Question #85945

A circular mesoscopic ring of radius $r$ is placed in a uniform magnetic field $B(t)$ perpendicular to its plane. The magnetic field varies with time $t$ as follows: it increases linearly from $0$ to $B_0$ in time $T_0$, and then immediately decreases linearly from $B_0$ back to $0$ in time $T_0/2$. Due to the discrete nature of the lattice, continuous eddy currents do not flow. Instead, charge transport occurs via discrete quantum hops. The number of electrons $dN$ that complete a full revolution around the ring during a time interval $dt$ is given by $dN = \frac{1}{\Phi_0} \left| \frac{d\Phi}{dt} \right| dt$, where $\Phi(t)$ is the magnetic flux and $\Phi_0$ is a fundamental flux constant. The energy dissipated by each electron during one revolution is equal to the work done on it by the induced electromotive force (EMF). Which of the following expressions gives the total energy dissipated in the ring during the entire time interval $t=0$ to $t=1.5T_0$?