Question #43322

Consider an LC circuit with an ideal inductor of inductance $L$ and a non-linear capacitor. The potential difference $V$ across the capacitor is related to the charge $q$ on its plates by $V(q) = \frac{q}{C_0} - \beta q^3$, where $C_0$ and $\beta$ are positive constants, and $\beta$ is sufficiently small such that $\beta C_0 q^2 \ll 1$. The circuit oscillates with a maximum charge $q_0$ on the capacitor. By mapping this system to an analogous mechanical oscillator (where charge $q$ corresponds to angular displacement $\theta$, and $L$ corresponds to moment of inertia $I$), one can analyze its anharmonic behavior similar to a large-angle physical pendulum. Let $T_0 = 2\pi\sqrt{LC_0}$ be the period of the linear system (where $\beta = 0$). Which of the following gives the fractional change in the time period $\frac{\Delta T}{T_0} = \frac{T - T_0}{T_0}$ to the first order in $\beta$, and correctly identifies the geometric feature of the potential energy $U(q)$ curve responsible for this change?