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

[Zacarias Nason]

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

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It displays all of the formulas in the form of energy per unit volume.

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

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:

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

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?

 

[TSny]

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.

 

[Zacarias Nason]

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?

 

[TSny]

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.

 

[TSny]

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.

 

[Zacarias Nason]

Awesome, thank you!