The Traveling Electromagnetic Wave, Qualitatively — Question 7

Question 7

Counterpropagating waves can cancel E while reinforcing B at an event

Two vacuum plane waves meet. At the chosen event wave 1 travels +x with electric field $E_*\hat y$, while wave 2 travels -x with electric field $-E_*\hat y$.

1. Find each magnetic field and the total E and B.

2. Evaluate the total E cross B and explain why the total fields need not obey the single-wave relation E=cB.

Question 7 - Solution

Counterpropagating waves can cancel E while reinforcing B at an event

(a) Method and calculation

$$\mathbf B_1=\hat x\times E_*\hat y/c=(E_*/c)\hat z,\quad\mathbf B_2=(-\hat x)\times(-E_*\hat y)/c=(E_*/c)\hat z.$$$$\boxed{\mathbf E_{\mathrm{tot}}=0,\quad\mathbf B_{\mathrm{tot}}=(2E_*/c)\hat z}.$$

(b) Interpretation and further calculation

$$\boxed{\mathbf E_{\mathrm{tot}}\times\mathbf B_{\mathrm{tot}}=0}.$$$$|E_{\mathrm{tot}}|=0\ne c|B_{\mathrm{tot}}|=2|E_*|.$$

Physical check

The total is a superposition traveling in two directions, so a pointwise identity valid for one traveling wave cannot be imposed on it.