And at the same time the concept was that the light is emitted by
oscillators. The oscillator has the average energy <E>
= kT having the temperature T. The energetic radiation density u(ν)dν multiplied
by the spectral range [ν, ν+dν]
is proportional to the oscillators amount per volume unit in this range
multiplied by kT:
u(ν)dν = <E> dN(ν) = kT dN(ν)
u(ν)dν = Energy of radiation in the range [ν,ν+dn]/Volume |
dN(ν) was calculated by Rayleigh
and Jeans: dN(ν)
= 8pn²/c³
dν
u(ν) = 8pn²/c³ kT | The experiments prove the Rayleigh-Jeans law
in IR region ! |
Since u(ν) increases proportionally to ν2 , so it leads to the "Ultraviolet Catastrophe" because there would be much more energy irradiated with the increasing frequency. The glance at the dark lecture room shows that it doesn't emit in UV, i.e. the energy distribution law is wrong (at least in UV region).
1900 Max Planck supposed
that the energy are not irradiated continuously but by small fractions (or
energy quanta): the electron should emit the energy quantum
hν or nothing moreover h is the proportionality
constant. So, it can be irradiated only integer of hν
, but not for instance, 3/2hν
or 1,7754hn.
Planck derived the following formula for the spectral energy density u(ν)dν
= <E> ·
dN(ν) in the frequency range [ν,
ν+dν]
(Derivation):
<E> | dN | |
u(ν)dν = |
|
|
u(ν) = 8πhν³/c³.1/ehν/kT− 1 |
This black body irradiation formula perfectly coincide with the experiments.
From the maximum of the distribution one can find νmax (differentiating upon ν and putting to zero) and the meaning of h can be determined from the experiment:
νmax = 2,8214 kT/h
h = 6,626176·10-34 Js |
Integration of the Planck equation gives the
radiation law of Stefan
and Boltzmann:
(a = 7,56·10-16 Jm-3K-4) |
The intensity (radiated energy per surface
and time) is I = 1/4 c ·
a · T4
so » 5,6697 · 10-8 [Wm-2K-4] |
The radiated energy increases with the fourth power of the absolute temperature T.
Because of the scale accuracy of Planck constant h we hasn't took into account it. Below you can find two numerical examples: