Question #42979

A stationary loop of inductance $L = 2.0 \text{ H}$ and zero resistance is placed in a uniform magnetic field. The initial magnetic flux through the loop is $\Phi_0 = 20 \text{ Wb}$. A thermally insulated resistor is connected in series with the loop, but is initially short-circuited by a superconducting switch. At $t=0$, the external magnetic field is abruptly reduced to zero. Immediately after, the switch is opened so that the induced current flows through the resistor. The resistor has an initial resistance $R_0 = 1.0 \ \Omega$ and a constant heat capacity $C = 5.0 \text{ J/K}$. Its resistance varies with its temperature $T$ as $R(T) = R_0 [1 + \alpha (T - T_0)]$, where $T_0$ is the initial temperature and $\alpha = 0.05 \text{ K}^{-1}$. Assuming no heat is lost to the surroundings, find the time $t$ it takes for the current to drop to half of its initial value.