blackbody radiation / Planck's law

[ptc-4495199]

Hello,

I am just trying to integrate the planck's law with the wavelenght as variable, but I do not get anything I can make sense of.

The task is the calculation of a photon density.

I am using Mathcad 15.

I looked around and there is quite a few old documents available but actually I can not reproduce them.

I am missing something.

I would appreciate a clarification on the right way to do it.

My files is "Planck radiation".

Integrations with density vs. E fails

Integration with density vs. lamdba give an odd number which does not seem right.

Planck.mcad (and pdf) is a legacy document from the internet

It shows that the integration is doable, but I can not reproduce it.

I look in the PTC discussion, but it seems that the attachments are no longer available.

Thanks,

Pierre

[Werner_E · Accepted solution]

Beginning with Mathcad 14 the symbolic engine was changed. In older versions its was Maple but now its muPad. Many people consider Maple being more efficient and capable. Anyhow, thats the reason the integral in the old file you posted does not evaluate as it should and as it is shown in the pdf of that file. The "assume" does not work as shown in the sheet, you would first have to clear those four variables for symbolic evaluation and then put the assume ine the symbolic eval - but unfortunately even then muPad does not find a solution for the integral.

Concerning your file and the integral in your formula 1. When I simplify the integrand I get

The denominator in the range of exp(E*10^19) gets really big for positive values of E. Mathcads symbolic interprets it a infinity and yields 0.0 as result, while the numeric fails for values greater than 10^307.

You may consider to chekc and rewrite your formula.

Concerning the result with formula 2, you misinterpreted the symbolics result. 1.23e4 is mupads way to say 1.23*10^4, no reference to Eulers constant.

You do not need tpo retype the symbolcs result - you may either assign ito a variable wich you can evaluate numerically or use inline evaluation.

When I google for Planck & Mathcad I get a lot of findings. Maybe this random pick (also old and from Maple times, I didn't bother to retype) is of help

https://www.cv.nrao.edu/~demerson/osfholo/Planck3.pdf

[Werner_E]

Does anyone know if there is a limit for the number of pictures you are allowed to insert in a posting? Whatever I try in the above reply, the fourth pic is shown correctly first but when I send/update, all I get is a non working link!

I'll try it again here:

[AlanStevens]

Werner Exinger wrote:

... muPad does not find a solution for the integral.

Mathcad sometimes needs a little help with symbolic integrals:

Alan

[Werner_E]

Great done, as usual. I already wondered if you would jump in with something like this 😉 http://communities.ptc.com/message/192562#192562

[AlanStevens]

Here's how you might tackle the E integral (NB NN isn't a function of E):

Alan

[ptc-4495199]

Thanks all for the help.

I have got the calculation going.

I did not get the symbolic result (pi^4/...) but a rational number (but also the good one).

How come ?

Pierre

[AlanStevens]

Pierre TOMASINI wrote:

I did not get the symbolic result (pi^4/...) but a rational number (but also the good one).

How come ?

Can't tell without seeing your worksheet! Upload it.

Alan

[ptc-4495199]

Here you go.

Thanks again for your comments.

Pierre

[AlanStevens]

If you multiply my value for NN by DR_current you get your value for Nph_current.

One word of warning: Be careful with the float command. It doesn't just affect the final result, but all intermediate calculations as well. Better to use a larger value than float, 4 generally.

Alan

[ptc-4495199]

Thanks.

Pierre