how can integrate black body spectrum in frequency with lowest limit has energy as 13.6 eV in matlab

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How can integrate black body equation in frequency with lowest limit has energy 13.6 eV

Accepted Answer

William Rose
William Rose on 17 Jun 2022
You have to specify the temperature of the object and the limits of integration. Your quesiton makes it sound like the lower limit for integraiton is the frequency or wavelength corresponding to 13.6 eV. If that is correct, then what is the temperature of the body? (The body would have to be very, very hot, to emit a significant amount of photons with energy exceeding 13.6 eV.)
Let's suppose the body temperature is 1000 degrees K.
Convert the lower limit energy to the equivalent frequency, and integrate the black body law from the lower limit to infiinity:
h=6.626e-34; c=2.998e8; kB=1.381e-23; T=1e3;
E1=13.6*1.602e-19; %convert 13.6 eV to Joules
nu1=E1/h; %frequency (Hz) that corresponds to 13.6 eV
fun = @(nu) (2*h*nu.^3/c^2).*(exp(h*nu/(kB*T))-1).^-1; %define the BB function
q=integral(fun,nu1,Inf) %evaluate the integral
q = 4.7646e-65
Try it.
  1 Comment
William Rose
William Rose on 18 Jun 2022
If you change the lower limit for integration from 13.6 eV to 0.02 eV or anything smaller, you will capture virtually all of the black body radiation energy, for an object at 1000 K. The result of the integral in this case is q=1.80E+04 W/(m^2-sr). Compare that to the radiance predicted by the Stefan-Boltzmann law:
which evaluates to 1.80E+04 W/(m^2-sr), using . This gives us confidence that the integral calculation is correct.

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