Maxwell's Rainbow — Question 7

Question 7

A spectrum uniform per hertz is not uniform per unit wavelength

Over a finite band a source has constant spectral power density $P_f=P_0$ watts per hertz. Define $P_\lambda$ by equal power in corresponding positive intervals: $P_f|df|=P_\lambda|d\lambda|$. In vacuum $f=c/\lambda$.

1. Derive the wavelength spectral density.

2. For two equal narrow wavelength intervals centered at lambda and 2lambda within the band, compare their powers to leading order in interval width.

Question 7 - Solution

A spectrum uniform per hertz is not uniform per unit wavelength

(a) Method and calculation

$$\frac{df}{d\lambda}=-\frac{c}{\lambda^2},\quad P_\lambda=P_f\left|\frac{df}{d\lambda}\right|.$$$$\boxed{P_\lambda=P_0c/\lambda^2}.$$

(b) Interpretation and further calculation

$$\Delta P_1\simeq P_0c\Delta\lambda/\lambda^2,\quad\Delta P_2\simeq P_0c\Delta\lambda/(2\lambda)^2.$$$$\boxed{\Delta P_1/\Delta P_2\simeq4}.$$

Physical check

A spectral-density graph depends on the horizontal variable; the integrated physical power is invariant.