Planck's law is

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Photodetection, photonic sources and quantum optics · Photodetection, photonic sources and quantum optics

Optics and Photonics, Ch The photon

Problem

Planck's law is Lλ(T)=2hc2λ51ehc/(λkBT)−1L_\lambda(T) = \dfrac{2hc^2}{\lambda^5}\dfrac{1}{e^{hc/(\lambda k_BT)}-1}. Writing x=hc/(λkBT)x=hc/(\lambda k_BT), the wavelength at which Lλ(T)L_\lambda(T) is greatest, at fixed TT, is fixed by finding the maximum of x5/(ex−1)x^5/(e^x-1) over xx. ((a)) Show that setting ddx[x5ex−1]=0\dfrac{d}{dx}\left[\dfrac{x^5}{e^x-1}\right]=0 leads to the condition (5−x)ex=5.(5-x)e^x = 5. ((b)) The root of this equation is supplied: xmax⁡=4.965x_{\max}=4.965. Use it to show that the wavelength of the peak obeys λmax⁡T=b≡hc4.965 kB,\lambda_{\max}T = b \equiv \frac{hc}{4.965\,k_B}, and evaluate bb numerically using h=6.626×10−34h=6.626\times10^{-34} J s, kB=1.381×10−23k_B=1.381\times10^{-23} J K−1^{-1}, c=3.00×108c=3.00\times10^8 m s−1^{-1}. ((c)) A tungsten lamp filament runs at a steady 30003000 K. Find the wavelength at which its spectrum peaks, and state whether this lies inside the visible band (about 400400-700700 nm).

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