Question #43584

A linearly polarized plane electromagnetic wave of uniform intensity $I_0$ and wavelength $\lambda$ is incident normally on a perfectly opaque, flat screen lying in the $x$-$y$ plane. The screen has a long, narrow slit of width $a$ parallel to the $y$-axis.

Due to diffraction, the transmitted wave spreads out. In the far-field ($r \gg a$), the time-averaged Poynting vector is directed radially outward from the slit. Its magnitude at a distance $r$ and angle $\theta$ (measured from the $z$-axis in the $x$-$z$ plane) is given by: $|\langle \vec{S} \rangle| = \frac{2 P_0}{\pi r} \cos^2\theta \quad \text{for } -\frac{\pi}{2} \le \theta \le \frac{\pi}{2}$ where $P_0$ is the total transmitted power per unit length of the slit (in $\text{W/m}$).

A perfectly absorbing, rigid semi-cylindrical surface of radius $R$ ($R \gg a$) is placed symmetrically behind the slit (centered on the $y$-axis) to completely intercept the transmitted wave. What is the net radiation force per unit length exerted on this semi-cylindrical surface? (Let $c$ be the speed of light in vacuum)