Homework Help: Planck's Law Problem

1. Jan 20, 2008

1. The problem statement, all variables and given/known data

Find the frequency (Vmax) at which energy density is at a maximum. This requires simple calculus and numerical solution of a simple transcendental equation.
You only need to find the answer to 3 significant digits.

2. Relevant equations

3. The attempt at a solution

So i derived the equation and set it equal to 0 and i got

V = [3kT / h][1 - 1/ (e^(hv/kT))]

But i cant seem to finish it because of the V in the exp....
also we know that
e^x = 1 + x + x^2/2! + x^3/3!+ ... and that somehow has to play a role in solving this...I think...

2. Jan 20, 2008

Dick

You have to do this numerically. Call hv/kT=X. Then your last equation is X/3=1-e^(-X). Write this as f(X)=1-X/3-e^(-X). You want to solve f(X)=0. Plot it. For values of X around 1 f(X) is positive, for values around 4 it's negative. So there must be a root in between. You could just solve it by bisection.

3. Jan 20, 2008

x/3 = 1 - e^(-x)

why is it 'x/3' ?, and not just x

4. Jan 20, 2008

Dick

Because you have 3kT/h. Note the '3'.

5. Jan 20, 2008

wont it be 3x/v ??
since we have

hv / kt = x

andd in this part we have

3h/kt...wont that equal 3x / v

6. Jan 20, 2008

Dick

Yes. And 3kt/h=3v/x. ????

7. Jan 20, 2008

hv / Kt = x
kt/hv = 1/x
3kt/hv = 3/x
3kt/h = 3v/ x

...i still dont understand where you get "x/3" from, please clarify

8. Jan 20, 2008