How do I work out an expression for the total energy

[bon]

Homework Statement

Black body radiation inside a cavity at temperature T may be thought of as a gas of photons with an energy distribution given by the function n(E) given by

n(E)dE = A [E^2 / (e^(E/kT) - 1) ] dE

Where A is independent of E and T. (k is Boltzmann constant)

The function n(E) describes the number of photons with energy between E and E + dE. There is no upper limit on E (although n(E) -> 0 as E -> 0)

Show that

a) the total number of photons is proportional to T^3 and
b) the total energy is proportional to T^4

Homework Equations

The Attempt at a Solution

So I'm a bit confused!

To work out the total number of photons, do I just do the integral of n(E)dE with E from 0 to infinity? How is this proportional to T^3?

How do I work out an expression for the total energy (since n(E) just describes the number of the photons..) ?

Thanks

 

[fzero]

To work out the total number of photons, do I just do the integral of n(E)dE with E from 0 to infinity? How is this proportional to T^3?

You don't actually have to do the integral to show the proportionality. Change variables to

$$x = E/kT$$

and you can factor the T dependence out of the integral completely.

How do I work out an expression for the total energy (since n(E) just describes the number of the photons..) ?

If n(E) is the total energy of the photons with energy between E and E+dE, can you write an expression for the energy of all photons with energy between E and E+dE?

If you can't, suppose there were 4 photons with energy between E and E+dE. What is the total energy of these 4 photons?

 

[bon]

Ah great thanks! Ok so is it okay to just say: integral from 0 to infintiy of n(E) dE = A k^3 T^3 times integral from 0 to infnity of x^2 / e^x -1 dx? (because i can't actually see any way of solving the integral..)

Oh is the energy just nE? so i integrate En(E) between 0 and infinity..hence the extra T power? yay i think i see

great help thanks//