Quiz 1.9

Let us consider the plane waves in complex notation of Eq. (1.25) with

$$\vec{E}_0=E_0\uvec{e},$$   and$$\qquad \uvec{e}=\frac{1}{\sqrt{2}}\left(\uvec{x}\pm i\uvec{y}\rig... ...1}{\sqrt{2}}\left(\uvec{x}+\uvec{y} \exp\left(\pm i\frac{\pi}{2}\right)\right)$$

Then

$$\vec{E}(\vec{r},t)=\frac{E_0}{\sqrt{2}} \left(\uvec{x}\exp(i\phi)... ...ght)\\ =\frac{E_0}{\sqrt{2}} \left(\uvec{x}\cos\phi\mp\uvec{y}\sin\phi\right)$$

and since $\omega\vec{B}_0=\uvec{z}\cross\vec{E}_0$

$$\vec{B}(\vec{r},t)=\frac{B_0}{\sqrt{2}} \left(\uvec{y}\exp(i\phi)... ...right) =\frac{B_0}{\sqrt{2}} \left(\pm\uvec{x}\sin\phi+\uvec{y}\cos\phi\right)$$

We check that $\vert\vec{E}\vert=E_0$, $\vert\vec{B}\vert=B_0$ and $\vec{E}\cdot\vec{B}=0$, therefore $\vec{E}$ and $\vec{B}$ are orthogonal and describe circles in the $(x,y)$-plane.