Question #43244

Two coherent, plane electromagnetic waves of equal amplitude propagate in a homogeneous, isotropic, linear dielectric medium of relative permittivity $\kappa = 4$ and relative permeability $\mu_r = 1$. Wave 1 propagates along the $+x$-axis and is linearly polarized with its electric field vector $\vec{E}_1$ oscillating parallel to the $+z$-axis. Wave 2 propagates along the $+y$-axis and is linearly polarized with its electric field vector $\vec{E}_2$ oscillating parallel to the $+z$-axis. The electric field amplitude of each wave in the medium is $E_0$. Consider a point in the medium where the electric fields of the two waves interfere constructively to yield a local maximum amplitude. At this point, what is the ratio of the time-averaged polarization energy density $\langle u_P \rangle$ to the time-averaged magnetic energy density $\langle u_B \rangle$?

(Note: The polarization energy density is defined as $u_P = \frac{1}{2} \vec{P} \cdot \vec{E}$, where $\vec{P}$ is the macroscopic polarization vector).