The Traveling Electromagnetic Wave, Qualitatively — Question 1

Question 1

A simultaneous field reversal preserves the direction of a traveling wave

A vacuum plane wave travels along +x. At one event its electric field is $\mathbf E=-E_*\hat y$ with positive E*. Use $\mathbf B=\hat n\times\mathbf E/c$, where n is the propagation unit vector.

1. Find B and the direction of the cross product E cross B.

2. Compare reversing both fields with reversing only B. Which change preserves propagation along +x?

Question 1 - Solution

A simultaneous field reversal preserves the direction of a traveling wave

(a) Method and calculation

$$\mathbf B=\hat x\times(-E_*\hat y)/c=\boxed{-(E_*/c)\hat z}.$$$$\mathbf E\times\mathbf B=(E_*^2/c)\hat x.$$

(b) Interpretation and further calculation

$$(-\mathbf E)\times(-\mathbf B)=\mathbf E\times\mathbf B\Rightarrow\boxed{\text{same direction}}.$$$$\mathbf E\times(-\mathbf B)=-\mathbf E\times\mathbf B\Rightarrow\boxed{\text{opposite direction}}.$$

Physical check

In a sinusoidal traveling wave both fields reverse every half-cycle; the energy-transport direction remains fixed.