Question #43443

A closed electrical circuit consists of a capacitor of capacitance $C$ and two wire loops, Loop 1 and Loop 2, connected in series. The total self-inductance of the two loops is $L_0$. Loop 1 is a small planar loop of area $A_1$ fixed at the origin in the $xy$-plane. Loop 2 is a wire bent into the shape of an open hemispherical dome of radius $a$. The circular base of the dome has area $A_2 = \pi a^2$. The center of this base is located at a distance $r$ from the origin along the $z$-axis, where $r \gg a$ and $r \gg \sqrt{A_1}$. Loop 2 is forced to rotate about an axis parallel to the $x$-axis passing through the center of its base, with a constant angular velocity $\omega$. At $t=0$, the circular base of Loop 2 is parallel to the $xy$-plane. Assume the resistance of the circuit is negligible and that at $t=0$, the magnetic fluxes produced by the two loops reinforce each other. Let $q(t)$ be the instantaneous charge on the capacitor, and define a constant $M_0 = \frac{\mu_0 A_1 A_2}{2\pi r^3}$. By applying Gauss's Law for Magnetism to determine the flux linkage, identify the correct differential equation governing $q(t)$.