Question #43165

A rigid non-conducting rod of length $L$ and negligible mass rotates in a horizontal plane about a frictionless vertical axis through one end (the pivot). A central hub at the pivot has a moment of inertia $I_0 = m L^2$. The system is given an initial angular velocity $\omega_0$. A uniform vertical magnetic field $B$ is present everywhere. A small conducting bead of mass $m$ slides outward along the rod at a constant speed $v_0$ relative to the rod, starting from the pivot at $t=0$. A fixed conducting plate of area $A$ is attached to the free end of the rod ($r=L$). The bead acts as the second plate of area $A$, forming a parallel-plate capacitor with a time-varying gap $L - r(t)$. To establish a potential difference, a thin conducting wire connects the pivot to the bead, and another separate thin conducting wire connects the pivot to the fixed end plate. Assume the wires have negligible mass, the electrical energy stored is negligible compared to the kinetic energy, and fringing effects are ignored. What is the magnitude of the displacement current in the gap at the instant the bead passes the midpoint of the rod ($r = L/2$)?