Planck's Law, Quantum mechanics help

[hxcguitar101]

Homework Statement

(2) The orbiting space shuttle moves around the Earth well above 99% of the atmosphere, yet it still accumulates an electric charge on its skin due (in part) to the loss of electrons caused by the photoelectric effect from sunlight. Suppose the skin of the shuttle is coated with nickel for which the work function is φ = 4.87 eV at the temperatures encountered while in orbit. (A) What is the maximum wavelength of solar radiation that can result in electron emission from the shuttle’s skin? (B) What is the maximum fraction of the total power falling on the shuttle that could potentially produce photoelectrons?

I Found the answer to part A to be 255 nm... how do you find part B?

Homework Equations

Planks law: u(λ) = 8πhcλ^-5)/(e^(hc/λkT)-1)

The Attempt at a Solution

I set up the integral: Int(8πhcλ^-5)/(e^(hc/λkT)-1)) from 0 to 255nm all over the same integral from 0 to infinity (planck's law) but I have NO IDEA how to solve it! THANKS

 

[mark726]

Hello!

To find the maximum fraction of the total power that could potentially produce photoelectrons, we can use the equation for the intensity of radiation, I = u(λ)c, where u(λ) is the spectral radiance and c is the speed of light. We can then integrate this equation from 0 to 255 nm to find the total power of radiation at that wavelength. Next, we can divide this value by the total power of radiation at all wavelengths (which can be found by integrating the same equation from 0 to infinity). This will give us the maximum fraction of the total power that could potentially produce photoelectrons.

I hope this helps! Let me know if you have any further questions.