- #1

spaghetti3451

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## Homework Statement

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.

## Homework Equations

## 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?