November 10, 2014

Jackson’s Electrodynamics

Michelle While

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Section 6.8 Poynting’s Theorem in Linear

Dispersive Media with Losses

• 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.

Summary

• 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 Characteristics1. Linear or non-linear in nature2. Isotropic or anisotropic3. Non-dispersive with “no” energy losses or

Dispersive with losses

Background

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

Energy Losses

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

Linear and Isotropic 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 re-acquired.

Once re-acquired, there is no dispersion.

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:

 

 

 

Derivation of for Dispersive Media

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

Dispersive Media-Energy Losses

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

Poynting’s Theorem

Jackson, John David, Classical Electrodynamics, 3rd Ed. John Wiley & Sons, Inc. (1999).

Griffiths, David J. Introduction to Electrodynamics, 4th Ed. Pearson, NY (2013)

Landau, L.D. and Liftshitz, E.M. Electrodynamics of Continuous Media Vol 8. 2nd Ed. Pergamon Press, NY (1984).

References