# Filament in a lightbulb, thermal radiation

I'm trying the following:

The filament inside a 100 W lightbulb has an absorption coefficient of 0.25, and while operating, it is at a temperature of 2,573 K. What's the size of the surface of the filament? (done with Stefan-Boltzmann) At what wavelength does it emit the highest intensity? (done with Wien displacement law). Relative to all emitted radiation, how much power is emitted as visible light (400 nm - 800 nm)?

The last one is causing me trouble. I have been trying to integrate Planck's radiation law for quite some time now, but I can't manage to do it(and neither can Mathematica). Is there another way of doing this?

• You could try by doing a high temperature expansion, which will simplify the situation, letting you integrate it. Oct 28, 2013 at 10:50

Here is how its done. The fraction of total energy emitted as wavelengths from 0 to a certain wavelength $\lambda$ can be obtained from the band emission table ($F_{(0-\lambda)}$). This way the emissions from 0 to 400 nm ($F_{(0-0.4\mu m)}$) and between 0 to 800 nm ($F_{(0-0.8\mu m)}$) can be obtained. The difference in these two quantities gives the emission from 400 nm to 800 nm and is the required answer.
The energy fractions $F_{(0-\lambda)}$ as a function of $\lambda$ and $T$ are given in the table
Here $T$ = 2,573 K, while $\lambda$ is the upper limit of wavelength.