Quantum Mechanics - Comparing Rayleigh-Jeans formula with Planck's

In summary, the question is asking for the frequency range where the Rayleigh-Jeans formula gives a result within 10% of the Planck blackbody spectrum. The two formulas provided are the Planck Formula and the Rayleigh-Jean Formula. The approach to solving this question involves finding an equation expressing the difference between the two distributions and simplifying it in terms of a variable x=h\nu/kT. The goal is to find a good approximate solution for x without using Mathematica.
  • #1
Cimster
1
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Homework Statement



Over what range in frequencies does the Rayleigh-Jeans formula give a result within 10% of the Planck blackbody spectrum?


Homework Equations



Planck Formula:

[itex]p(v)dv = \frac{8πh}{c^3}(\frac{v^3}{e^(hv/kT)-1})dv[/itex]


Rayleigh-Jean Formula:

[itex]p(v)dv = \frac{8πkT}{c^3}(v^2)dv[/itex]


The Attempt at a Solution



I've used the Rayleigh-Jean and Planck blackbody formulas, so I'm not unfamiliar with them. But I'm not even sure where to start with this question. The only two approaches I can think of are to start arbitrarily picking frequency values, solving both equations, and test for the percent errors... or to combine the two formulas along with the percentage error formula in a massively complex equation that I would need Mathematica to solve, which can't possibly be the correct way of going about it.

I have a feeling that I'm going to kick myself over this, but can someone provide me with some guidance on how to go about this?
 
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  • #2
Write an equation expressing that the difference between the two distributions is 10% of the Planck distribution. Simplify as far as possible.

For convenience let x = h[itex]\nu[/itex]/kT and write the equation in terms of x.

I think you'll end up with an equation that cannot be solved exactly for x, as you suspected. See if you can think of a way to find a good approximate solution for x without resorting to Mathematica.
 

Related to Quantum Mechanics - Comparing Rayleigh-Jeans formula with Planck's

1. What is the Rayleigh-Jeans formula?

The Rayleigh-Jeans formula is a classical theory used to describe the emission of radiation from a blackbody. It predicts the energy distribution of electromagnetic radiation at different wavelengths based on classical physics principles.

2. What is Planck's law?

Planck's law is a quantum mechanical theory that describes the emission of radiation from a blackbody. It takes into account the discrete nature of energy and predicts the energy distribution of electromagnetic radiation at different wavelengths based on quantum mechanics principles.

3. How does the Rayleigh-Jeans formula differ from Planck's law?

The Rayleigh-Jeans formula and Planck's law differ in their predictions of the energy distribution of electromagnetic radiation. The Rayleigh-Jeans formula is based on classical physics and predicts that the energy of radiation will increase with shorter wavelengths. However, Planck's law, based on quantum mechanics, predicts that the energy of radiation is not continuous but rather comes in discrete packets or quanta, and that the energy distribution will decrease with shorter wavelengths.

4. Why is Planck's law considered a breakthrough in quantum mechanics?

Planck's law was a breakthrough in quantum mechanics because it was the first theory to successfully explain the energy distribution of blackbody radiation. It took into account the discrete nature of energy, which was a major departure from classical physics principles. This paved the way for further developments in quantum mechanics and our understanding of the physical world.

5. How is Planck's law used in modern science?

Planck's law is used in many areas of modern science, including astrophysics, cosmology, and nanotechnology. It is used to study the emission of radiation from different sources, such as stars and galaxies, and is essential in understanding the properties of light and matter at a microscopic level. Planck's law is also used in the development of new technologies, such as quantum computers, which rely on the principles of quantum mechanics.

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