Physics 214b-2008 Walter F. Smith Assignment 3

Due: 4 pm Wednesday 2-13-08

Reading: Finish Ch. 2

Exercises

All problems are group problems, unless otherwise marked; you are encouraged to work in small groups on these.. For the individual problems, you are not allowed to consult any other students, but you may ask me for help.

2A. The Planck distribution, part two.

a. On your last assignment, you showed that an electromagnetic standing wave with a frequencyn in a cavity at temperature T has an excitation level corresponding to an average number of photons given by

.

Show that this simplifies to . This is called the Planck Distribution. Hint: if you use Mathematica, you should be able to do this fairly quickly.

b. As we discussed in class, one can show that, for a three-dimensional cavity, the frequency spacing between electromagnetic normal modes gets smaller as you go higher in frequency. We quantify this using the “density of states” , where is the number of normal mode frequencies in the range n to . (For example, for a one-dimensional string stretched between two walls, the normal mode frequencies are given by , where n₁ is the frequency of the fundamental mode. Thus, the frequency spacing between normal modes isn₁, so that the number of states in a frequency interval is .)

For a cubical cavity, one can show that

, (1)

where V is the volume of the cavity.

The “energy density” of electromagnetic standing modes in a cavity at temperature T is , where is the energy per volume due to modes in the frequency range n to . Use (1) and your results from part a to show that

c. Usually, experimental results are quoted in terms of wavelength rather than frequency. To convert, note that , where the minus sign comes from the fact that an increase in frequency corresponds to a decrease in wavelength. Use this to show that

.

2A d. Use Mathematica to make two plots of your function from part c, each of which is to be compared with the corresponding plot shown near the bottom of this page.:

i. A single plot at 1595 K.

ii. A graph with curves for 6000 K, 5000 K, 4000 K, and 3000 K.

It will be smart for you to define a function of two variables (wavelength and temperature). Here’s an example:

Notes:

1) I was taught to use underscores after the variable names when defining the functions (as shown). This tells Mathematica that they are dummy variables. However, it doesn’t seem to make any difference if you omit the underscores.

2) The “PlotRange” option gives you control over the vertical scale. (I found that Mathematica did not properly choose the scale for one of my plots.)

(Figure from Eisberg & Resnick)

Note, for your multiple temperature plot, how the peak wavelength shifts down, and the total amount of radiation increases, as you increase the temperature.

Townsend 1.18 (Individual problem), 1.22 (Individual problem), 1.28, 1.36, 2.9, 2.14 (individual problem)