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I don't see how it would make sense physically to integrate over temperature.TeslaPow said:Want to integrate the total energy density over all photon energies between two
temperature values from 500K to 5800K, but not sure how to proceed.
DrClaude said:I don't see how it would make sense physically to integrate over temperature.
In any case, you have ##N/V = C T^3## [see the last line of the solution of 40. (a)], which shouldn't be too hard to integrate
That should be ##k^3## and you forgot the ##\Gamma(3) \zeta(3) \approx 2.40## factor.TeslaPow said:By C you mean: 8 *pi * k / (hc)^3 ?
DrClaude said:That should be ##k^3## and you forgot the ##\Gamma(3) \zeta(3) \approx 2.40## factor.
I don't understand. You said youTeslaPow said:So I also have to multiply the constant outside the integral with 2.40? What value will I use for T in the same equation on the left side if I integrate between 500K and 5500K ?
which I take to meanTeslaPow said:Want to integrate the total energy density over all photon energies between two
temperature values from 500K to 5800K, but not sure how to proceed.
As I said above, I do not understand what an integral over temperature physically means here. Usually one is interested in the value at a given ##T## or in the change with respect to ##T##.TeslaPow said:When looking at Stefan-Boltzmann law, and how the procedure is done there,
Not if you are interested in the energy density.TeslaPow said:is this the equation to use if I want to find the integral between 500K and 5500K?
Everything comes from Planck's law, which gives the energy density of a photon gas per wavelength/frequency. Wien's displacement law is simply the maximum of that distribution. Stefan's law gives, which is indeed Planck's law multiplied by ##C/4## if you use the correct ##C##, gives you the power per unit area emitted by a blackbody.TeslaPow said:I thought maybe that an integration was necessary on the energy density, but it seems that the Wien displacement law is used to find the peak curve and then you use Stefan Boltzmann law to integrate between wavelengths within that peak. Stefan law is the Planck radiation formula multiplied by C/4. Does this seem reasonable?
Yes.TeslaPow said:Is the procedure for the numerical integration for Planck's radiation law the same for the energy density
as it is for the intensity?
There is no integration here, you simply plug in the values of λ and T.TeslaPow said:How is the value calculated here in b) ?
DrClaude said:Yes.
So the value for the constant outside the integral in the energy density is
8 * pi * c^(2) * h ?
DrClaude said:It depends what you are integrating. Are you integrating over wavelength, frequency, temperature?
In the second case I presented, yes. But in your post #10, problem 24(a) is calculated using a direct integration as a function of λ, as in the first case I presented.TeslaPow said:View attachment 272035
So this is used for the integration?
Again, I have never seen Planck's law being integrated over temperature. I do not understand what this would mean physically. Planck's law gives the energy density of a photon gas at equilibrium , hence at a given temperature.TeslaPow said:The integration I outlined is used for temperature?
You can write Planck's law as a function of frequency or wavelength. How to go from energy density of a photon gas to emission from a black body is explained in statistical physics textbooks. A couple of online references that might be useful:TeslaPow said:What I don't understand quite is why wavelength is used in both energy density and intensity as the factor C/4 is used to convert between these.
I don't understand what you are doing here. You seem to be calculating I(400nm,T) / (integral of I over all λ) instead of I(400nm,T)/I(966nm,T).TeslaPow said:Last question, as for 24 b) in #10, the answer for the first intensity should be I(400nm,T) = 335289 W/m^2
https://www.wolframalpha.com/input/?i=solve(x/(4.593*10^(6))=0.073,x)
This is correct.TeslaPow said:Just trying to plug in these values as in thread #11 but don't come up with the same answer.
https://www.wolframalpha.com/input/?i=(2*pi*(299792458)^2*6.62607004*10^(-34))/(4.0*10^(-7))^(5)*e^(-6.62607004*10^(-34)*(299792458)/((4.0*10^(-7)*(1.38064852*10^(-23)*(3000)))))
Bose-Einstein numerical integration is a computational method used to numerically solve equations that involve Bose-Einstein statistics. It is based on the principles of quantum mechanics and is used to calculate the behavior of particles at very low temperatures.
Bose-Einstein numerical integration involves discretizing the energy levels of a system and using a numerical algorithm to calculate the probability of particles occupying each energy level. This is then used to calculate the total energy and other properties of the system.
One advantage of using Bose-Einstein numerical integration is that it allows for the calculation of properties of a system at very low temperatures, where classical statistical methods are not applicable. It is also a more accurate method compared to other numerical techniques.
Bose-Einstein numerical integration can be computationally intensive and may not be suitable for systems with a large number of particles. It also assumes that the particles in the system are in thermal equilibrium, which may not always be the case.
Bose-Einstein numerical integration is commonly used in the field of condensed matter physics to study the behavior of particles in systems such as superfluids and Bose-Einstein condensates. It also has applications in astrophysics, where it is used to model the behavior of particles in extreme environments such as neutron stars.