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=vφ(ϱ,ν)dν=−Eβν{lg⁡Evαν3dν−1}.S = v \varphi(\varrho, \nu) d \nu = -\frac{E}{\beta \nu} \left\{ \lg \frac{E}{v \alpha \nu^3 d \nu} - 1 \right\}.
In wordsThe entropy of radiation of energy E, in a narrow band of frequency and filling the volume v, is v times the entropy density times the band's width. With the energy density written as the energy over the volume and the width, it is minus the energy over the constant β times the frequency, times the natural logarithm of the energy over the volume, the constant α, the frequency cubed and the width, less one.OverviewThe entropy of a given amount of radiation energy in a narrow band of frequencies, filling a volume v: the entropy density times the volume and the band's width, written through the energy instead of the density.Full explanationThe paper now takes a definite amount of radiation: energy EE, with frequencies between ν\nu and ν+dν\nu + d \nu, filling the volume vv. Its entropy is the entropy per unit volume and frequency, times the volume, times the width of the band. The density is the energy per unit volume and frequency, so ϱ=Evdν\varrho = \frac{E}{v d \nu}, and putting that into the preceding line turns everything into the energy, the volume and the band. The formula now shows how the entropy depends on the volume, which the next line isolates.Every stepS=vφ(ϱ,ν)dνS = v \varphi(\varrho, \nu) d \nuThe entropy of section 3's first display, for a band so narrow that φ\varphi is the same across it: the integral over frequency becomes φ\varphi times the width dνd \nu.ϱ=Evdν\varrho = \frac{E}{v d \nu}The energy EE fills the volume vv and the band dνd \nu, so the energy per unit volume and unit frequency is EE divided by both.vφdν=−vdνϱβν{lg⁡ϱαν3−1}v \varphi d \nu = -v d \nu \frac{\varrho}{\beta \nu} \left\{ \lg \frac{\varrho}{\alpha \nu^3} - 1 \right\}Put in the entropy density of the preceding display.vdν⋅ϱ=Ev d \nu \cdot \varrho = EThe factor in front becomes the energy.S=−Eβν{lg⁡Evαν3dν−1}S = -\frac{E}{\beta \nu} \left\{ \lg \frac{E}{v \alpha \nu^3 d \nu} - 1 \right\}Replace ϱ\varrho inside the logarithm by Evdν\frac{E}{v d \nu} as well: the printed line.Also

Read it in the paper