Phys 312 - Assignment 10 - Due 9 Apr 98

1.

Radiation and scattering: why is the sky blue? As one application of the Larmor radiation formula we will study the excitation of an atom by an incident plane EM wave. The interaction of the electric field of the incident plane wave with the electrons and the nucleus causes them to oscillate at the same frequency as the wave, resulting in an oscillating electric dipole (the magnetic forces are entirely negligible except for very intense incident waves). The accelerations of the electrons are large compared to the acceleration of the nucleus, due to the very large mass ratio. The oscillating dipole then causes the atom to radiate electric dipole radiation, as discussed in class.

(a)

Consider a simple model in which one electron is bound harmonically to the nucleus (i.e., the electron is subject to a restoring force $-m\omega _{0}^{2}z$, with m the electron mass, $\omega _{0}$ a natural atomic frequency, and z the displacement from the origin). The electric field of the incident plane wave at the atom has the form $E_{z}(t)=E_{0}\cos \omega t$ (we can manage well without complex notation in this problem). Assuming that $\omega \ll \omega _{0}$, show that the electron moves in phase with the electric field with an acceleration  

$$a=\frac{eE_{0}\omega ^{2}}{m\omega _{0}^{2}}\cos \omega t.$$ (1)

(b)

Use the Larmor radiation formula to show that the time-averaged power radiated by the charge is  

$$P_{{\rm av}}={\frac{1}{4\pi \epsilon _{0}}}\frac{e^{4}E_{0}^... ...3m^{2}c^{3}}\left( {\frac{\omega }{\omega _{0}}}\right) ^{4}.$$ (2)

Express this in terms of the wavelength $\lambda$ of the incident wave to show that $P_{{\rm av}}\propto \lambda ^{-4}$, which is a famous law derived by Lord Rayleigh. This result shows that short wavelength radiation is scattered more effectively by atoms than long wavelength radiation. This result is valid for $\lambda \gg a$, with a the characteristic size of the atom.

(c)

Use the above result to explain (i) why the sky is blue, (ii) why sunsets are red, and (iii) why it is easier to get sunburned at midday.

(d)

Finally, think a bit about the polarization of the scattered radiation. Suppose you take some Polaroid sunglasses and look northward as the sun sets in the west. If you rotate the sunglasses you'll notice marked intensity variations (try it). Why? Please be as specific as possible. Figure 4.3(c) in Melissinos may help. See also the lecture notes.

2.

The plasma frequency and the ionosphere.

(a)

Note that eq. 4.34 in Melissinos becomes equivalent to eq. 4.42 in the appropriate limit. What is this limit and what does it mean, physically?

(b)

The plasma frequency for the ionosphere is given on page 135 by Melissinos. Electrons in a metal behave much like those in the ionosphere, but their density is much higher. Estimate the plasma frequency of copper and explain why many metals become transparent in the ultraviolet.

3.

Physics vocabulary. The following terms (which are used by Melissinos) have specific meanings in the physical sciences, although they may seem equivalent or vague to the uninitiated. Give a short definition and an example of each.

(a)

Diffusion

(b)

Dispersion

(c)

Diffraction

(d)

Refraction

(e)

Birefringence

(f)

Scattering

4.

Lifetime of the sun. The flux of radiant energy from the sun at the earth's surface is approximately 1.4 kW/m².

(a)

From this estimate the total power produced by nuclear reactions in the sun; from your knowledge of the proton-proton cycle, estimate the rate of hydrogen burning in the sun; finally, estimate the amount of time before the sun ``burns out.''

(b)

Now assume that the sun's energy comes from chemical reactions. Estimate the lifetime of this ``chemical'' sun. [Note: exactly this calculation was carried out by Lord Kelvin at the end of the 19th century. His estimate was substantially larger than ``biblical'' estimates, but much smaller than geological estimates based on sedimentation rates.]