Section 6.8 Poynting S Theorem in Linear Dispersive Media with Losses

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.

2.  The rate of doing work on a single charge by EXTERNAL electromagnetic fields E and B is:

dWdt=qv∙E

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

dWdt=VJ∙Ed3x

This is the electromagnetic energy converted into mechanical or thermal energy. Of course, that means that energy has been removed from the electromagnetic fields E and B.

Jackson Equation 6.105

Vd3xJ∙E=-Vd3x∇∙E X H+E∙∂D∂t+H∙∂B∂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 means the relationship between D and E is linear as is the relationship between H and B.

Jackson Equation 6.63

D=ϵE+P

H=1μB-M

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.

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 (transformations) given in the text for D and E and H and B exhibit this wave nature.

Ex,t=-∞∞dωEx,ωe-iωt

Dx,t=-∞∞dωDx,ωe-iωt

Hx,t=-∞∞dωHx,ωe-iωt

Bx,t=-∞∞dωBx,ωe-iωt

Additionally, dispersion affects the relationship between D and E and H and B.

Jackson Equation 7.105 reveals the nonlocality in time condition that occurs with dispersion.

Basically, the value of D at time t depends upon the value of the electric field at times other than t.

This temporal/spatial adjustment is given in

Jackson Equation 7.106.

Gτ=12π-∞∞dωϵωε0-1e-iωτ

Clearly when ϵω is independent of ω Gτ is directly proportional to the change in time δτ and the instantaneous connection between D and E is re-acquired.

Dx,t=ε0Ex,t

Once re-acquired, there is no dispersion.

5.  Linearity and Isotropism implies:

Dx,ω=ϵωEx,ω

Bx,ω=μωHx,ω

Other useful relations are provided in our text:

Ex,-ω=E*x,ω

Dx,-ω=D*x,ω

Bx,-ω=B*x,ω

Hx,-ω=H*x,ω

ϵ-ω=ϵ*ω

6.  Two terms in Equation 6.105 are affected by the frequency dependence of dielectric susceptibility and magnetic susceptibility.

E∙∂D∂t and H∙∂B∂t

First we will write out E∙∂D∂t in terms of the Fourier integrals with spatial dependence implicit.

Fourier integrals with spatial dependence:

Et=dωEωe-iωt

Dt=dωDωe-iωt

Take the partial derivative ∂D∂t

∂∂tDωe-iωt=dωDω-iωe-iωt

Substitute Dx,ω=ϵωEω'.

∂D∂t=dω'ϵω-iωEω'e-iωt

∂D∂t=dω'ϵω-iωE-ω'eiωt

∂D∂t=dω'ϵω-iωE*ω'eiωt

E∙∂D∂t=dωEωe-iωtdω'ϵω-iωE*ω'eiωt

Some Re-arrangement here

E∙∂D∂t=dωdω'E*ω'-iωϵω∙Eωe-iω-ω't

Second, split the integral into two equal parts

E∙∂D∂t=12dωdω'E*ω'-iωϵω∙Eωe-iω-ω't+12dωdω'E*ω'-iωϵω∙Eωe-iω-ω't

In the second integral make the following substitutions:

ω→-ω'and ω'→-ω

E∙∂D∂t=12dωdω'E*ω'-iωϵω∙Eωe-iω-ω't+12d-ω'd-ωE*-ω-i-ω'ϵ-ω'∙E-ω'e-i-ω'+ωt

E∙∂D∂t=12dωdω'E*ω'-iωϵω∙Eωe-iω-ω't+12dω'dωEωiω'ϵ*ω'∙E*ω'e-iω-ω't

E∙∂D∂t=12dωdω'E*ω'-iωϵω∙Eωe-iω-ω't+12dω'dωEωiω'ϵ*ω'∙E*ω'e-iω-ω't

Grouping Terms Results in Jackson Equation 6.124

E∙∂D∂t=12dωdω'E*ω'-iωϵω+iω'ϵ*ω'∙Eωe-iω-ω't

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

E∙∂D∂t=12dωdω'E*ω'∙Eωω Im ϵωe-iω-ω't+∂∂t12dωdω'E*ω'∙Eωddωωϵ*ωe-iω-ω't

Thus exposing this more complex expression as compared with non-dispersive media in Section 6.7

∂E∂t∙D=∂D∂t∙E

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:

Ht=dωHωe-iωt

Bt=dωBωe-iωt

Take the partial derivative ∂B∂t

∂∂tBωe-iωt=dωBω-iωe-iωt

Substitute Bx,ω=μωHω'.

H∙∂B∂t=dωHωe-iωtdω'μω-iωH*ω'eiωt

After Re-arrangement:

H∙∂B∂t=dωdω'H*ω'-iωμω∙Hωe-iω-ω't

Second, split the integral into two equal parts

H∙∂B∂t=12dωdω'H*ω'-iωμω∙Hωe-iω-ω't+12dωdω'H*ω'-iωμω∙Hωe-iω-ω't

In the second integral make the following substitutions:

ω→-ω'and ω'→-ω

H∙∂B∂t=12dωdω'H*ω'-iωμω∙Hωe-iω-ω't+12dω'dωHωiω'μ*ω'∙H*ω'e-iω-ω't

Grouping Terms Results in:

H∙∂B∂t=12dωdω'H*ω'-iωμω+iω'μ*ω'∙Hωe-iω-ω't

Expand iω'μ*ω'

=2ω Im μω-iω-ω'ddωωϵ*ω+…

Jackson Equation 6.125 Magnetic Analog

H∙∂B∂t=12dωdω'H*ω'∙Hωω Im μωe-iω-ω't+∂∂t12dωdω'H*ω'∙Hωddωωμ*ωe-iω-ω't

8.  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=ω0 Imϵω0Ex,t∙Ex,t+ω0 Imμω0Hx,t∙Hx,t+∂ueff∂t

9.  The Effective Electromagnetic Energy Density is:

ueff=12Redωϵdωω0Ex,t∙Ex,t+12Redωμdωω0Hx,t∙Hx,t

Section 6.7 counterpart to EM Energy Density is Jackson Equation 6.106

10.  Conservation of Energy in dispersive media with losses (Poynting’s Theorem) reads:

Jackson Equation 6.127

∂ueff∂t+∇∙S=-J∙E-ω0 Imϵω0Ex,t∙Ex,t-ω0 Imμω0Hx,t∙Hx,t

-J∙E is apparently ohmic (resistance) losses

-ω0 Imϵω0Ex,t∙Ex,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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