Black body radiation, Plancks' Law question?

[applestrudle]

Homework Statement

A cylinder of length 0.02m emits 100W and is at 2000K. It is a perfect black body. What is it's diameter?

Homework Equations

Planck's Law:

B(λ,T) =

http://en.wikipedia.org/wiki/Planck's_law

solid radian = A/r^2 (not sure if needed or not)

The Attempt at a Solution

surface area = 0.02 x 2 pi r + 2 pi r^2

solid radians of the cylinder = (pi r^2) / (r^2 + 0.01^2)

r is radius of the cylinder

frequency of peak radiation = 2.07 x 10^14

(I used Wien's displacement law to get the peak frequency)

B x surface area x solid radians x frequency = 100W

(I'm using dimensional analysis here, not really sure what's going on)

100 = 9.104 x 10^-10 x (0.04 pi r + 2 pi r^2) x (pi r^2)/(r^2 x 10^-6) x 2.07 x 10^14

which rearranges to...

5.303 x 10^-14 (r^2 + 1x10^-6) = 0.04 (pi)^2 r^3 + 2 (pi)^2 r^4

I can't solve for r :(

any help or explanation is much appreciated

 

[dextercioby]

Don't you have to use the Stefan-Boltzmann law ??