Solution to Set 1

Problem 1

(a)

It is a common misconception that Max Planck derived his now-famous law, Planck's Law, in order to resolve the so-called "ultraviolet catastrophe," which predicts, from classical physics, that a blackbody will emit greater and greater intensity radiation at shorter wavelengths, thus outputting infinite power as electromagnetic radiation. In fact, this problem was not noticed until five years after Planck derived his law.

What actually motivated Planck was his desire to improve on the Wien approximation, which fit known blackbody radiation spectra only at short wavelengths. Expressed as spectral radiance:

$I(\lambda ,T)={\frac {2hc^{2}}{\lambda ^{5}}}e^{-{\frac {hc}{\lambda kT}}}$

Conversely, the Rayleigh-Jeans law fit the data only at long wavelengths:

$I(\lambda ,T)={\frac {2ckT}{\lambda ^{4}}}$

Planck derived his function to fit the data at all wavelengths:

$I(\lambda ,T)={\frac {2hc^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{\lambda kT}}-1}}$

Planck derived his law via a consideration of various ways in which electromagnetic energy can be distributed over the different modes of oscillation of charged oscillators in matter (today known to be atoms). He found that when he assumed the energy to be quantized, the above law emerged and fit the data very well.