Rayleigh-Jeans law

The Rayleigh-Jeans law is a flawed attempt in physics to describe the spectral radiance of electromagnetic radiation as a function of wavelength from a blackbody at a given temperature.

Consider a cube of side , filled with electromagnetic radiation in thermal equilibrium at temperature . If a tiny aperture exists in one of the walls, the radiation that it emits would exhibit the properties of an ideal blackbody. The walls, regarded as perfect conductors, provide boundary conditions for the electromagnetic waves inside the cube.

In a perfect conductor, an idealised model for real metals, electric charges can move without any resistance. An electromagnetic wave propagating in the -direction, with its electric field vector parallel to the wall, causes electrons in the conductor to move in response to the field. The movement of these electrons creates an opposing electric field that cancels out the external electric field. Given that the electric field is zero at the walls, the boundary conditions can only be satisfied by standing waves. These are waves that oscillate in place and have nodes at the cube’s walls. For example, standing waves oscillating in the -direction have wavelengths of , where is an integer (see below diagram).

The amplitude of each of the above standing waves can be expressed as , or in terms of the wave vector pointing in the -direction:

where .

Similarly, the wavelengths of the oscillation modes of the electromagnetic field in the -direction and -direction are and  respectively. Correspondingly, the magnitudes of the wave vectors are given by and  respectively. It follows that the wave vector  of the oscillation modes of the electromagnetic field in an arbitrary direction in the cube lies in a -space, which contain a cubical array of points that are apart (see Figure I below).

Since the magnitude of the wave vector is , boundary conditions are satisfied by each ordered triple . In other words, every mode of oscillation of an arbitrary wave oscillating in the cube is defined by a point in the -space. We can also plot the array of modes in an -space, where the points are now unit length apart from one another (see Figure II above). Substituting in , we have

where is the radius of an octant (one-eighth of the volume of a sphere) in the -space.

For a certain value of , ordered triples of that satisfy eq1 lie on the surface of the octant. Consider the volume of the shell of the octant enclosed by the surfaces at  and . Substituting eq1 and  in , we have

Since the density of the number of modes in the -space is one per unit volume, the number of modes within the shell is the product of and :

As there are two independent polarisations for each mode of an electromagnetic wave (see above diagram),

Lord Rayleigh used the equipartition theorem to assume that each mode has an average energy of . Multiplying the above equation by the average energy per mode, we have

where is the energy density (average energy per mode between and per unit volume of the cube).

Eq2 is known as the Rayleigh-Jeans law, which fails to describe the spectrum of blackbody radiation at high frequencies because it implies that radiated energy would increase without limit as increases, leading to what is called an ultraviolet catastrophe (see diagram below).


Next article: Planck radiation law
Previous article: Reduction of the two-particle problem to one-particle problems
Content page of quantum mechanics
Content page of advanced chemistry
Main content page

Leave a Reply

Your email address will not be published. Required fields are marked *