Why doesn't any coloured body get infinitely hot?

[Shreya]

Homework Statement

Let's consider an orange coloured body. It is going to absorb all wavelength of the EM spectrum and reflect only the wavelength corresponding to the orange colour. So why doesn't it get hotter and hotter or why doesn't it start emitting radiation. And if the orange body stops absorbing radiation once it is at thermal equilibrium with surroundings, is it reflecting all wavelengths (then why doesn't it appear white?)

Relevant Equations

E=hf

Sorry if the question is dumb. Please be kind to help

 

[haruspex]

Just because it looks orange doesn't mean it only reflects orange.. but suppose it does do that.
At any wavelength it can absorb it can equally easily emit. How much it emits at each of those wavelengths depends on its temperature. As it gets hotter it will emit more power in total and more towards the shorter wavelengths.
Being at thermal equilibrium is not the same as emitting exactly the same spectrum as it absorbs. The Earth is/was in thermal equilibrium with power arriving from the sun, but the power emitted is mostly in the IR band.

 

[Shreya]

Does that mean that the absorbed EM radiation from visible spectrum is released as IR radiation? And the energy that is absorbed but not emitted make the body in thermal equilibrium with the hot body

 

[OmCheeto]

Does that mean that the absorbed EM radiation from visible spectrum is released as IR radiation? And the energy that is absorbed but not emitted make the body in thermal equilibrium with the hot body

Yes.

 

[Shreya]

Can I also get some intuition on why classical theory predicted that blackbody would radiated a lot of UV rays? (Also referred to as the UV Catastrophe)

 

[phystro]

There is a theoretical explanation based on quantized photons according to the new theory, however in my opinion it would be easier to understand it if you simply look at the plots of I (Radiation Intensity) against λ (wavelength) or f(frequency) according to 1) the predictions of classical theory: where I ∝ λ^-4 and 2) according to Planck's law which is an empirical law but has a more complex dependence on λ.

Mathematics actually make it easier to explain why something happens.

 

[hutchphd]

I think Wikipedia does a good short synopsis:https://en.wikipedia.org/wiki/Ultraviolet_catastrophe

 

[Shreya]

if you simply look at the plots of I (Radiation Intensity) against λ (wavelength) or f(frequency) according to 1) the predictions of classical theory: where I ∝ λ

I think Wikipedia does a good short synopsis

Thanks @hutchphd and @phystro, it makes more sense now!

 

[vela]

Can I also get some intuition on why classical theory predicted that blackbody would radiated a lot of UV rays? (Also referred to as the UV Catastrophe)

The classical theory predicts the average energy per mode is $kT$. This is just a result of the equipartition theorem which relies on the assumption that the energy of a mode can take on any value. The problem with this is that the number of modes increases with frequency, so the total energy increases with frequency.

Planck assumed the energy was quantized. At low frequency, the energy of a mode was essentially continuous—that is, to a good approximation, the energy could take on any value—so you recover the classical result of $kT$ for the average energy. At high frequency, however, the allowed energies are widely separated. When you calculate the average energy of these discrete values, you find the average energy goes to 0 as $f \to \infty$. It goes to 0 faster than the rate at which the number of modes increases, so the total energy goes to 0 as $f \to \infty$, in agreement with observations.

 

[phystro]

It is perhaps also interesting to look at the Planck's law on Wikipedia. According to Wikipedia:

"The colourful term "ultraviolet catastrophe" was given by Paul Ehrenfest in 1911 to the paradoxical result that the total energy in the cavity tends to infinity when the equipartition theorem of classical statistical mechanics is (mistakenly) applied to black-body radiation. But this had not been part of Planck's thinking, because he had not tried to apply the doctrine of equipartition: when he made his discovery in 1900, he had not noticed any sort of "catastrophe". It was first noted by Lord Rayleigh in 1900, and then in 1901 by Sir James Jeans; and later, in 1905, by Einstein when he wanted to support the idea that light propagates as discrete packets, later called 'photons', and by Rayleigh and by Jeans. In 1913, Bohr gave another formula with a further different physical meaning to the quantity hν. In contrast to Planck's and Einstein's formulas, Bohr's formula referred explicitly and categorically to energy levels of atoms. Bohr's formula was W_τ2 − W_τ1 = hν where W_τ2 and W_τ1 denote the energy levels of quantum states of an atom, with quantum numbers τ2 and τ1. The symbol ν denotes the frequency of a quantum of radiation that can be emitted or absorbed as the atom passes between those two quantum states. In contrast to Planck's model, the frequency v has no immediate relation to frequencies that might describe those quantum states themselves. Later, in 1924, Satyendra Nath Bose developed the theory of the statistical mechanics of photons, which allowed a theoretical derivation of Planck's law. The actual word 'photon' was invented still later, by G.N. Lewis in 1926, who mistakenly believed that photons were conserved, contrary to Bose–Einstein statistics; nevertheless the word 'photon' was adopted to express the Einstein postulate of the packet nature of light propagation."

 

[Shreya]

Thanks a lot Vela! That really helped.

 

[kuruman]

A simple explanation of why a coloured body cannot get infinitely hot is the second law of thermodynamics which puts a theoretical upper limit to the absorbing body's temperature. One may be comforted to know that basking in the Sun will not cause one's temperature to rise above that of the Sun's photosphere, approximately 6000 K.