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Blackbody radiation problem

  1. Jun 11, 2013 #1
    1. The problem statement, all variables and given/known data

    a) Show that for photons of frequency [itex]\nu[/itex] and wavelength [itex]\lambda[/itex] :

    1) [itex]d\nu = - c d\lambda / \lambda^{2}[/itex]
    2) [itex]u(\lambda)d\lambda = - u(\nu)d\nu[/itex]
    3) [itex]u(\lambda)d\lambda = u(\nu) c d\lambda / \lambda^{2}[/itex]

    b) Show that the Rayleigh-Jeans spectral distribution of blackbody radiation, [itex]u_{RJ}(\nu)[/itex], is of the form required by Wien's law, [itex]u_{W}(\nu) = \frac{W(\lambda T)}{\lambda ^{5}}[/itex]

    c) Obtain the correct form of Wien's undetermined function [itex]W(\lambda T)[/itex] from Planck's formula.

    2. Relevant equations

    3. The attempt at a solution

    Solution to a):

    1) [itex]d\nu = \frac{d\nu}{d\lambda}d\lambda = - \frac{c}{\lambda^{2}}d\lambda[/itex]
    2) can't do
    3) substitute [itex]d\nu[/itex] in 1) to [itex]d\nu[/itex] on the RHS of 2)

    Solution to b):

    [itex]u_{RJ}(\nu)d\nu = \frac{8\pi\nu^{2}}{c^{3}}kTd\nu[/itex]
    [itex]- u_{RJ}(\lambda)d\lambda = (\frac{8\pi\frac{c^{2}}{\lambda^{2}}}{c^{3}}kT)(-\frac{c}{\lambda^{2}}d\lambda)[/itex]
    [itex]u_{RJ}(\lambda)d\lambda = \frac{8\pi k(\lambda T)}{\lambda^{5}}d\lambda[/itex]
    [itex]So, W(\lambda T) = 8\pi k(\lambda T)[/itex]

    Solution to c):

    [itex]u(\nu)d\nu = \frac{8\pi h \nu^{3}}{c^{3}} \frac{d\nu}{e^\frac{h\nu}{kT} - 1}[/itex]
    [itex]- u(\lambda)d\lambda = \frac{8 \pi h \frac{c^{3}}{\lambda^{3}}}{c^{3}} \frac{- \frac{c}{\lambda^{2}}d\lambda}{e^{\frac{hc}{\lambda kT}} - 1}[/itex]
    [itex]u(\lambda)d\lambda = \frac{8 \pi hc}{\lambda^{5}} \frac{d\lambda}{e^{\frac{hc}{\lambda kT}} - 1}[/itex]
    So, [itex]W(\lambda T) = \frac{8\pi hc}{e^{\frac{hc}{k(\lambda T)}} - 1}[/itex]

    Need help with a)2). Also, can you check the rest, please?
  2. jcsd
  3. Jun 12, 2013 #2
    You know what you get when you integrate u(λ) dλ and u(v) dv. You can for example integrate first integral from 0 to some λ, and second one from v to ∞. Then these integrals are equal, right?
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