Question #43894

An RL circuit consists of a battery of EMF $V_0$, a switch $S$, a resistor $R$, and an inductor $L$ in series. A long cylindrical capacitor with inner radius $a$ and outer radius $b$ is connected in parallel with the inductor. The capacitance is negligible and does not affect the RL circuit's transient current. A small biconvex electro-optic lens, with both radii of curvature equal to $R_c$, is placed between the cylindrical plates at a radial distance $r_0$ from the central axis. The refractive index of the lens material varies with the local electric field magnitude $E$ as $n(t) = n_0 + \beta E(t)$, where $n_0 = 1.5$ and $\beta$ is a positive constant. Given that $\frac{\beta V_0}{r_0 \ln(b/a)} = 0.5$. At $t=0$, the switch $S$ is closed. A stationary point object is placed on the principal axis at a distance $3 R_c$ in front of the lens. Assuming the lens is small enough that the electric field is uniform across it, find the speed of the image at time $t = \frac{L}{R} \ln 2$.