1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Stefan-Boltzman Law Proof with Regards to Solar Radiation

  1. Sep 18, 2012 #1
    1. The problem statement, all variables and given/known data

    The unit area intensity of radiation from the Sun at the photosphere is 6.33*107 W/m2.

    a) Check this value using the calculus of the Stefan-Boltzman Law, assuming the Sun is a blackbody emitter ([itex]\epsilon[/itex] = 1) with a surface temperature of 5777K.

    2. Relevant equations

    Stefan-Boltzmann Law:
    [tex]E_b = \int_0^∞ \! E_{\lambda,b} \, \mathrm{d} \lambda = σT^4[/tex]
    Planck's Law:
    [tex]E_{\lambda, b} = {\frac{2\pi hc^2}{\lambda^5[e^{(hc/\lambda kT)}-1]}} [/tex]


    [itex]c = 2.998*10^{14} \ \mu m \ s^{-1}[/itex]
    [itex]h = 6.626 * 10^{-34} \ J \ s[/itex]
    [itex]k = 1.381 * 10^{-23} \ J/K [/itex]
    3. The attempt at a solution

    I'm not sure how to solve the integral of the Stefan-Boltzmann Law. I know I can substitute [itex]E_{\lambda, b}[/itex] from Planck's law into the Stefan-Boltzmann law, but I have no idea how to integrate it then. Integration by substitution fails here and I have to prove using calculus, that the sun's [itex]E_b[/itex] is equal to 6.33*107 W/m2. Thanks.
  2. jcsd
  3. Sep 18, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper

    It's not an elementary integral, as you've probably guessed. It involves zeta functions and stuff like that. You can reduce it to a dimensionless integral which contains the hard part. See http://en.wikipedia.org/wiki/Stefan–Boltzmann_law But I'm not sure you need to do that. Doesn't it just ask you to use the Stefan-Boltzann constant? You can look that up. Do they really mean you should derive it from Planck's Law?
    Last edited: Sep 18, 2012
  4. Sep 18, 2012 #3
    Thanks for the reply. It's certainly not. The Stefan-Boltzmann constant comes from the integration, right? No, it doesn't say I need to derive it from Planck's law. However, I don't see how to integrate it without the substitution.
  5. Sep 18, 2012 #4


    User Avatar
    Science Advisor
    Homework Helper

    I don't think you need to do any integration. The Stefan-Boltzmann constant already includes the integral. Just put numbers in. It's a hard integral. Involves functions that aren't elementary. Stuff beyond just substitution and integration by parts.
  6. Sep 18, 2012 #5
    Alright thanks. I'll give it a shot and see if I can come up with anything.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook