Question #92558

A monochromatic light beam of wavelength $\lambda$ and total power $P_0$ travels along the $+y$-direction and passes through a rectangular slit of width $a$ (parallel to the $z$-axis). The beam falls on a detector plate of mass $m$ and length $L_z$ (along the $z$-axis) placed at a distance $D$ in the $x-z$ plane. Assume the optical intensity $I(x)$ of the central diffraction maximum is uniform over its width $2x_0$ (where $x_0 = \frac{\lambda D}{a}$) and zero outside. The light generates a local surface current density on the detector given by $\vec{K} = \eta I(x) \hat{i}$ (in A/m), where $\eta$ is a positive constant. The detector is suspended by a spring of stiffness $k$ and is constrained to move only along the $z$-axis. A non-uniform magnetic field $\vec{B} = B_0 \left(1 - \frac{x^2}{c^2}\right) (1 + \beta z) \hat{j}$ exists in the region, where $B_0, c,$ and $\beta$ are positive constants, and $z$ is the coordinate of the detector's center of mass. Assuming $k$ is sufficiently large and neglecting damping, what is the angular frequency $\omega$ of the small vertical oscillations of the detector along the $z$-axis?