The Traveling Electromagnetic Wave, Qualitatively — Question 4

Question 4

Oblique propagation fixes the transverse magnetic-field components

A vacuum plane wave propagates along $\hat n=(3\hat x+4\hat y)/5$. At an event its electric field is $E_*\hat z$. Use the right-handed Cartesian convention.

1. Find the vector magnetic field and verify that it is transverse.

2. Find the direction of E cross B and check the magnitude relation.

Question 4 - Solution

Oblique propagation fixes the transverse magnetic-field components

(a) Method and calculation

$$\mathbf B=\frac{\hat n\times\mathbf E}{c}=\boxed{\frac{E_*}{5c}(4\hat x-3\hat y)}.$$$$\hat n\cdot\mathbf B=\frac{E_*}{25c}(12-12)=0.$$

(b) Interpretation and further calculation

$$\mathbf E\times\mathbf B=\frac{E_*^2}{5c}(3\hat x+4\hat y)=\frac{E_*^2}{c}\hat n.$$$$\boxed{|\mathbf B|=|E_*|/c}.$$

Physical check

Transversality refers to the propagation direction, which need not coincide with a coordinate axis.