Question #43325

A plane electromagnetic wave propagating in a vacuum has a wave vector $\vec{k}$ lying in the $xy$-plane. The wave vector makes a small angle $\alpha$ with the positive $x$-axis. The wave is linearly polarized with its electric field vector parallel to the $z$-axis, and its root-mean-square (RMS) electric field magnitude is $E_{\text{rms}} = 10\pi \text{ V/m}$. The wave is incident upon an opaque screen at $x = 0$ containing a single narrow slit of width $b$, extending from $y = -b/2$ to $y = b/2$. According to Huygens' principle, the slit acts as a continuous array of secondary wavelet sources. Assuming the small angle approximation ($\sin\alpha \approx \alpha$) is valid and $\alpha = \frac{\lambda}{2b}$ (where $\lambda$ is the wavelength), calculate the time-averaged intensity $I$ of the electromagnetic wave at a point $P$ located on a distant screen at $x = D$ ($D \gg b$) exactly on the $x$-axis ($y = 0$). Let $c$ be the speed of light and $\epsilon_0$ be the vacuum permittivity. Assume standard Fraunhofer diffraction conditions where the secondary wavelets have uniform amplitude across the slit.