For the thermal radiation from an ideal blackbody radiator with a surface temper

Question

For the thermal radiation from an ideal blackbody radiator with a surface temperature of 2200 K, let Ic represent the intensity per unit wavelength according to the classical expression for the spectral radiancy and Ip represent the corresponding intensity per unit wavelength according to the Planck expression. What is the ratio Ic/Ip for a wavelength of (a) 410 nm (at the blue end of the visible spectrum) and (b) 210 m (in the far infrared)? (c) Does the classical expression agree with the Planck expression in the shorter wavelength range or the longer wavelength range?

Explanation / Answer

Rayleigh-Jeans Law: Ic = 2ckT/
Planck's Law: Ip = 2hc²/[(e^(hc/kT) 1)]

h = Planck's constant
c = Speed of light
k = Bolzmann's constant

For the shorter wavelength in part a) the ratio Ic/Ip is about 5.7 million to 1, since the classical
approach predicts a much higher intensity at shorter wavelengths

In part b), for the longer wavelength, this ratio is essentially 1

The classical prediction thus agrees with Planck's law at long wavelengths only. At shorter wavelengths, the classical theory predicts very high energies, leading to an "ultraviolet catastrophy"

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