝜖 𝜔 𝜕𝑡 −𝑖𝜔 𝐸 −𝜔′ 𝑒 𝑖𝜔𝑡 Note that 𝐸 −𝜔′ = 𝐸 ∗ 𝜔′ and make substitution 𝜕𝐷 = 𝑑𝜔′ 𝜖 𝜔 𝜕𝑡 Multiply through by 𝐸 −𝑖𝜔 𝐸 𝐸 ∙ 𝜕𝐷 𝜕𝑡 = 𝑑𝜔 𝐸 𝜔 𝑒 −𝑖𝜔𝑡 𝑑𝜔′ 𝜖 𝜔 ∗ 𝜔′ 𝑒 𝑖𝜔𝑡 −𝑖𝜔 𝐸 ∗ 𝜔′ 𝑒 𝑖𝜔𝑡
Derivation of
𝐸 ∙ 𝜕𝐷 𝜕𝑡 Some Re-arrangement here
for Dispersive Media
𝐸 ∙ 𝜕𝐷 𝜕𝑡 = 𝑑𝜔 𝑑𝜔′ 𝐸 ∗ 𝜔′ −𝑖𝜔 𝜖 𝜔 ∙ 𝐸 𝜔 𝑒 −𝑖 𝜔−𝜔′ 𝑡 Second, split the integral into two equal parts 𝐸 ∙ 𝜕𝐷 𝜕𝑡 = 1 2 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 𝐸 ∙ 𝜕𝐷 𝜕𝑡 = 1 2 𝑑𝜔 𝑑𝜔′ 𝐸 ∗ 𝜔′ −𝑖𝜔𝜖 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 𝐸 ∙ 𝜕𝐷 𝜕𝑡 = 1 2 𝑑𝜔 𝑑𝜔′ 𝐸 ∗ 𝜔′ −𝑖𝜔𝜖 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 𝐸 ∙ 𝜕𝐷 𝜕𝑡 = 1 2 𝑑𝜔 𝑑𝜔′ 𝐸 ∗ 𝜔′ −𝑖𝜔𝜖 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
Dispersive Media-Energy Losses
Jackson Equation 6.124
𝐸 ∙ 𝜕𝐷 𝜕𝑡 = 1 2 𝑑𝜔 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 ∙ 1 𝜕𝑡 2 𝑑𝜔 𝑑𝜔 ′ 𝐸 ∗ 𝜔 ′ ∙ 𝐸 𝜔 𝜔 𝐼𝑚 𝜖 𝜔 𝑒 −𝑖 𝜔−𝜔 ′ 𝑡 + 𝜕 𝜕𝑡 1 2 𝑑𝜔 𝑑𝜔 ′ 𝐸 ∗ 𝜔 ′ ∙ 𝐸 𝜔 𝑑 𝑑𝜔 𝜔𝜖 ∗ 𝜔 𝑒 −𝑖 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 2 𝑑𝜔 𝑑𝜔 ′ 𝐻 ∗ 𝜔 ′ ∙ 𝐻 𝜔 𝜔 𝐼𝑚 𝜇 𝜔 𝑒 −𝑖 𝜔−𝜔 ′ 𝑡 + 𝜕 𝜕𝑡 1 2 𝑑𝜔 𝑑𝜔 ′ 𝐻 ∗ 𝜔 ′ ∙ 𝐻 𝜔 𝑑 𝑑𝜔 𝜔𝜇 ∗ 𝜔 𝑒 −𝑖 𝜔−𝜔 ′ 𝑡 Now we can take the average of 𝐸 ∙ 𝜕𝐷 𝜕𝑡 𝑎𝑛𝑑 𝐻 ∙ 𝜕𝐵 𝜕𝑡
Jackson Equation 6.126a
𝐸 ∙ 𝜕𝐷 𝜕𝑡 𝑎𝑛𝑑 𝐻 ∙ 𝜕𝐵 𝜕𝑡 = 𝜔 0 𝐼𝑚𝜖 𝜔 0 + 𝜔 0 𝐼𝑚𝜇 𝜔 0 + 𝜕𝑢 𝑒𝑓𝑓 𝜕𝑡 Effective Electromagnetic Energy Density is: 𝑑 𝜔𝜖 𝑢 𝑒𝑓𝑓 = 1 2 𝑅𝑒 𝑑𝜔 𝜔 0 + 1 2 𝑅𝑒 𝑑 𝜔𝜇 𝑑𝜔 𝜔 0
Section 6.7 counterpart to EM Energy Density is Jackson Equation 6.106
Poynting’s Theorem
Jackson Equation 6.127
𝜕𝑢 𝑒𝑓𝑓 𝜕𝑡 + 𝛻 ∙ 0 𝐼𝑚𝜖 𝜔 0 − 𝜔 0 𝐼𝑚𝜇 𝜔 0 − 𝐽 ∙ 𝐸 represent the ohmic (resistance) losses −𝜔 0 𝐼𝑚𝜖 𝜔 0 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 𝜕𝑡
References
Jackson, John David, Classical Electrodynamics, 3 rd & Sons, Inc. (1999).
Ed. John Wiley Griffiths, David J. Introduction to Electrodynamics, 4 th Pearson, NY (2013) Ed. Landau, L.D. and Liftshitz, E.M. Electrodynamics of Continuous Media Vol 8. 2 nd Ed. Pergamon Press, NY (1984).