Purcell Problem 9.19

Plane electromagnetic waves are self-consistent patterns where electric and magnetic fields oscillate together, perpendicular to each other and to the direction of travel. Since your wave travels in $-\hat{x}$ and $\vec{E}$ must avoid $\hat{z}$, it must point in the $\hat{y}$ direction—then $\vec{B}$ automatically points in $\hat{z}$ by the right-hand rule. The key is using the wave number $k = 2\pi f/c$ and phase $kx + \omega t$ (note the $+$ sign for $-\hat{x}$ propagation) to write both fields with the same frequency and a fixed amplitude ratio $E_0/B_0 = c$.

Plane electromagnetic waves are self-consistent patterns where electric and magnetic fields oscillate together, perpendicular to each other and to the direction of travel. Since your wave travels in $-\hat{x}$ and $\vec{E}$ must avoid $\hat{z}$, it must point in the $\hat{y}$ direction—then $\vec{B}$ automatically points in $\hat{z}$ by the right-hand rule. The key is using the wave number $k = 2\pi f/c$ and phase $kx + \omega t$ (note the $+$ sign for $-\hat{x}$ propagation) to write both fields with the same frequency and a fixed amplitude ratio $E_0/B_0 = c$.