Question #43391

A long non-conducting solid cylinder of radius $R$ has a time-varying volume charge density given by $\rho(r, t) = \rho_0 \left(\frac{r}{R}\right) t$, where $\rho_0$ is a positive constant, $r$ is the radial distance from the central axis, and $t$ is time. The cylinder rotates about its central longitudinal axis with a constant angular velocity $\omega_1$. A closed conducting loop is formed in a plane perpendicular to the cylinder's axis (far from the ends). The loop consists of a stationary radial wire from the axis to $r=R$, a stationary circular arc of radius $R$, and a conducting rod of length $R$ pivoted at the axis. The rod rotates with a constant angular velocity $\omega_2$ in the same sense as $\omega_1$, continuously expanding the sector area of the loop. At $t=0$, the rod coincides with the stationary radial wire. Find the magnitude of the total induced electromotive force (EMF) in the loop at time $t$. Assume all quantities are in SI units and ignore self-inductance.