In part 1 of this blog, I showed that the 3-dimensional wave equation for an electromagnetic (EM) wave can be written (ignoring the magnetic component as it is much smaller than the electric component ) as
In part 2 I showed that, for EM waves in a cubic cavity with sides of length the only modes which can exist have to satisfy the equation
where is the number of modes allowed in the cavity in the x-direction, etc. This is the so-called “standing wave solution to the wave equation for a cubical cavity with sides of length L”.
In this third part and final part of the derivation of the Rayleigh-Jeans law, I will calculate the total number of allowed modes in the cavity given this standing wave solution, secondly I will calculate the number of modes per unit wavelength in the cavity, and finally I will calculate the energy density per unit wavelength and per unit frequency of the EM waves. This final part is the famous Rayleigh-Jeans law.
The total number of modes in our cubic cavity
In order to calculate the total number of allowed modes in our cubic cavity we need to sum over all possible values of . To do this we use a mathematical trick of working in “n-space”, that is to say we determine the volume of a sphere where the x-axis is given by , the y-axis by and the z-axis by . The value of . We can determine the value of by considering the volume of a sphere with radius , which is of course just
But, as we can see in the diagram below, if we sum over for an entire sphere we will be including negative values of , whereas we only have positive values of each.
To correct for this, to only consider the positive values of , we just need to divide the volume above by 8, as the part of a sphere in the positive part of the diagram is one eighth of the total volume. But, we also need to make another correction. Light can exist independently in two different polarisations at right angles to each other, so we need to double the number of solutions to our standing wave equation to account for this .We therefore can write
which gives the number of modes in the cavity as
The number of modes per unit wavelength
The expression above is the total number of modes in the cavity summed over all wavelengths. The number of modes per unit wavelength can be found by differentiating this expression with respect to , i.e. we find .
The minus sign is telling is that the number of modes decreases with increasing wavelength.
We can also derive the number of modes per unit wavelength in the cavity volume by dividing by the volume of the cavity
The energy per unit volume per unit wavelength and per unit frequency
Because the matter and radiation are in thermal equilibrium with each other, we can say that the energy of each mode of the EM radiation is where is Boltzmann’s constant and is the temperature in Kelvin of the radiation. This comes from the principle of the Equipartition of Energy.We write the energy per unit volume (also called the energy density) with the symbol so we have that the energy per unit volume per unit wavelength is given by
To write this in terms of frequency we remember that
and, from the chain rule we can write that
(the minus sign is just telling is that as .
This gives us that
So, finally we have the Rayleigh-Jeans law, that the energy density of the radiation is given by
So, using Classical Physics, we find that the energy density is proportional to the frequency squared (), which means the energy density plotted as a function of frequency should look like the purple curve below.
Of course, physicists already knew that the energy density of blackbodies followed the blackcurve, not the purple curve. If it were to follow the purple curve (the Rayleigh-Jeans law) the blackbody would get brighter and brighter at shorter and shorter wavelengths, the so-called ultraviolet catastrophe. In a future blog I will outline how Max Planck resolved this problem, and in so doing heralded in the dawn of Quantum Mechanics, an entirely new way of thinking about the sub-atomic world.