# Thick as a planck and making a spectra of myself

In summary: Thanks for the help! I'll give it a try. In summary, the Rayleigh-Jeans equation is larger than the Planck equation for frequencies where hv>>kT. However, the Planck equation can be expanded about zero in the variable x to give a smaller result.
for hv>>kT how does exp(hv/kT) compare to 1?
I understand hv >>KT leads to an exponential fall in brightness but why did Planck introduce 1 in his equation.
and only for values hv<<kT can this exponential be expanded!
Thanks for any help!

I have no clue how to help you, however that might be one of the catchiest titles I've ever seen XD

I'm not sure about the question, but you can always expand the exponential in
$$\exp(\hbar\nu / k T) = 1 + \hbar\nu / kT + \mathcal O((\hbar\nu / k T)^2)$$
where the last term indicates the error you would make if you would just terminate after the first two terms.
Now what happens if $\hbar\nu \gg k T$ and if $\hbar\nu \ll k T$?

BTW I agree, the title is not very descriptive but definitely caught my eye.

Thanks for looking; I have been looking at Rayleigh Jeans Formula and I need to understand what happens for hv>>kT

Catastrophe?

So, Rayleigh-Jeans says that the energy density is $$U(\nu) = \frac{8 \pi k T \nu^4}{(c^4)}$$ and Planck says $$U(\nu)d\nu = \frac{4 \hbar \nu^3}{c^3 (\exp(\hbar\nu / k T)-1)}$$.

When $$\hbar \nu >> kT, exp(\hbar \nu / kT) >> 1$$, so you can treat it as
$$U(\nu)d\nu = \frac{4 \hbar \nu^3}{c^3 \exp(\hbar\nu / k T)}$$

Now you can do a limit as $$\nu$$ goes to infinity for the Rayleigh-Jeans and the Planck equations and compare what happens at really high frequencies.

Thanks 82
I actually thought that the result would be of little consequence;
I'll look closely at the differentiation!
Obviously now that I see it, with h=constant and K=constant, any overall increase over 1 must mean the value of v increases against T which I assume doesn't change either because I should be looking at Dv against T. (D is supposed to be delta).
Is the 1 there because of the nature of exp(1)?
Will any of the differentiation has to employ the exponential rules or product rule?
Thanks Red

for hv>>kT how does exp(hv/kT) compare to 1?
it's much larger than 1
I understand hv >>KT leads to an exponential fall in brightness but why did Planck introduce 1 in his equation.
to account for the fact that photons are bosons.
and only for values hv<<kT can this exponential be expanded!
in a taylor expansion about zero... true. But for hv>>kT the Planck formula itself can be expanded about zero in the variable $x=e^{-hv/kT}$ which is small and the first term of the expansion is given in klile82's post.

Thanks 82
I actually thought that the result would be of little consequence;
I'll look closely at the differentiation!
what differentiation? do you mean to say "derivation"?
Obviously now that I see it, with h=constant and K=constant, any overall increase over 1 must mean the value of v increases against T which I assume doesn't change either because I should be looking at Dv against T. (D is supposed to be delta).
Is the 1 there because of the nature of exp(1)?
Will any of the differentiation has to employ the exponential rules or product rule?
Thanks Red

Thanks to all you good guys!

Is the Planck function mentioned above, valid for high frequency range where v>>kt/h?
What would a graph look like; v against kt/h?...an exponential rising upwards and rapidly from 0. How would the range on the y-axis appear 10^-1 to 10^-10 for example or the otherway about?

Red

Thanks to all you good guys!

Is the Planck function mentioned above, valid for high frequency range where v>>kt/h?
What would a graph look like; v against kt/h?...an exponential rising upwards and rapidly from 0.

what the heck? have you read any of the previous posts? what was Planck's whole point?

Why the 1?

Is the 1 there because of the nature of exp(1)?

The 1 isn't a hack, it's there because of an identity used in the derivation. Basically, as you go through the derivation you wind up with a term that looks like the sum from n=0 to infinity of $$exp(-nh\nu / kT)$$. There's an identity that says that the sum from n=0 to inifity of x^n is \frac{1}{1-x} [/tex] as long as x< 1. That's where the 1 in the denominator comes from.

It's really helpful, if you're comparing the Rayleigh-Jeans equation to the Planck equation, to graph them as a function of v on the same graph. You should be able to see the difference right away. Just choose a value for T and use the standard values for c, k and h. You can do this easily in something like Mathematica. You can do this fairly easily in excel, too.

## 1. What does it mean to be "thick as a Planck"?

Being "thick as a Planck" refers to the concept of Planck length, which is the smallest possible unit of length in the universe according to the theory of quantum mechanics. Therefore, being "thick" in this context means having a very small or infinitesimal size.

## 2. How is a spectra of oneself created?

A spectra of oneself is created by using a spectroscopy technique, which involves passing light through a sample and analyzing the resulting spectrum of wavelengths that are absorbed or emitted by the sample. In this case, the sample would be the person's body, and the spectrum would provide information about the chemical composition of their body.

## 3. What can a spectra of oneself tell us?

A spectra of oneself can provide information about the chemical elements present in the body, as well as their abundance and the chemical bonds they form. This can give insights into a person's overall health, diet, and environment, as well as potential genetic or environmental factors that may affect their health.

## 4. Is creating a spectra of oneself possible?

Yes, creating a spectra of oneself is possible using advanced spectroscopy techniques. However, it would require highly specialized equipment and expertise, as well as ethical considerations and consent from the individual.

## 5. What are the potential applications of creating a spectra of oneself?

The potential applications of creating a spectra of oneself include medical diagnostics, monitoring of health and wellness, and understanding the effects of environmental factors on the body. It could also have implications for personalized medicine and precision healthcare.

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