Plank's law

Plank obtained a formula to explain how a body emits different lights depending of its temperature.

$$ u(\nu, T) = \frac{8\pi h \nu^3}{c^3} \left(\frac{1}{e^{\frac{h\nu}{kT}} - 1}\right) $$

In this formula:

$u(\nu, T)$ is the energy density per unit volume per unit frequency at frequency $\nu$ and temperature $T$.

$h$ is Planck's constant ($6.62607015 \times 10^{-34}$ m$^2$ kg / s).

$\nu$ is the frequency of the electromagnetic radiation.

$c$ is the speed of light in a vacuum ($3.00 \times 10^8$ m/s).

$k$ is Boltzmann's constant ($1.380649 \times 10^{-23}$ m$^2$ kg s$^{-2}$ K$^{-1}$). See Boltzmann distribution.

$T$ is the absolute temperature of the black body in kelvins (K).

To understand, first, the problem solved by Plank: video of the ultraviolet catasthope

In this video also appears the key step in the reasoning of Plank:

Energy levels are discrete and can be represented as $E = 0, \epsilon, 2\epsilon, 3\epsilon, 4\epsilon, \ldots$ or $E = n\epsilon$ where $n = 1, 2, 3, \ldots$

To calculate average energy, replace integrals with sums.

The continuous form for average energy is (taking into account Boltzmann distribution):

$$ \bar{E} = \frac{\int_{0}^{\infty} Ee^{-E/kT} dE}{\int_{0}^{\infty} e^{-E/kT} dE} $$

This is transformed into a discrete sum for quantized energy:

$$ \bar{E} = \frac{\sum_{n=0}^{\infty} n\epsilon e^{-n\epsilon/kT}}{\sum_{n=0}^{\infty} e^{-n\epsilon/kT}} $$

Another derivation of Plank's law, here

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

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