dispersion of the permittivity section 77. polarization involves motion of charge against a...
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Dispersion of the permittivity
Section 77
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Polarization involves motion of charge against a restoring force.
When the electromagnetic frequency approaches the resonance frequency, new physics appears.
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An EM field that varies in time, varies in space too, due to finite propagation speed.
The spatial periodicity of the field is l ~ c/w
As w increases, l approaches interatomic distances “a”.
Then macroscopic electrodynamics makes no sense., since averaging over interatomic distances would produce a macroscopic field E= 0.
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Frequencies close to polarization resonances, but where the macroscopic description of the fields still applies, must exist.
The fastest possible motion is electronic.
The electron relaxation time ~ a/v.
Atomic velocities v<<c.
At resonance, l ~ c/w ~ c/(v/a) = (v/c) a >> a, so that the macroscopic field description holds.
At high frequencies near atom-electron resonances, metals and dielectrics behave the same (see section 78)
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Macroscopic maxwell’s equations
Electrically neutral matter without extraneous charge
Always true
Faraday’s law for macroscopic fields
Dielectric, without free current
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Those maxwell equations are not enough to solve for E,D,B,and H
The solution requires constitutive relations
But these relations are not as simple as in the static and quasistatic cases
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D and B depend on E and H, not only at the present time, but also on their values at earlier times.
The polarization lags the changes in the fields.
Microscopic charge density
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• The derivation of divP = -<r>r is independent of the time dependence of the field. • Interpretation of P is the same.
The total electric dipole moment of a body =
The interpretation of the polarization P = (D-E)/4p is electric moment per unit volume, regardless of the variation of the field.
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Rapidly varying fields are usually weak (except for laser fields in non-linear optics)
D = D(E) is usually linear in E.
Present instant
Previous instants
A function of time and the nature of the medium
Some linear integral operator
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• A field with arbitrary time dependence can be decomposed into Fourier components
• Because equations are linear, we can treat each monochromatic term in the expansion independently
• Each has time dependence e-iwt
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For wth Fourier component
A frequency dependent material property
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Dispersion relation, generally complex
Real part Imaginary part
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An even function of frequency
An odd function of frequency
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Well below resonances in the material, the dispersion effects are small.
There we can expand e(w) in powers of w
Can have only even powers of w, namely w0, w2, w4, etc.
Can have only even powers of w, namely w, w3, w5, etc.
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Static case: Limit w -> 0
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Conductors are a special case
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Expansion of ( )e w begins with 4pis/w term for metals
• First term is an imaginary odd function of w • The next term is a real constant• This term is unimportant if effects of spatial variations (skin effect) are
more important than effects of time variations.
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The real part of the permittivity is an1. Even function of w2. Odd function of w3. Exponential function of w
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The imaginary part of the permittivity is1. An even function of w2. An odd function of w3. Independent of w
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Which curve seems most likely to belong to a metal?