Question #43342

Two identical closed electrical loops, each comprising a series combination of a resistor of resistance $R$ (in $\Omega$), an inductor of inductance $L$ (in $\text{H}$), and a capacitor of capacitance $C$ (in $\text{F}$), are magnetically coupled with a mutual inductance $M = kL$ (in $\text{H}$), where $0 < k < 1$ is the dimensionless coupling coefficient. The resistance is sufficiently small ($R \ll \sqrt{L/C}$) such that the system exhibits two distinct underdamped normal modes of oscillation. Let $\Delta\omega_L$ and $\Delta\omega_H$ (both in $\text{rad s}^{-1}$) be the half-power bandwidths of the lower-frequency and higher-frequency normal modes, respectively. If $\Delta\omega_0 = R/L$ (in $\text{rad s}^{-1}$) is the bandwidth of a single uncoupled loop, which of the following expressions correctly represents the difference $\Delta\omega_H - \Delta\omega_L$?