Stefan’s law, DYK Fact #592

[Ebudae]

I have just read about energy distribution in the spectrum of a black body in the chapter on the beginnings of the quantum theory in my A-level textbook on relativity and quantum mechanics. Did you know? Stefan’s law states that the intensity of the radiation emitted by a black body is proportional to the fourth power of its absolute temperature: E = σT⁴, where σ is the Stefan–Boltzmann constant.

The book actually describes E as “total energy radiated per unit time per unit surface area” – but I know that this is technically called intensity all right.

I also read about Wien’s displacement law: the wavelength of the radiation corresponding to the maximum amount of energy radiated by a black body is inversely proportional to its absolute temperature.

[algebraic topology]

Did you know?

. In fact, , where k is Boltzmann’s constant, h is Planck’s constant, and c is the speed of light in vacuum.

[Ebudae]

Did you know? For a non-black body, E = εσT⁴, where the total emissivity ε of the body has a value between 0 and 1.

[Athene_noctua]

Did you know? The intensity of the radiation emitted by a star is related to the brightness of the star. Magnitude is a measure of the brightness of a star based on a decreasing logarithmic scale – so a star of magnitude N is 10^0.4 ≈ 2.512 times brighter than one of magnitude N+1. (In other words, the greater the magnitude, the less bright it is.) Apparent magnitude is the brightness of a star as seen from Earth; this obviously depends on the distance of the star from Earth. The absolute magnitude of a star is its brightness at a fixed distance of 10 parsecs. Thus, the Sun (whose surface temperature is about 6 000 K) has an apparent magnitude of −26.8 and an absolute magnitude of +4.8. I have just read about magnitude in the spread “Stars and Galaxies” under the section The Nature of the Universe of The Guinness Encylopedia.

[Ebudae]

Did you know? Joseph Stefan (1835–93) was a Slovenian physicist.

[JaneFairfax]

Did you know? The luminosity L of a star regarded as a spherical black body of radius r and surface temperature T is given by

http://www.mathisfunforum.com/viewtopic.php?pid=89452#p89452

[algebraic topology]

Did you know?

The brightness β of a star of luminosity L at distance d from Earth is given by

[Ebudae]

I have now read about Planck’s radiation law in the chapter on the beginnings of the quantum theory in my A-level textbook on relativity and quantum mechanics. Did you know? The radiation law proposed by Max Planck in December 1900 states that the intensity per unit wavelength λ of the radiation of a black body at temperature T is given by E_λ = A/[λ⁵(e^B/λT−1)] where A and B are constants. Indeed, A = 2πhc² and B = hc/k where c is the speed of light in a vacuum, h is Planck’s constant and k is Boltzmann’s constant. From this formula, both Stefan’s law and Wien’s displacement law can be derived.

Planck also proposed that energy of a black body is not emitted continuously but only in discrete packets, or quanta, of the form hf where f is the frequency of the radiation.

[algebraic topology]

Did you know?

It can be shown that , where . This is Stefan’s law. Wien’s displacement law may be obtained by setting .

[JaneFairfax]

Did you know? A sufficient condition for E_λ to have a local maximum at λ = λ₀ is that and at λ = λ₀.

[algebraic topology]

Did you know?

Intensity per unit wavelength is actually a function of two variables: wavelength (λ) and thermodynamic temperature (T):

[Athene_noctua]

Did you know? Planck presented his results in a seminar to the German Physical Society on 14 December 1900. He admitted that he considered the quantization of energy only as a mathematical trick to make his equations work – he himself believed that energy was continuous (indeed he believed that matter was not composed of atoms but was infinitely divisible) and would later labour in vain to find a way round the mathematical trick he had introduced. In any case, Planck did not consider all energy as quantized – only the energy emitted by a black body. It was Albert Einstein who, in 1905, extended the quantization hypothesis to light in his celebrated explanation of the photoelectric effect – for which, after it was experimentally verified by Robert Millikan, Einstein received the 1921 Nobel Prize in Physics.

This is what I have learnt from the book Quantum by Jim Al-Khalili.

[Ebudae]

Incidentally, we’ve been talking about “black body” without having defined it. So, did you know? A black body is a body that absorbs all of the radiation incident on it. The radiation it emits is characteristic of its temperature and does not depend on the nature of its surface.

[Athene_noctua]

Did you know? While Planck used the quantum method to explain black-body radiation and Einstein used it to explain the photoelectric effect, neither of them developed quantum theory any further. The first person to make a significant contribution in this direction was the Danish physicist Niels Bohr – who, according to Jim Al-Khalili, is the one who should be credited with the appellation of father of quantum physics.

For I have just been reading about Bohr’s model of the hydrogen atom in the second chapter of Al-Khalili’s book Quantum.

[algebraic topology]

Did you know?

Bohr based his model of the hydrogen atom on the assumption that, basically, such an atom can only absorb and emit energies in discrete quanta rather than continuously. While this explains the stability of the atom and also accounts for the existence of atomic spectra, it is not really an accurate quantum description of atoms in general – in fact the Bohr model breaks down for atoms of elements heavier than hydrogen.

[Ebudae]

Did you know? Niels Bohr was born on this day (07 October) in 1885.

[Athene_noctua]

Did you know? The next step in the advancement of quantum theory was made in 1924 by Louis Victor de Broglie, who suggested in his PhD thesis that if electromagnetic waves could sometimes behave like particles, then particles such as electrons should also be associated with wave behaviour.

This is what I’ve just read about in the book Quantum.

[algebraic topology]

Did you know?

A particle of momentum p has de Broglie wavelength λ = h/p (h being Planck’s constant).

[Ebudae]

Did you know? De Broglie’s hypothesis was confirmed in 1927 by the Davisson–Germer experiment, and de Broglie received the Nobel Prize in Physics in 1929.

[algebraic topology]

Did you know?

It is worth noting that the p in the formula for the de Broglie wavelength of a particle is relavistic momentum – thus, if the particle has rest mass m₀ and is moving with speed v, its de Broglie wavelength is

[shygeorge]

Did you know? Today (24 March 2010) is the 175^th anniversary of the birth of Joseph Stefan.