Question #43297

A rigid, rectangular superconducting loop of mass $m$, self-inductance $L$, and area $S$, with its sides parallel to the $x$ and $y$ axes, is attached to a horizontal ideal spring of spring constant $k$. The loop is constrained to oscillate on a frictionless horizontal surface along the $x$-axis. A non-uniform, time-independent magnetic field is present in the region, given by $\vec{B} = B_0 (1 + \alpha x) \hat{k}$, where $B_0$ and $\alpha$ are positive constants. The plane of the loop is parallel to the $xy$-plane. At $t = 0$, the loop is at the equilibrium position of the spring ($x = 0$) with velocity $v_0$, and the current in the loop is exactly zero. Which of the following expressions represents the angular frequency $\omega$ of the subsequent harmonic oscillations?