Planck's Law and Wien's Law

Planck's Radiation Law Problem

• Planck's radiation law is: $B_{\lambda}(T) = \frac{2hc^2}{\lambda^5}\frac{1}{e^{\frac{hc}{\lambda kT}}-1}$
• Wien's radiation law is: $B_{\lambda}(T) = \frac{2hc^2}{\lambda^5}e^{-\frac{hc}{\lambda kT}}$
• The problem is to show that Planck's law reduces to Wien's law in the limit of short wavelength.

Solution Steps

• Start with Planck's radiation law: $B_{\lambda}(T) = \frac{2hc^2}{\lambda^5}\frac{1}{e^{\frac{hc}{\lambda kT}}-1}$
• In the limit of short wavelength ($\lambda \rightarrow 0$), $\frac{hc}{\lambda kT} >> 1$.
• Therefore, $e^{\frac{hc}{\lambda kT}} >> 1$, allowing us to neglect 1 in the denominator of Planck's radiation law.

Approximation

• Approximating Planck's law: $B_{\lambda}(T) \approx \frac{2hc^2}{\lambda^5}\frac{1}{e^{\frac{hc}{\lambda kT}}}$

Result

• This simplifies to Wien's radiation law: $B_{\lambda}(T) = \frac{2hc^2}{\lambda^5}e^{-\frac{hc}{\lambda kT}}$