Derivation of Wien's+Reyleigh-Jean's Laws from Planck's Law

In summary: So, I can't even really use the formulas in the form of energy per unit volume to check for equality, since Wien's and Rayleigh-Jeans are in that form as well, right?That is correct. The energy per unit volume formulas are just different representations of the same underlying concept, but one is more accurate in certain situations than the others.
  • #1
Zacarias Nason
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Homework Statement



After reading the forum stickies I'm not entirely sure where to put this question since it involves using math to solve a question, but is informally stated and isn't a book problem, either-I just started reading Fong's Elementary Quantum Mechanics, and in the first few pages the relationship between the Plank, Wien and Rayleigh-Jean formulas are discussed. I'm assuming that since Rayleigh-Jean and Wien are special cases of Planck's Law that they can be derived from Planck's Law, but I'm having trouble getting from one to the other/proving they are equivalent under some circumstances, assuming only algebra is really necessary for this.

Homework Equations


[/B]
It displays all of the formulas in the form of energy per unit volume.

[tex] u(v)dv = \frac{8 \pi v^2 k T}{c^3} dv \ \ \ \text{(Rayleigh-Jean's)} [/tex]
[tex] u(v)dv \ \text{~} \ v^3e^{-hv/kT} dv \ \ \ \text{(Wien's)}[/tex]
[tex] u(v)dv = \frac{8 \pi h v^3}{c^3} \frac{1}{e^{hv/kT}-1} \ \ \ \text{(Planck's)} [/tex]

The Attempt at a Solution



I tried to equate the Wien and Planck formula and ran into some trouble, but may very well be doing this wrong from the outset. Also, I'm not really sure how or if the tilde/asymptote approx. symbol may affect this. Here's my work:

[tex] v^3 e^{-hv/kT} dv = \frac{8 \pi h v^3}{c^3} \frac{1}{e^{hv/kT}-1} dv[/tex]
[tex] e^{-hv/kT} dv = \frac{8 \pi h}{c^3} \frac{1}{e^{hv/kT}-1} dv[/tex]
[tex] e^{-hv/kT} = \frac{8 \pi h}{c^3} \frac{1}{e^{hv/kT}-1} [/tex]
[tex] e^{-hv/kT} = \bigg(\frac{8 \pi h}{c^3}\bigg) (e^{-hv/kT})\bigg( \frac{1}{1-1/e^{hv/kT}}\bigg) [/tex]
[tex] 1 = \bigg(\frac{8 \pi h}{c^3}\bigg) \bigg( \frac{1}{1-1/e^{hv/kT}}\bigg) [/tex]
[tex] 1-1/e^{hv/kT} = \frac{8 \pi h}{c^3} [/tex]
[tex] e^{hv/kT} = 1 - \frac{c^3}{8 \pi h} [/tex]

This...obviously can't be right. Apart from the crazy huge number, I can't take the natural log of a negative number. What am I doing wrong? Does the necessary approach require integral calc or something?
 
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  • #2
Welcome to PF!

In going from your next to last equation to your last equation, you did not solve for ##e^{hv/kT} ## correctly. But you don't need to go through all of this.

Wien's law is an approximation to Planck's law for low temperatures (or high frequencies); i.e., for ##kT << h \nu##.
The key to showing this is to approximate ##\frac{1}{e^{hv/kT}-1}## in Planck's law for low temperatures (or high frequencies).

The tilde in your statement of Wien's law is just indicating that the expression on the right side does not include an overall constant factor. It just shows the functional dependence on various parameters.
 
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  • #3
So you can't really set them equal whatsoever, you just sort of have to set some parameters for where it is roughly equal?

And, for the Rayleigh-Jeans Law compared to Planck's, despite the equals sign-is there any range of values for which both formulas are exactly equal, or no-can they be equated?
 
  • #4
No two of the three laws are exactly equal to each other for any value of ##\nu## in the range ##0 < \nu < \infty## for any specified temperature.
 
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  • #5
Zacarias Nason said:
So you can't really set them equal whatsoever, you just sort of have to set some parameters for where it is roughly equal?

Yes. In your analysis, it will be important for you to see under what conditions the Wien and Rayleigh laws are "good" approximations to the Planck law.
 
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  • #6
Awesome, thank you!
 

Related to Derivation of Wien's+Reyleigh-Jean's Laws from Planck's Law

1. What is the significance of Wien's and Rayleigh-Jeans' laws?

Wien's and Rayleigh-Jeans' laws are important in understanding the behavior of blackbody radiation, which is the electromagnetic radiation emitted by a perfect absorber and emitter of energy. These laws help us understand the distribution of energy in this type of radiation and have implications in fields such as astrophysics and thermal physics.

2. How are Wien's and Rayleigh-Jeans' laws related to Planck's law?

Wien's and Rayleigh-Jeans' laws are derived from Planck's law, which is a more general expression for the distribution of energy in blackbody radiation. These two laws are limiting cases of Planck's law, which can be derived from the principles of quantum mechanics.

3. Can Wien's and Rayleigh-Jeans' laws be applied to all types of radiation?

No, these laws are only applicable to blackbody radiation, which is a theoretical concept that does not exist in nature. They are not valid for other types of radiation, such as thermal radiation from a real object.

4. What are the main differences between Wien's and Rayleigh-Jeans' laws?

The main difference between these two laws lies in the regions of the electromagnetic spectrum they describe. Wien's law is valid for shorter wavelengths, while Rayleigh-Jeans' law applies to longer wavelengths. Additionally, Wien's law accounts for the peak wavelength of the radiation, while Rayleigh-Jeans' law gives an overall distribution of energy.

5. How accurate are Wien's and Rayleigh-Jeans' laws compared to Planck's law?

Wien's and Rayleigh-Jeans' laws are good approximations of Planck's law in their respective regions of the electromagnetic spectrum. However, as the wavelength approaches zero or infinity, these laws diverge from the more accurate Planck's law. This is due to the limitations of classical physics, which cannot fully account for the behavior of electromagnetic radiation at the quantum level.

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