Set 11
Assignments
Set 9
Phys 312 - Assignment 10 - Due 15 April 99
- 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
, with m the electron mass,
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
(we can manage well without complex notation in this problem). Assuming
that
, show that the electron moves in phase with
the electric field with an acceleration
| ![\begin{displaymath}
a=\frac{eE_{0}\omega ^{2}}{m\omega _{0}^{2}}\cos \omega t.
\end{displaymath}](img5.gif) |
(1) |
- (b)
- Use the Larmor radiation formula to show that the time-averaged power
radiated by the charge is
| ![\begin{displaymath}
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}.
\end{displaymath}](img6.gif) |
(2) |
Express this in terms of the wavelength
of the incident wave to
show that
, 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
, 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.
- Light sources. Briefly describe the characteristics of
- Light emitted from a sodium lamp (looks yellow)
- Natural light from the sun
- Synchrotron radiation
- Laser light
In each case, is the light broad-spectrum or narrow-spectrum, polarized (at
least partly) or unpolarized, coherent (at least partly) or incoherent?
Vittorio Celli
4/6/1998