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Homework Statement
The dependence on wavelength [tex] \lambda [/tex] of the intensity [tex]I(\lambda)d\lambda[/tex] of the radiation emitted by a body which is in thermal equilibrium with its surroundings at temerature T is given by:
[tex]I(\lambda)d\lambda = \frac{2 \pi h c^{2}/\lambda^5}{e^{hc/kT\lambda}-1}d\lambda[/tex]
in the interval of wavelength between [tex] \lambda [/tex] and [tex] \lambda+d\lambda [/tex]. In this expression, h is Planck's constant, k is Boltzmann's constant, and c is the velocity of light.
Sketch and clearly label on one figure the dependences of [tex]I(\lambda)d\lambda[/tex] on [tex] \lambda [/tex] for three different temperatures [tex] T_{1} < T_{2} < T_{3} [/tex].
Simplify the above expression in the limit of (i) short wavelength ([tex]\lambda\rightarrow0[/tex]) and (ii) long wavelength ([tex]\lambda\rightarrow\infty[/tex]).
(The binomial expansion [tex]e^{x} = 1+x+x^{2}/2+...[/tex] may be useful.)
Homework Equations
All given in the problem i think.
The Attempt at a Solution
I found Planck's Radiation Law was almost exactly the same as this i searched for it on wikipedia for more information:
http://en.wikipedia.org/wiki/Planck's_law
On that page is a graph which i thought was showing what the first part of the question is asking but i don't understand what the question means when it says "clearly label on one figure the dependences of [tex]I(\lambda)d\lambda[/tex] on [tex] \lambda [/tex]"?
For the second part i tried to make the formula look simpler first:
[tex]\frac{A}{\lambda^{5}(e^{B/\lambda} - 1)}[/tex]
I think as [tex]\lambda\rightarrow0[/tex], [tex]e^{B/\lambda} - 1[/tex] can be simplified to [tex]e^{B/\lambda}[/tex] because the latter expression will be very large giving:
[tex]\frac{A}{\lambda^{5}e^{B/\lambda}}[/tex]
I'm having some trouble posting the rest of my thread but i thought for the last part as lambda goes to infinity the expression would simplify to A/lambda^5 but I'm not sure how to work these out for definite i think this is probably wrong.
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