Planck's law states that the energy density of blackbody radiation of wavelength $x$ is given by $$f(x)=\frac{8 \pi h c x^{-5}}{e^{h c /(k T x)}-1}$$ Use the linear approximation in exercise 44 to show that $f(x) \approx 8 \pi k T / x^{4},$ which is known as the Rayleigh-Jeans law.
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The linear approximation mentioned in exercise 44 is the approximation $e^u \approx 1 + u$ for small values of $u$. Show more…
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Planck's radiation law, expressed in terms of energy per unit range of wavelength instead of frequency, becomes $$ E_{\lambda}=\frac{8 \pi c h}{\lambda^{5}\left(e^{c h / \lambda k T}-1\right)} $$ Use the variable $x=c h / \lambda k T$ to show that the total energy per unit volume at temperature $T^{\circ}$ absolute is given by $$ \int_{0}^{\infty} E_{\lambda} \mathrm{d} \lambda=a T^{4} \mathrm{~J} \mathrm{~m}^{-3} $$ where $$ a=\frac{8 \pi^{5} k^{4}}{15 c^{3} h^{3}} $$ (The constant $c a / 4=\sigma$, Stefan's Constant in the Stefan-Boltzmann Law.) Note that $$ \int_{0}^{\infty} \frac{x^{3} \mathrm{~d} x}{e^{x}-1}=\frac{\pi^{4}}{15} $$
The energy density of black body radiation (ρ) at temperature T is given by Planck's formula ρ(λ) = (8πhc/λ^5)(e^{hc/λkT} - 1)^{-1} where λ is the wavelength, h is Planck's constant, and c is the speed of light., show that the formula reduces to the classical Raleigh-Jeans law ρ(λ) = 8πkT/λ^4 for large wavelengths λ (λ → ∞)
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