light quanta

The equation explained

\varrho_\nu = \frac{\alpha}{\beta} \nu^2 T.

In wordsThe radiation's energy per unit volume and unit frequency equals the constant α divided by the constant β, times the frequency squared, times the absolute temperature.OverviewFor long waves and high radiation densities, Planck's formula goes over into a density proportional to the square of the frequency and to the temperature: the form section 1 derived.Full explanationWhere

T/\nu

is large, the exponent

\frac{\beta \nu}{T}

is small, and the exponential of a small number is very nearly one plus that number. The denominator of Planck's formula becomes

\frac{\beta \nu}{T}

, and dividing

\alpha \nu^3

by it gives this line. The paper reads large

T/\nu

as 'large wavelengths and radiation densities'. The next sentence observes that this is the form section 1 derived from Maxwell's theory and the electron theory; only the way the coefficient is written differs,

\frac{\alpha}{\beta}

here and

\frac{R}{N} \frac{8 \pi}{L^3}

there.Every step

\varrho_\nu = \frac{\alpha \nu^3}{e^{\frac{\beta \nu}{T}} - 1}

Planck's formula, the first display of section 2.

e^{\frac{\beta \nu}{T}} \approx 1 + \frac{\beta \nu}{T}

For large

T/\nu

the exponent is small, and the exponential of a small number is very nearly one plus the number.

e^{\frac{\beta \nu}{T}} - 1 \approx \frac{\beta \nu}{T}

Subtract the one: the denominator is, very nearly, the exponent itself.

\varrho_\nu \approx \frac{\alpha \nu^3}{\frac{\beta \nu}{T}} = \frac{\alpha \nu^3 T}{\beta \nu}

Put that into the formula. Dividing by a fraction is multiplying by its reciprocal.

\varrho_\nu = \frac{\alpha}{\beta} \nu^2 T

Cancel one factor

\nu

between the numerator and the denominator. The approximation becomes exact in the limit of large

T/\nu

, the paper's 'in der Grenze': the printed line.AlsoOverviewFull explanationEvery step

Read it in the paper