How Do You Calculate Photon Number Density in a Specific Wavelength Range?

[xdrgnh]

Homework Statement

In Problem 1 find the number density of photons in the sun’s core for the region of wavelengths between 599 nm and 601 nm. In problem 1 temperature is given.

Homework Equations

Planck blackbody formula and number density= constant times T^3 where T is temperature.

The Attempt at a Solution

Everyone is telling me to use the constant times T^3 formula but that makes no sense in this problem. The constant times T^3 is only when all frequencies are integrated. However only a slice of frequency is considered. Is this question an error or something. A lot of questions and slides had errors in this class so far.

 

[TSny]

You are right, you can't use that formula. You need to use Planck's distribution function for the energy per unit volume for radiation that has a wavelength between λ and λ + dλ. See equation 5 here: http://www.oocities.org/vhsatheeshkumar/ModernPhysics.pdf.

Then think about how you would use this to get the number of photons per unit volume with wavelengths in the specified range.

 

[xdrgnh]

I know I got to integrate the wavelengths to get the energy density. However I'm having a problem finding a way to get from energy density to photon density. I know the energy of each photon is hc/(wavelength).

 

[TSny]

You won't need to integrate. Note that you are considering a range of wavelengths Δλ that is very small. So, to a very good approximation you can let dλ ≈ Δλ.

To go from energy density to photon number density, consider the approximate energy of anyone photon in the range of wavelengths that your are dealing with.

 

[xdrgnh]

So once I apply the approximation and plug in 599nm with my dλ ≈ Δλ equaling 2nm I divide by hc/599nm

 

[TSny]

Yes, that should do it. (You could also use 600 nm as a good average wavelength in the interval you are working with, but it won't make much difference.)

 

[xdrgnh]

Thanks a lot. I think what got me was that I never had a expression with df in it before where I did not integrate. I thought a infinitesimal was so small that you can't put a number on it. I knew I had to find the energy density and then divide by something. However because I thought I was was integrating from 601 nm to 599 nm and the wavelength was changing I didn't really have a hc/v to divide by and that got me hooked up for hours today.