Why Infrared

Dear Mr. Malone

When we met some time back, you asked me why the human body emits infrared radiation. I gave you an answer that was my intelligent guess, and a very unsatisfactory explanation (at least, I thought so). The intelligent guess turned out to be the correct answer. After having spent time leisurely reading in the summer (of which half that time is spent on physics), I have understood more concepts related to this question, and I feel I can attempt a more satisfying explanation. Some parts of the explanation require me to qualitatively explain the mathematics, but this is far from a ‘dumbed-down’ explanation, because in as much as mathematics can be translated to words, I believe that the translation is mostly correct. I am not sure whether it’s possible to try to explain this to your class, but I thought you may be interested intellectually. So here goes.

(In the following, I may use the terms ‘light’ and ‘electromagnetic radiation’ interchangeably.)

First, I will introduce the concept of a blackbody. A blackbody absorbs all light that falls upon it. Once the light is absorbed, it will bounce around inside the blackbody. Light will be continually absorbed and re-emitted by the atoms inside the blackbody. Because the light cannot escape and is eventually absorbed/re-emitted by ALL the atoms, there is a continual exchange of energy between all the atoms. Any net flow of energy stops, on average, when the entire blackbody is in thermal equilibrium, i.e. we can define the blackbody to have an exact temperature.

A perfect blackbody is sure to achieve thermal equilibrium, if we give it some time. But we cannot expect that the human body is a perfect blackbody, because we only absorb some of the light that falls on us. Hence, the human body is not exactly in thermal equilibrium, but we shall make the approximation that in fact, it is! 310K, which is about the normal temperature you see when you use a thermometer on yourself. Of course, we know that the temperature of our liver is probably slightly higher than that of our earlobes, but I make no apologies about the blatant approximation, because we usually need them to make any prediction at all. Most dense, opaque objects can be approximated as blackbodies, because the light interacts strongly with the atoms in this objects, and the exchange of energy through light also results in an approximate thermal equilibrium.

Second, I must introduce the concept of the simple harmonic oscillator (SHO), as applied to electromagnetic fields. The classic example of a SHO is a mass on a spring, or a pendulum. As you probably already know, the energy of the system, which depends on two variables (position and momentum), will oscillate between kinetic and potential energy.

Any electromagnetic field in a cavity/object/blackbody can be described in the following manner: at an infinite set of points (or locations) inside the cavity, we measure the electric (E) and magnetic (B) fields in the x,y,z directions for each point. In a sense, this is the most obvious way to describe the e-m field. However, the mathematics offers us another equivalent way of looking at the problem. The alternative description is that the field is really a sum of an infinite number of standing waves which vibrate at different frequencies. The following may sound familiar to you: a string clamped at both ends, any arbitrary deformation of the string can be alternatively described as a sum of many standing waves of different wavelengths, and vibrating at different frequencies. The technique I have described for the string is called Fourier decomposition, and the mathematics is simpler but not unlike that for the electromagnetic field.

Depending on the nature of the e-m field, standing waves of certain frequencies may have a stronger “presence” or amplitude of oscillation, as compared to other frequencies. We quantify the strength of each standing wave by a number N. Since we have an infinite number of standing waves, we really have an infinite number of N’s. Now it’s pretty amazing, but each N obeys an equation that is identical to that of a simple harmonic oscillator. The same equation that describes how the position of the mass/pendulum evolves with time, also describes N. This means that the e-m field is mathematically equivalent to a set of independent harmonic oscillators. In the case of the mass/pendulum, the total energy oscillates between kinetic and potential energy (at a certain time, the kinetic may be higher than the potential, but at all times the total energy is constant); in the case of the e-m field, the total energy sloshes between energy stored in the electric and the magnetic fields.

We need quantum mechanics to quantize these harmonic oscillators. In effect, we are allowing the energy of each harmonic oscillator to assume only certain values that occur at discretely-spaced levels. No longer do we have a continuous spread of energies.

Now suppose this harmonic oscillator is in thermal equilibrium at temperature 310K. The temperature tells us, on average, what the energy of the oscillator would be. We expect that at higher temperatures, the oscillator has more energy.

After we sum the average energies of all the harmonic oscillators in the human body, we find that the total energy is spread out among all the possible frequencies. Again, at certain frequencies, the electromagnetic energy is “stronger” than at other frequencies. When we actually plot it out (for 310K), the energy spread peaks at the infrared. And that is why the human body emits light in the infrared. It is more accurate to say that it emits light at all frequencies, but mostly in the infrared. Incidentally, using the same analysis on the sun (which has a surface temperature of 5800K), the energy spread peaks nicely in the region of visible light, hence we earthlings see only in the visible spectrum. The most perfect blackbody known to man is the entire universe (during a time, billions of years ago, when it was still opaque), and the Cosmic Microwave Background presents the most exact fit to this blackbody model.

I hope this explanation was more satisfying. It is certainly very mathematical, in the sense that you could not intuit the arguments here by simple physical principles, but that can’t be helped (for now). The math does not lie, and we do our best in interpreting it. Tell me if you want a greater depth of understanding for any of the ideas I wrote about. I wish you well.