Photon flux from blackbody source

[AStaunton]

Had a quantum assignment recently and one of the problems was:

What is the photon flux (photon cm^-2 s^-1) at a wavelength of 600nm at a distance of 1km from a blackbody soiurce at 2000k emitting 50W of radiation between 400nm-700nm.

My thinking was:

This involves Planck's radiation Equation:

$$I(\lambda)=\frac{2\pi hc^{2}}{\lambda^{5}(e^{hc/\lambda kT}-1)}$$

we are interested in 600nm but have info about 400-700nm..Also know that $$I\propto P$$ so should find area under Plank's equation between limits of 400-700nm and then get area under plank's function from $$600nm$$ to $$600_{nm}+\Delta\lambda$$ as limit $$\Delta\lambda$$ tends to 0, express this latter area as a fraction of the former:

$$\frac{P_{600nm}}{P_{400-700nm}}$$

now, find intensity for the 400-700nm range at a distance of 1km and then can find intensity for 600nm by:

$$I_{600nm}=I_{400-700nm}\frac{P_{600nm}}{P_{400-700nm}}$$

now divide this intensity value by energy of 600nm photon: $$E=\frac{h}{\lambda}$$ and we have photon flux...

firstly I would be grateful if someone could confirm for me if I have gotten the concept right and this is the correct method to use...

I also have a technical question:

My ability with calculus is quite poor and so was not able to integrate Planck's function and so had to find area under curve between 400-700nm numerically, which was abviously less precise and quite time consuming! similar problem for area under curve at 600nm...although I think it can be done analytically by: $$600nm$$ to $$600_{nm}+\Delta\lambda$$ as limit $$\Delta\lambda$$ tends to 0, my calculus let me down and so I did it again numerically and found area between limits of 599.9 and 600.1.

So my question is, can what is the method to integrate plank's function and also is the idea of $$600nm$$ to $$600_{nm}+\Delta\lambda$$ as limit $$\Delta\lambda$$ tends to 0 a correct method to use.

Any advice appreciated.

 

[mark726]

Thank you for sharing your approach and thought process for solving this problem. Your understanding of using Planck's radiation equation and the relationship between intensity and power is correct. Your method of finding the photon flux at 600nm by dividing the intensity at that wavelength by the energy of a 600nm photon is also correct.

As for your technical question, there are a few ways to integrate Planck's function. One method is to use numerical integration, like you did, by dividing the wavelength range into small intervals and calculating the area under the curve for each interval. Another method is to use analytical integration, which involves using mathematical techniques such as substitution or integration by parts. You can find step-by-step instructions for integrating Planck's function online or in a calculus textbook.

The idea of using 600nm to 600_{nm}+\Delta\lambda as the limit as \Delta\lambda tends to 0 is a valid approach. This is known as a limit definition of a derivative and is commonly used in calculus. However, it may not be necessary to use this method in this particular problem as you can integrate Planck's function directly using the techniques mentioned above.

I hope this helps and good luck with your quantum assignment!