Answer to Question #310347 in Quantum Mechanics for Duke Nyankieya ombasa

Question

Show that low-frequency limit of Planck’s Law reduces to the Rayleigh-Jeans Law

and in the high-frequency limit reduces to Wien’s Law

Answer 1

The Planck’s Law says

$$B(\nu,T)=\frac{2h\nu^3}{c^2}\frac{1}{e^{h\nu/kT}-1}$$

At low-frequency

$$e^{h\nu/kT}-1=1+h\nu/kT-1=h\nu/kT$$

Hence

$$B(\nu,T)=\frac{2h\nu^3}{c^2}\frac{1}{h\nu/kT}=\frac{2\nu^2}{c^2}kT$$

At high-frequency

$$e^{h\nu/kT}-1=e^{h\nu/kT}$$

Hence

$$B(\nu,T)=\frac{2h\nu^3}{c^2}\frac{1}{e^{h\nu/kT}}=\frac{2h\nu^3}{c^2}e^{-h\nu/kT}$$