Question #43260

A microscopic magnetic dipole with magnetic moment $\mu$ and gyromagnetic ratio $\gamma$ is located at the origin. A uniform static magnetic field $\vec{B} = B_0 \hat{k}$ is present in the region. The dipole's axis makes a constant angle $\theta$ with the $z$-axis and precesses around it. A stationary circular coil of radius $R$ ($R$ is much larger than the dipole's dimensions), having $N$ turns and self-inductance $L_0$, is placed in the $xz$-plane, centered at the origin. The coil is connected to a capacitor of capacitance $C$ to form a closed LC circuit with negligible resistance. The capacitance $C$ is perfectly tuned such that the natural frequency of the LC circuit matches the Larmor precession frequency of the dipole. Assuming the circuit is completely unenergized at $t=0$, which of the following expressions represents the total electromagnetic energy $U(t)$ accumulated in the LC circuit as a function of time $t$? (Assume $t \gg \frac{1}{\gamma B_0}$ and the dipole's precession angle $\theta$ remains constant).