Electromagnetism II

Cristina Lazzeronic.lazzeroni@bham.ac.uk

Lecture 11

Last lecture

Also Poynting vector:

Lecture 11:

Qualitatively examine light scattered by free electrons

Qualitatively examine light scattered by bound electrons

Examine the cross section for scattering as a function ofthe frequency of EM waveHighlight the regions of Rayleigh, Resonance and Thomson scattering

Explain the blueness and redness of the sky under variousconditions

Free electrons

Two cases:

Different mathematical approach

Thomson scattering

Interaction of oscillating field with free (unbound) electron

We can ignore the magnetic field:

Magnetic force Electric force

Electron velocity

Plane polarized EM wave:

Since E is sinusoidal, electron will oscillate up and downat frequency ωAs a result of the acceleration, electron will radiate at the driving frequency but not in the original direction of polarization

Electron at z=0 experiences acceleration:

Oscillating charge that re-radiates in all directions

Radiation field similar to ideal dipole (also called Hertzian dipole)

Plane through the axis of dipole,For r >>λ

Radiation pattern that shows dependence of field strengthon angle at fixed distance - length of blue arrow is proportional to strength

Solution: with

Mean power radiated by accelerating charge:(derivation beyond the scope of this course)

Note dependenceon mass

Larmor formula

Lecture 9

Here v = c, free space:

Power per unit area of incident wave

Define scattering cross section:

Classical radius

Consider cross section as a target of area σ (not a realmaterial target !). If all the light energy that impinged onthat area were to be spewed out in all directions, then thatis the energy that would be scattered by the electron.

σT is very small - scattering is weak (atomic cross section~10-20 m2) σT is much smaller for proton than electronσT is independent of frequency

Result correct if

For

Cross section for scattering decreases with increasing photon energy - Compton Scattering region

The main difference is that the momentum of the photon is now significant, meaning that the electron has some non negligible recoil due to the scattering

Bound electrons, elastic scattering

Scattering of radiation by neutral atoms without change in frequency (elastic scattering).Classical treatment, similar approach as before.

Atom with a single electron which can oscillate at natural angular frequency ω0, and subject to a plane EM wave:

Dumped simple harmonic oscillator

Loss of energy as EM radiation

Taking oscillating electronas a dipole (Hertzian dipole)it can be shown that:

Steady state solution:

Real part of x:

with

Total power radiated by charge:(Larmor formula)

Incident power per unit area:

Cross section:

Elasticscattering

I. Rayleigh scattering (tightly bound electrons)

ω << ω0

Response is controlled by restoring force

Atmospheric molecules, mostly N2 : natural frequencies are in the ultraviolet so electrons are tightly bound for visible light.

Blue light is scattered far more than the red.

Mean free path:

II. Resonance scattering

III. Thomson scattering

Response controlled by electron inertia

When a simple harmonic oscillator is driven at frequencymuch higher than resonance frequency, the inertia, not therestoring or dumping force, control the response.So electron behaves as if free - Thomson scattering (or Compton)

Summary

Recommended readings:Grant+Phillips: 13.1, 13.2

Next Lecture:

Propagation of EM waves in dielectric materials