Show that the wavelength of maximum spectral radiant emittance is inversely proportional to the absolute temperature and find the proportionality constant.

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Accepted Answer

To find the maximum, we set the derivative of the function of [latex]Eq. (14.4-5):η(λ)dλ=\frac{2πhc^{2}}{λ^{5}(e^{hc/λk_{B}T}−1)} dλ[/latex] equal to zero:

[latex]\frac{dη}{dλ} =2πhc^{2}\frac{(hc/λk_{B})Te^{hc/λk_{B}T}-5(e^{hc/λk_{B}T}-1)}{λ^{6}[e^{hc/λk_{B}T} − 1]} =0[/latex]

This expression can vanish only if the numerator vanishes, which is equivalent to

[latex]\frac{hc}{λ_{max}k_{B}T} =5\Bigl(1 − e^{−hc/λ_{max}k_{B}T}\Bigr)[/latex]

or, if we let [latex]x=hc/λ_{max}k_{B}T [/latex], then

[latex]x=5(1−e^{−x})[/latex]

This equation must be solved by numerical approximation. The result is that x=4.965, so that

[latex]λ_{max}=\frac{1}{4.965} \frac{hc}{k_{B}T}=\frac{2.898 × 10^{−3} mK}{T} [/latex] (14.4-6)

This result agrees with Wien’s law and with the experimental value of the proportionality constant.

Question 14.2: Show that the wavelength of maximum spectral radiant emittan...

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