The Traveling Electromagnetic Wave, Quantitatively — Question 5

Question 5

Transversality reconstructs an unknown component of an oblique wave

A source-free vacuum plane wave has direction $\hat n=(\hat x+2\hat y+2\hat z)/3$ and electric amplitude vector $\mathbf E_0=a\hat x+b\hat y$, with known positive b and no z component.

1. Find a from the electric divergence constraint.

2. Find the magnetic amplitude vector and verify its magnitude is E0/c.

Question 5 - Solution

Transversality reconstructs an unknown component of an oblique wave

(a) Method and calculation

$$\hat n\cdot\mathbf E_0=(a+2b)/3=0\Rightarrow\boxed{a=-2b}.$$$$\boxed{\mathbf E_0=b(-2\hat x+\hat y),\quad|E_0|=\sqrt5b}.$$

(b) Interpretation and further calculation

$$\mathbf B_0=\hat n\times\mathbf E_0/c=\boxed{\frac{b}{3c}(-2\hat x-4\hat y+5\hat z)}.$$$$|B_0|=\frac{b}{3c}\sqrt{4+16+25}=\boxed{\sqrt5b/c=|E_0|/c}.$$

Physical check

An arbitrary missing component cannot be set to zero; the divergence law constrains the entire amplitude vector.