Annus MirabilisEinstein in 1905
Reading preferences

Reading preferences

Keep every explanation and worked example on the page. Nothing moves or plays by itself, and the larger instruments load only when you ask.

Line length
Type size
Contrast
Paragraph spacing

light quanta

The equation explained

S=v∫0∞φ(ϱ,ν)dνS = v \int_0^\infty \varphi(\varrho, \nu) d \nu
In wordsThe radiation's entropy equals the volume it fills times the sum, over every frequency, of the entropy per unit volume and unit frequency, which depends on the energy density at that frequency and on the frequency itself.OverviewIf the radiation's energy density at every frequency fixes its properties, its entropy is the volume times a sum over frequencies of an entropy density that depends on the density and the frequency. The entropies of different frequencies add, because radiations of different frequencies can be separated without work or heat.Full explanationSection 3 takes a radiation filling the volume vv and assumes that its perceptible properties are completely fixed once its density ϱ(ν)\varrho(\nu) is given at every frequency. Radiations of different frequencies can be separated from one another without doing work and without supplying heat, so their entropies add, and the entropy of the whole can be written as the volume times an integral over frequency of a function φ\varphi of the density and the frequency. The paper remarks that φ\varphi could be reduced to a function of one variable by using the fact that adiabatic compression between reflecting walls leaves the entropy unchanged, but it does not take that route; it goes straight to finding φ\varphi from the law of black radiation.Every stepφ(ϱ,ν)dν\varphi(\varrho, \nu) d \nuThe entropy per unit volume of the radiation between ν\nu and ν+dν\nu + d \nu: a function of the density there and of the frequency, because the paper assumes the density at every frequency fixes the radiation's properties.∫0∞φ(ϱ,ν)dν\int_0^\infty \varphi(\varrho, \nu) d \nuRadiations of different frequencies can be separated without work and without heat, so their entropies add: sum over every frequency.S=v∫0∞φ(ϱ,ν)dνS = v \int_0^\infty \varphi(\varrho, \nu) d \nuThe density does not vary through the volume, so multiply the entropy per unit volume by the volume vv: the printed line.Historian's marginSection 3 opens by saying that the argument is contained in 'a famous paper by Mr. W. Wien' and is given here only for completeness (p. 137); the paper does not identify which paper of Wien's. Its footnote calls the assumption that the density at every frequency fixes the radiation's perceptible properties 'arbitrary', to be kept as the simplest until experiment forces it to be abandoned.Also

Read it in the paper