The Traveling Electromagnetic Wave, Quantitatively — Question 8

Question 8

An elliptically polarized wave has a time-varying field magnitude

A +z vacuum wave has $\mathbf E=E_0[2\cos\phi\,\hat x+\sin\phi\,\hat y]$, where $\phi=kz-\omega t$ and omega=ck. E0 is positive.

1. Find the magnetic vector field.

2. Find the minimum and maximum magnitudes of E and B and the phases where they occur.

Question 8 - Solution

An elliptically polarized wave has a time-varying field magnitude

(a) Method and calculation

$$\mathbf B=\hat z\times\mathbf E/c=\boxed{\frac{E_0}{c}[-\sin\phi\,\hat x+2\cos\phi\,\hat y]}.$$$$|E|^2=E_0^2(4\cos^2\phi+\sin^2\phi)=E_0^2(1+3\cos^2\phi).$$

(b) Interpretation and further calculation

$$\boxed{|E|_{\min}=E_0\text{ at }\phi=\pi/2+m\pi;\quad |E|_{\max}=2E_0\text{ at }\phi=m\pi}.$$$$\boxed{|B|_{\min}=E_0/c,\quad|B|_{\max}=2E_0/c}.$$

Physical check

Each Cartesian component is sinusoidal, but the vector magnitude is neither constant nor a signed sinusoid.