Question

Explain how the equipartition theorem leads to the ultraviolet catastrophe

Answer 1

ultraviolet catastrophe states that an ideal body at an equilbrium radiation emits radiation at all frequencies. as the value of frequency increases, the emission of energy also increases. when the total amount of radiated energy is calculated, it can be concluded that black body releases infinite amount of energy contradicting conservation principle.

ultraviolet catastrophe is basically derived from equipartition theroem which states that all harmonic oscillator modes at equillibrium state has an average energy of 0.5kT. the number of modes with respect to wavelength is expressed as,

the negative sign shows that number of modess with an increase in the wavelength. to get the value of number of modes per unit volume, the above equation will be changed as follws:

similarly if we do according to the quipartition theroem, we get.,

here U reperesents the energy density.

hence can be concluded that equipartition theorem leads to the ultraviolet catastrophe