Question #43282

An LC oscillator circuit consists of an ideal capacitor of capacitance $C$ and an inductor of inductance $L$. The inductor is a solenoid of length $l$, cross-sectional area $A$, and $N$ turns, completely filled with a perfectly laminated magnetic core of permeability $\mu$ (such that $L = \frac{\mu N^2 A}{l}$). The core material exhibits a temperature-dependent hysteresis B-H loop, transitioning from a soft to a hard magnetic material as the absolute temperature $T$ increases. The area of the B-H loop (representing the energy dissipated per unit volume per cycle) is given by $A_H = \frac{\alpha T}{\mu} B_{max}^2$, where $B_{max}$ is the peak magnetic field in the core during the cycle and $\alpha$ is a positive constant with units of $K^{-1}$. Assuming the damping is weak ($\alpha T \ll 1$), which of the following correctly gives the equivalent series resistance $R_{eq}$ of the inductor due to hysteresis loss, and the time $t_0$ taken for the current amplitude to decrease to $1/e$ of its initial value?