The Traveling Electromagnetic Wave, Qualitatively — Question 10

Question 10

A faster temporal oscillation requires a proportionally shorter vacuum wavelength

Two sinusoidal vacuum plane waves have equal electric amplitudes and frequencies f and 3f. Each travels along +x with the same polarization. Compare them at equal phase.

1. Compare their periods, wavelengths, wave numbers, and magnetic amplitudes.

2. Compare maximum time derivatives and maximum spatial derivatives of E. Explain how both derivative changes preserve the propagation speed.

Question 10 - Solution

A faster temporal oscillation requires a proportionally shorter vacuum wavelength

(a) Method and calculation

$$\boxed{T_2/T_1=1/3,\quad\lambda_2/\lambda_1=1/3,\quad k_2/k_1=3}.$$$$B_0=E_0/c\Rightarrow\boxed{B_{02}/B_{01}=1}.$$

(b) Interpretation and further calculation

$$|\partial_tE|_{\max}=\omega E_0,\quad|\partial_xE|_{\max}=kE_0\Rightarrow\boxed{\text{both ratios}=3}.$$$$\boxed{\frac{|\partial_tE|_{\max}}{|\partial_xE|_{\max}}=\omega/k=c\text{ for both}}.$$

Physical check

Greater frequency changes the space and time scales together; it does not require a larger field amplitude.