How does the spectral distribution of blackbody radiation relate to Wien's law?

In summary, the conversation discusses various equations and solutions related to photons and their properties, such as frequency and wavelength. It also explores the relationship between the Rayleigh-Jeans spectral distribution of blackbody radiation and Wien's law, and how the undetermined function W(λT) can be obtained from Planck's formula. Some help is needed with solving one of the equations and checking the rest of the solutions.
  • #1
spaghetti3451
1,344
33

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?
 
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  • #2
failexam said:
Need help with a)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?
 

Related to How does the spectral distribution of blackbody radiation relate to Wien's law?

1. What is blackbody radiation?

Blackbody radiation is the electromagnetic radiation emitted by a hypothetical object that absorbs all radiation incident upon it, regardless of the frequency or angle of incidence. It is a fundamental concept in thermodynamics and quantum mechanics.

2. Why is the blackbody radiation problem important?

The blackbody radiation problem is important because it played a crucial role in the development of quantum mechanics and our understanding of the behavior of matter and energy at a microscopic level. It also has practical applications in fields such as astrophysics, where it helps us understand the thermal radiation emitted by celestial bodies.

3. What is the relationship between temperature and blackbody radiation?

According to Planck's law, the intensity and wavelength distribution of blackbody radiation depend on the temperature of the blackbody. As the temperature increases, the peak of the wavelength distribution shifts to shorter wavelengths, and the total intensity of the radiation increases.

4. How do we solve the blackbody radiation problem?

The blackbody radiation problem can be solved using various theoretical models, such as the Planck's law, Wien's displacement law, and the Stefan-Boltzmann law. These models use mathematical equations to describe the relationship between temperature, wavelength, and intensity of blackbody radiation.

5. What are some real-life examples of blackbody radiation?

Some examples of blackbody radiation in everyday life include the thermal radiation emitted by a hot stove, the light emitted by an incandescent light bulb, and the heat radiated by the Sun. In astrophysics, blackbody radiation can be observed from various celestial bodies, such as stars and planets.

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