Electromagnetism II

Cristina Lazzeronic.lazzeroni@bham.ac.uk

Lecture 15

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

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

Neglect polarization

Propagation determined by much larger freeconduction currents

Current density in conductors

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

Solution:

Conductivity

Resistivity

In most conducting materials, E producesJ proportional to E

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

Wave equation in conductors (LIH)

Start with M4

With

Differentiate w.r.t time

Use M3

Hence

Since

And M1

Similarly

Conduction current

Displacementcurrent

Solution of the type:

Short cuts:

Wave numberIs complex

Define Ratio of displacement toconduction terms

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

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 :

Put into EM wave propagating in z direction:

Wave attenuatedin good conductor

Attenuated by factor ein traveling distance δ

Skin depth

Amplitude of magnetic field has same exponential decay

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)

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:

Magnetic field

Use

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.

Summary

conductivity

Good and bad conductors

Skin depth

Recommended readings:Grant+Phillips: 11.5 11.6.3

Next Lecture:

EM waves in conductors (II)