The Traveling Electromagnetic Wave, Qualitatively — Question 2

Question 2

Electric and magnetic zeros coincide in one vacuum traveling wave

At a fixed point in a +x vacuum plane wave, $E_y(t)=E_0\sin\omega t$, with positive E0 and omega. There are no other waves or static fields.

1. Find Bz(t) and the first positive time when both fields reach their positive maxima.

2. Find both fields at t=pi/omega and determine the sign of E cross B immediately before and immediately after that time.

Question 2 - Solution

Electric and magnetic zeros coincide in one vacuum traveling wave

(a) Method and calculation

$$\boxed{B_z(t)=E_y(t)/c=(E_0/c)\sin\omega t}.$$$$\boxed{t_{\max}=\pi/(2\omega)}.$$

(b) Interpretation and further calculation

$$t=\pi/\omega\Rightarrow\boxed{E_y=B_z=0}.$$$$\mathbf E\times\mathbf B=\frac{E_0^2}{c}\sin^2\omega t\,\hat x\Rightarrow\boxed{+\hat x\text{ on both sides}}.$$

Physical check

The fields are in phase at a point; a time delay between their maxima would contradict this single traveling-wave model.