Electromagnetic Waves in a Cubical CavityElectromagnetic standing waves in a cavity at equilibrium with its surroundings cannot take just any path. They must satisfy the wave equation in three dimensions:  Substituting this solution into the wave equation above gives  which simplifies to 
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How many modes in the cavity?From the standing wave solution to the wave equation we get the condition.
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How many modes per unit wavelength?Having developed an expression for the number of standing wave modes in a cavity, we would like to know the distribution with wavelength. This may be obtained by taking the derivative of the number of modes with respect to wavelength. The negative sign here reveals that the number of modes decreases with increasing wavelength. Now to get the number of modes per unit volume per unit wavelength, we can simply divide by the volume of the cubical cavity.  Note that this does not involve approximating a sphere with a cube! The sphere we used in calculating the number of modes was a sphere in "n-space", allowing us to count the number of possible modes. Also, the use of a cubical cavity for the calculation just allows us to reduce the geometrical complexity of the development, but the final result obtained is independent of cavity geometry.
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How much energy per unit volume?Assigning energy to the electromagnetic standing waves in a cavity draws on the principle of equipartition of energy. Each standing wave mode will have average energy kT where k is Boltzmann's constant and T the temperature in Kelvins. Letting u represent the energy density: This is an important relationship in classical electromagnetic cavity theory. It can also be expressed in terms of the frequency by making use of the chain rule and the wave relationship:  The minus sign here just reminds us that a decrease with wavelength implies an increase with increasing frequency. The magnitude of the energy density dependence on frequency is given by:  Note that this is the classical result which was used in the Rayleigh-Jeans Law, but led to the ultraviolet catastrophe. It produces good agreement in the low frequency limit, but for higher frequencies the Planck radiation formula must be used.
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Radiated Energy as a Function of WavelengthIf we consider energy radiated perpendicular to a small increment of area, then it must be noted that half of the energy density in the waves is going toward the walls and half is coming out if the system is in thermal equilibrium. Evaluating the power seen at a given observation point requires a consideration of the geometry: Having averaged over all angles, the calculated radiated power per unit wavelength is finally 
This is the Rayleigh-Jeans formula. The fact that it failed to predict the spectral distribution from hot objects was one of the major unresolved issues in physics at the beginning of the 20th century. 
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