Question #43335

A rigid rectangular coil of $N$ turns, area $A$, and mass $m$ is hinged along its top horizontal edge (aligned with the $z$-axis) and hangs vertically in the $yz$-plane under uniform gravity $g$ (acting in the $-y$ direction). The coil carries an alternating current $I(t) = I_0 \sin(\omega t)$. A uniform horizontal magnetic field $\vec{B} = B_0 \hat{i}$ is present in the region. The hinge is attached to a cart that accelerates horizontally in the $+x$ direction with an acceleration $\vec{a}(t) = a_0 \sin(\omega t) \hat{i}$. Assume the angular frequency $\omega$ is much smaller than the natural frequency of the coil as a pendulum, so the coil continuously maintains its quasi-static equilibrium position relative to the non-inertial frame of the cart. Also assume the magnetic torque is negligible in determining the angular position of the coil, and $a_0 \ll g$ so small angle approximations are valid. What is the magnitude of the time-averaged magnetic torque on the coil over one complete cycle?