Question #43294

A rigid wire frame consists of two identical square loops, Loop 1 (left) and Loop 2 (right), each of side length $L$, sharing a common vertical branch. The common branch contains an ideal parallel-plate capacitor of capacitance $C$. The left vertical branch of Loop 1 has a resistance $R$, and the right vertical branch of Loop 2 has a resistance $R$. All other wire segments have negligible resistance. The frame lies in the $x-y$ plane. A spatially uniform magnetic field $\vec{B}_1 = 4\alpha t \hat{k}$ is confined to the region of Loop 1, and $\vec{B}_2 = \alpha t \hat{k}$ is confined to the region of Loop 2, where $\alpha > 0$ is a constant and $t \ge 0$ is time. At $t=0$, the capacitor is uncharged. Assuming the self-inductance of the loops is negligible, what is the magnitude of the displacement current $I_d$ through the capacitor as a function of time $t$?