Calculus: Project 26

Hot bodies emit radiation. No, this isn't the start of a racy novel; we're referring to a blacksmith shop where iron comes out of the fire "white hot," cools to "red hot," and passes below the visible spectrum into the infrared. (Warm human bodies do emit infrared radiation, which is detected by the "motion" lights many of us have outside our houses.) Physicists wanted to be able to predict the "color" of radiation from the temperature.

Wien made measurements and found an empirical law that says: "When the absolute temperature of a radiating black body increases, the most powerful wavelength decreases in such a way that

is constant." The measured constant

(cm

K) is known as Wien's displacement constant.

Planck discovered a formula for the intensity of each frequency radiating from a black body. When we maximize intensity as a function of wavelength using Planck's formula we can derive Wien's law. This derivation was the first triumph of quantum mechanics. The classical theory made a wrong prediction and the new theory gave the right one and shows why Wien's law holds. We will show how Planck's law gives a theoretical derivation of Wien's law and how calculus, the computer and graphing interact in seeing this. We think this should show you why you should learn the rest of the chapter.

Planck began his study with a graph of the measured intensities, an empirical curve. He then fit a formula to this curve in what his 1920 Nobel Prize lecture described as "an interpolation formula which resulted from a lucky guess." The lucky guess led him to a derivation from physical laws. If you go to your local quantum mechanic's shop, just next door to the blacksmith, she will tell you that the intensity of radiation for the frequencies between and of a black body at absolute temperature T is

where

and

(erg sec) is called Planck's constant,

(cm/sec) is the speed of light, and

(erg/deg) is called Boltzmann's constant. Physicists treat this formula as a differential because you need to measure a small range of frequencies, not just the intensity at one frequency.

26.1 The Derivation of Planck's law of Radiation

We want to be sloppy about the units in this example and just emphasize the difficulty we encounter because of the scales of the real physical constants. Normally, we will not accept this practice ourselves or as part of your homework, but we don't want to get side-tracked with the physics. The Feynman Lectures on Physics derive Planck's law (in angular frequency form) with their usual mixture of brilliance, charm and mathematical giant steps (flim-flam?). We won't repeat that. If you wish to look that up, you can also straighten out the units and give careful definitions of the variables.

26.2 Wavelength Form and First Plots

We want to express Planck's law in terms of wavelength using the basic relation

and its differential, so

giving

where

and b=hc/k.

Let's take a hot object, say K (about F). The first lesson the computer learns from hand computations is what range to plot. Normally, in a textbook problem to plot a function like , you might pick a range like . This is nonsense in our problem, because the interesting range of wavelengths is orders of magnitude smaller. Let's try it with Mathematicain Figure 26.1 anyway:

Figure 26.1: A useless plot

The wavelength of red cadmium light in air is used for calibrations and equals

so let's try another range in Figure 26.2.

Figure 26.2: Also useless

This plot tells us something, at least. The interesting stuff must be between these ranges, so we try a range of {l,0,.002} in Figure 26.3.

Figure 26.3: Intensity for

Now we see what we want - the maximum. It seems to be around 0.0003 as well as we can read the graph. We didn't really use calculus, but a little physics to look for a reasonable range. We were lucky to find the peak. What we wanted was the place where the curve peaks, so we could have used calculus to find it. And we can do that now. Let , so and use the Product Rule and Chain Rule:

It is horribly messy to do this computation with the explicit values of the constants. We want you to learn to use parameters, even in cases like this where they only make the calculations neater and thus easier to follow. We will look at the computation with parameters below for a more important reason. Actually, we asked you to differentiate Planck's formula in the last chapter and showed you the answer in the computer program Dfdx. All the computations in this section are on the program PlanckL and we encourage you to compare that with your Planck's law exercise below.

The graph is horizontal when so the numerator must equal zero

If we try to solve using the computer, we encounter nasty numerical problems because of the scale. A change of variables makes the problem appear simpler and scales it in a form that the computer can handle. Let and rewrite the equation above to obtain the problem:

We need a starting guess at a root. Figure 26.4 shows a root near x=5.

Figure 26.4: y=5-(5-x)Exp[x]

The computer finds our root more accurately to be , so . The observed wavelength at 1000 K is 0.000289782 cm. Very nice start, but this is not the answer we are looking for. We have only shown what happens at T=1000 and we want to know what happens as we vary T.

We could try another value, say T=900, and work through the whole problem again, but that would just give us another specific number. We need a formula that tells us how the maximum point depends on the temperature T. We can find this by solving the problem as before, but keeping the letter T as an unknown constant or parameter rather than letting T=1000.

26.3 Maximum Intensity in Terms of a Parameter

The peak on each curve where T is constant (but not fixed at some specific number) still occurs the point where the slope is zero or