Question #42978

A plane electromagnetic wave with electric field $\vec{E}_{\text{inc}} = E_0 \cos(\omega t - kx) \hat{j}$ is normally incident on a thin, infinite conducting grid located at $x=0$. The grid has zero electrical resistance but possesses a constant surface self-inductance $L_s$ (in Henry). The electric field at the grid induces a surface current density $\vec{K}(t) = K(t) \hat{j}$ (in A/m). The dynamics of this current are governed by $E(0,t) = L_s \frac{dK}{dt}$, where $E(0,t)$ is the net electric field at $x=0$.

The induced current generates secondary electromagnetic waves propagating away from the grid in both directions. Assuming the surrounding medium is vacuum (permeability $\mu_0$, speed of light $c$), the reflected electric field for $x < 0$ is given by $\vec{E}_{\text{ref}} = E_r \cos(\omega t + kx + \theta) \hat{j}$, where $E_r > 0$.

For low frequencies such that $\omega L_s \ll \mu_0 c$, what is the approximate phase shift $\theta$ of the reflected wave?