Question #43886

A uniform plane electromagnetic wave of wavelength $\lambda$ propagates in vacuum along the $+z$-direction. It is normally incident on a non-ideal, reactive metallic surface located at the plane $z=0$. Unlike a perfect electric conductor, this surface does not produce a phase reversal of exactly $\pi$ for the electric field. Instead, the boundary conditions dictate that the effective plane of reflection—where the electric field amplitude is strictly zero—is located at an "end correction" depth of $\delta = \frac{\lambda}{12}$ inside the metal ($z > 0$). A standing wave pattern is formed in the vacuum region ($z < 0$). Let $d_1$ be the distance from the surface ($z=0$) to the first electric field antinode, and $d_2$ be the distance from the surface to the first electric field node, both located strictly in the vacuum region ($z < 0$). What is the ratio $\frac{d_2}{d_1}$?