GATE 2020 PH – Question 65
Planck’s spectral energy density is $$u(\omega)=\frac{\hbar\omega^3}{\pi^2c^3[e^{\hbar\omega/(k_BT)}-1]}.$$ At long wavelengths the spectral energy density varies as $T^\alpha$. Find α.
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Correct answer: 1 to 1
Explanation
Long wavelength means $\hbar\omega\ll k_BT$. Expand the exponential denominator as $e^{\hbar\omega/(k_BT)}-1\simeq\hbar\omega/(k_BT)$. Then $u(\omega)\simeq k_BT\omega^2/(\pi^2c^3)$, the Rayleigh–Jeans limit. At fixed long wavelength or frequency this is linear in T, so α = 1. It is a spectral statement, not the T⁴ scaling of the total integrated blackbody energy.
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