Maxwell's Rainbow — Question 1

Question 1

Widely separated spectral scales share one vacuum propagation speed

Three vacuum waves have wavelengths $\lambda_A=3.0$ m, $\lambda_B=30\,\mu$m and $\lambda_C=300$ nm. Use $c=3.00\times10^8$ m/s. Classify A as radio, B as infrared and C as ultraviolet for this problem.

1. Find the three frequencies and their ratios.

2. Find the travel time of each wave over 900 km and explain why the shortest wavelength does not arrive first.

Question 1 - Solution

Widely separated spectral scales share one vacuum propagation speed

(a) Method and calculation

$$f=c/\lambda\Rightarrow\boxed{f_A=10^8,\ f_B=10^{13},\ f_C=10^{15}\ \mathrm{Hz}}.$$$$\boxed{f_B/f_A=10^5,\quad f_C/f_B=10^2}.$$

(b) Interpretation and further calculation

$$t=L/c=\frac{9.00\times10^5}{3.00\times10^8}\,\mathrm s.$$$$\boxed{t_A=t_B=t_C=3.00\,\mathrm{ms}}.$$

Physical check

Wavelength and frequency differ across the spectrum; the vacuum speed is common to all three.