(i) A plane electromagnetic wave in a vacuum has an electric field

$$\mathbf{E}=\left(E_{1}, E_{2}, 0\right) \cos (k z-\omega t),$$

referred to cartesian axes $(x, y, z)$. Show that this wave is plane polarized and find the orientation of the plane of polarization. Obtain the corresponding plane polarized magnetic field and calculate the rate at which energy is transported by the wave.

(ii) Suppose instead that

$$\mathbf{E}=\left(E_{1} \cos (k z-\omega t), E_{2} \cos (k z-\omega t+\phi), 0\right),$$

with $\phi$ a constant, $0<\phi<\pi$. Show that, if the axes are now rotated through an angle $\psi$ so as to obtain an elliptically polarized wave with an electric field

$$\mathbf{E}^{\prime}=\left(F_{1} \cos (k z-\omega t+\chi), F_{2} \sin (k z-\omega t+\chi), 0\right),$$

then

$$\tan 2 \psi=\frac{2 E_{1} E_{2} \cos \phi}{E_{1}^{2}-E_{2}^{2}} .$$

Show also that if $E_{1}=E_{2}=E$ there is an elliptically polarized wave with

$$\mathbf{E}^{\prime}=\sqrt{2} E\left(\cos \left(k z-\omega t+\frac{1}{2} \phi\right) \cos \frac{1}{2} \phi, \sin \left(k z-\omega t+\frac{1}{2} \phi\right) \sin \frac{1}{2} \phi, 0\right) .$$