Planck's law is with

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)=2hc2λ51ex−1L_\lambda(T) = \dfrac{2hc^2}{\lambda^5}\dfrac{1}{e^{x}-1} with x=hc/(λkBT)x = hc/(\lambda k_BT), and the classical Rayleigh-Jeans form is Lλ=2ckBT/λ4L_\lambda = 2ck_BT/\lambda^4. ((a)) Show that the ratio of Planck's law to the Rayleigh-Jeans form is x/(ex−1)x/(e^x - 1), and, using ex≈1+xe^x \approx 1 + x for small xx, that Planck's law becomes the Rayleigh-Jeans form at long wavelength. ((b)) Keep one more term, ex≈1+x+x2/2e^x \approx 1 + x + x^2/2, and show that the ratio is about 1−x/21 - x/2 when xx is small. ((c)) A cavity is held at room temperature, T=300T = 300 K. Find the wavelength beyond which the classical form overestimates Planck's law by less than 1010 per cent, and say where in the spectrum it lies.

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