Physics

Blackbody Radiation

The wave theory of light, which Maxwell’s equations captured so well, became the dominant light theory in the 1800’s (surpassing Newton’s corpuscular theory, which had failed in a number of situations). The first major challenge to the theory came in explaining thermal radiation, which is the type of electromagnetic radiation emitted by objects because of their temperature.

Testing Thermal Radiation

An apparatus can be set up to detect the radiation from an object maintained at temperature T₁. (Since a warm body gives off radiation in all directions, some sort of shielding must be put in place so the radiation being examined is in a narrow beam.) Placing a dispersive medium (i.e. a prism) between the body and the detector, the wavelengths (lambda) of the radiation disperse at an angle (theta). The detector, since it’s not a geometric point, measures a range delta-theta which corresponds to a range delta-lambda, though in an ideal set-up this range is relatively small.

If I represents the total intensity of the electromagnetic radiation at all wavelengths, then that intensity over an interval delta-lambda (between the limits of lambda and delta-lamba) is:

delta-I = R(lambda) delta-lambda

R(lambda) is the radiancy, or intensity per unit wavelength interval. In calculus notation, the delta-values reduce to their limit of zero and the equation becomes:

dI = R(lambda) dlambda

The experiment outlined above detects dI, and therefore R(lambda) can be determined for any desired wavelength.

Radiancy, Temperature, and Wavelength

Performing the experiment for a number of different temperatures, we obtain a range of radiancy vs. wavelength curves, which yield significant results:

The total intensity radiated over all wavelengths (i.e. the area under the R(lambda) curve) increases as the temperature increases.
This is certainly intuitive and, in fact, we find that if we take the integral of the intensity equation above, we obtain a value that is proportional to the fourth power of the temperature. Specifically, the proportionality comes from Stefan’s law and is determined by the Stefan-Boltzmann constant (sigma) in the form:
I = sigma T⁴
The value of the wavelength lambda_max at which the radiancy reaches its maximum decreases as the temperature increases.
The experiments show that the maximum wavelength is inversely proportional to the temperature. In fact, we have found that if you multiply lambda_max and the temperature, you obtain a constant, in what is known as Wein’s displacement law:
lambda_max T = 2.898 x 10^-3 mK

Blackbody Radiation

The above description involved a bit of cheating. Light is reflected off objects, so the experiment described runs into the problem of what is actually being tested. To simplify the situation, scientists looked at a blackbody, which is to say an object that does not reflect any light.

Consider a metal box with a small hole in it. If light hits the hole, it will enter the box, and there’s little chance of it bouncing back out. Therefore, in this case, the hole, not the box itself, is the blackbody. The radiation detected outside the hole will be a sample of the radiation inside the box, so some analysis is required to understand what’s happening inside the box.

The box is filled with electromagnetic standing waves. If the walls are metal, the radiation bounces around inside the box with the electric field stopping at each wall, creating a node at each wall.
The number of standing waves with wavelengths between lambda and dlambda is
N(lambda) dlambda = (8pi V / lambda⁴) dlambda
where V is the volume of the box. This can be proven by regular analysis of standing waves and expanding it to three dimensions.
Each individual wave contributes an energy kT to the radiation in the box. From classical thermodynamics, we know that the radiation in the box is in thermal equilibrium with the walls at temperature T. Radiation is absorbed and quickly reemitted by the walls, which creates oscillations in the frequency of the radiation. The mean thermal kinetic energy of an oscillating atom is 0.5kT. Since these are simple harmonic oscillators, the mean kinetic energy is equal to the mean potential energy, so the total energy is kT.
The radiance is related to the energy density (energy per unit volume) u(lambda) in the relationship
R(lambda) = (c / 4) u(lambda)
This is obtained by determining the amount of radiation passing through an element of surface area within the cavity.