Question

Derive Wien's displacement law from Planck's law. Proceed as follows. In Active Figure $40.3,$ notice that the wavelength at which a black body radiates with greatest intensity is the wavelength for which the graph of $I(\lambda, T)$ versus $\lambda$ has a horizontal tangent. From Equation 40.6 , evaluate the derivative $d I / d \lambda .$ Set it equal to zero. Solve the resulting transcendental equation numerically to prove $h c / \lambda_{\max } k_{\mathrm{B}} T=4.965 \ldots$ or $\lambda_{\max } T=h c / 4.965 \mathrm{k}_{\mathrm{B}}$ . Evaluate the constant as precisely as possible and compare it with Wien's experimental value.

          Derive Wien's displacement law from Planck's law. Proceed as follows. In Active Figure $40.3,$ notice that the wavelength at which a black body radiates with greatest intensity is the wavelength for which the graph of $I(\lambda, T)$ versus $\lambda$ has a horizontal tangent. From Equation 40.6 , evaluate the derivative $d I / d \lambda .$ Set it equal to zero. Solve the resulting transcendental equation numerically to prove $h c / \lambda_{\max } k_{\mathrm{B}} T=4.965 \ldots$ or $\lambda_{\max } T=h c / 4.965 \mathrm{k}_{\mathrm{B}}$ . Evaluate the constant as precisely as possible and compare it with Wien's experimental value.

Added by Victor P.

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