Show that Wien’s displacement law follows from Planck’s radiation law. Strategy Wien’s law, Equation (3.14), refers to the wavelength for which l(λ, T) is a maximum for a given temperature. From calculus we know we can find the maximum value of a function for a certain parameter by taking the

Question

Show that Wien’s displacement law follows from Planck’s radiation law. Strategy Wien’s law, Equation (3.14), refers to the wavelength for which $\ell(\lambda, T)$ is a maximum for a given temperature. From calculus we know we can find the maximum value of a function for a certain parameter by taking the derivative of the function with respect to the parameter, set the derivative to zero, and solve for the parameter.

$\lambda_{\max } T=2.898 \times 10^{-3} m \cdot K$ (3.14)

[latex]\lambda_{\max } T=2.898 imes 10^{-3} m \cdot K[/latex] (3.14)

Accepted Answer

Therefore, to find the value of the Planck radiation law for a given wavelength we set [latex]d \ell / d \lambda=0[/latex] and solve for λ.

[latex]\frac{d\ell(\lambda, T)}{d \lambda}=0 \quad ext { for } \lambda=\lambda_{\max }[/latex]

[latex]\left.2 \pi c^{2} h \frac{d}{d \lambda}\left[\lambda^{-5}\left(e^{h c / \lambda k T}-1 ight)^{-1} ight] ight|_{\lambda_{\max }}=0[/latex]

[latex]-5 \lambda_{\max }^{-6}\left(e^{h c / \lambda_{\max } k T}-1 ight)^{-1}-\lambda_{\max }^{-5}\left(e^{h c / \lambda_{\max } k T}-1 ight)^{-2}[/latex]

[latex] imes\left(\frac{-h c}{k T \lambda_{\max }^{2}} ight) e^{h c / \lambda_{\max } k T}=0[/latex]

Multiplying by [latex]\lambda_{\max }^{6}\left(e^{h c / \lambda_{\max } k T}-1 ight)[/latex] results in

[latex]-5+\frac{h c}{\lambda_{\max } k T}\left(\frac{e^{h c / \lambda_{\max } k T}}{e^{h c / \lambda_{\max } k T}-1} ight)=0[/latex]

Let

[latex]x=\frac{h c}{\lambda_{\max } k T}[/latex]

Then

[latex]-5+\frac{x e^{x}}{e^{x}-1}=0[/latex]

and

[latex]x e^{x}=5\left(e^{x}-1 ight)[/latex]

This is a transcendental equation and can be solved numerically (try it!) with the result [latex]x \approx 4.966[/latex], and therefore

[latex]\frac{h c}{\lambda_{\max } k T}=4.966[/latex]

[latex]\lambda_{\max } T=\frac{h c}{4.966 k}=\frac{1240 eV \cdot nm }{4.966\left(8.617 imes 10^{-5} \frac{ eV }{ K } ight)} \frac{10^{-9} m }{ nm }[/latex]

and finally,

[latex]\lambda_{\max } T=2.898 imes 10^{-3} m \cdot K[/latex]

which is the empirically determined Wien’s displacement law.

Question 3.6: Show that Wien’s displacement law follows from Planck’s radi...

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