Energy Transport and the Poynting Vector — Question 6

Question 6

Circular polarization has constant instantaneous energy flow

Compare a +z circular wave $\mathbf E_C=E_0(\cos\phi\,\hat x+\sin\phi\,\hat y)$ with a linear wave $\mathbf E_L=E_0\cos\phi\,\hat x$. Both propagate in vacuum.

1. Find the instantaneous magnitude of the Poynting vector for each wave.

2. Compare their average intensities and find the circular component amplitude needed to give the same intensity as the stated linear wave.

Question 6 - Solution

Circular polarization has constant instantaneous energy flow

(a) Method and calculation

$$\mathbf B=\hat z\times\mathbf E/c\Rightarrow S=\epsilon_0c|E|^2.$$$$\boxed{S_C=\epsilon_0cE_0^2,\quad S_L=\epsilon_0cE_0^2\cos^2\phi}.$$

(b) Interpretation and further calculation

$$\boxed{I_C=\epsilon_0cE_0^2=2I_L}.$$$$\epsilon_0cE_{C,\mathrm{eq}}^2=\tfrac12\epsilon_0cE_0^2\Rightarrow\boxed{E_{C,\mathrm{eq}}=E_0/\sqrt2}.$$

Physical check

Equal component amplitudes do not mean equal total mean-square field when one wave has two orthogonal components.