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Previous Lectures:
Dielectric: response of bound electrons Displacement from equilibrium position caused by E field in EM wave is much smaller of radiation wavelength.Spatial variation of E can be ignored.
By considering what happens when a sinusoidal plane waveis incident on the surface of a dielectric, and using boundaryconditions, obtain the laws of optics and determine the fraction of transmitted / reflected light.
Describe total internal reflection and what is meant by evanescent waveDescribe significance of the critical angle, and Brewster angleDerive Fresnel’s equations, and determine intensity of reflectedand transmitted beams
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Lecture 15
Drude model for conductors
Good and bad conductors in terms of displacement andconduction currents
Model the propagation of plan monochromatic waves in a conducting medium
Derive expression for skin depth
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Neglect polarization
Propagation determined by much larger freeconduction currents
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Current density in conductors
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Drude model
Free electrons in conductor are subject to:
Dumping force due to collisionswith lattice at frequency ν
Mean time between collisions
Solve differential eq. by multiplying by integrating factor
where
for copper
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Solution:
Conductivity
Resistivity
In most conducting materials, E producesJ proportional to E
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Static conductivity
Frequency dependence of EM field is not important Average drift motion in static electric field:
Copper at 20C:
Valid for ω << 1 / τc
In this lecture, we restrict ourselves to this case
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Wave equation in conductors (LIH)
Start with M4
With
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Differentiate w.r.t time
Use M3
Hence
Since
And M1
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Similarly
Conduction current
Displacementcurrent
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Solution of the type:
Short cuts:
Wave numberIs complex
Define Ratio of displacement toconduction terms
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Poor conductor
If ω > σ / ε0 the material behaves as poor conductor Absorption but oscillation is “normal” - plenty of oscillationsin the medium like dielectricWeakly dumped single harmonic oscillator
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Good conductor
If ω < σ / ε0 the material behaves as good conductor and the EM wave is severely attenuated inside the conductor
It can be shown that this corresponds to Q :
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Put into EM wave propagating in z direction:
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Wave attenuatedin good conductor
Attenuated by factor ein traveling distance δ
Skin depth
Amplitude of magnetic field has same exponential decay
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Attenuation is very rapid: compare δ with λ, free space wavelength
Since
Distance that EM wave penetrates in conductor is much lessthan free space wavelength.Attenuation is so large that wav is hardly discernable.Therefore good conductors are highly opaque to EM waves(except for extremely thin films)
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Copper: skin depth = 1 µm at 1 GHz; 1 cm at 50 Hz
Copper wire
E penetrates distance δ into surface.At 1 GHz current is carried in thin outer layer.
Resistance at 1 GHz:
Resistance at 50 Hz:
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Magnetic field
Use
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In good conductor:
Electric field leads the magnetic field by 45 degrees in goodconductors.Current that gives rise to H is conduction current, not displacement current.
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Summary
conductivity
Good and bad conductors
Skin depth
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Recommended readings:Grant+Phillips: 11.5 11.6.3
Next Lecture:
EM waves in conductors (II)
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