Section 6.8 Poynting's Theorem in Linear Dispersive Media ...
Section 6.8 Poyntings Theorem in Linear Dispersive Media with Losses November 10, 2014 Jacksons 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 () susceptibilities dependent upon frequency of the EXTERNAL EM energy Poyntings 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 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
Represents the EM energy converted into mechanical or thermal energy. EM energy is being removed from the fields 3. Energy Losses are described by Jackson Equation 6.105 Linear and Isotropic Media 4. Familiar Relationships Jackson Equation 6.63 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 Dispersive Media 6. Energy losses within the media affect the relationships between and . Jackson Equation 7.105 reveals the nonlocality in time condition that occurs with dispersion. Basically, the value of at time t depends upon the value of the electric field at times other than t.
Jackson Equation 7.106 the Temporal/Spatial Adjustment: Clearly when is independent of is directly proportional to the change in time and the instantaneous connection between is reacquired. Once re-acquired, there is no dispersion. Derivation of for Dispersive Media Jackson Equation 6.105 First we will write out in terms of the Fourier integrals with spatial dependence implicit. Fourier integrals with spatial dependence:
Take the partial derivative Derivation of for Dispersive Media Substitute . Note that and make substitution Multiply through by Derivation of for Dispersive Media Some Re-arrangement here
Second, split the integral into two equal parts In the second integral make the following substitutions: Dispersive Media-Energy Losses Jackson Equation 6.124 Recall that the changes wrt to frequency so those terms must be expanded Jackson Equation 6.125
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 Now we can take the average of Jackson Equation 6.126a
Effective Electromagnetic Energy Density is: Section 6.7 counterpart to EM Energy Density is Jackson Equation 6.106 Poyntings Theorem Jackson Equation 6.127 represent the ohmic (resistance) losses 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 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. 2 nd Ed. Pergamon Press, NY (1984).
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