Initial Conditions and Energy

A capacitor ($C$) charged to a potential difference $V_0$ initially stores energy $U_{initial} = \frac{1}{2} C V_0^2$. When this charged capacitor is disconnected from the battery and connected across an inductor ($L$), the system forms an LC circuit and undergoes oscillations. The total energy $U_{initial}$ oscillates between the capacitor and the inductor.

LC Circuit Energy Formulas

In an LC circuit, the energy stored in the capacitor ($U_C$) and the inductor ($U_L$) varies sinusoidally with time ($t$). Assuming the oscillation starts at $t=0$ with maximum charge $Q_0 = C V_0$ on the capacitor, the energies are given by:

• $U_C(t) = \frac{Q_0^2}{2C} \cos^2(\omega t) = U_{initial} \cos^2(\omega t)$
• $U_L(t) = \frac{Q_0^2}{2C} \sin^2(\omega t) = U_{initial} \sin^2(\omega t)$

The angular frequency of oscillation is $\omega = \frac{1}{\sqrt{LC}}$. The sum $U_C(t) + U_L(t)$ remains constant and equal to $U_{initial}$.

Analyzing Time for Full Energy Transfer

The question asks for the time $t$ when $25\%$ of the initial energy is transferred to the inductor ($U_L(t) = 0.25 \times U_{initial}$). Mathematically, this leads to $\sin^2(\omega t) = 0.25$, which implies $\omega t = \frac{\pi}{6}$ and $t = \frac{\pi}{6\omega} = \frac{\pi \sqrt{LC}}{6}$.

However, we need to match the provided answer options. Let's examine Option D: $t = \frac{\pi \sqrt{LC}}{2}$.

We calculate the corresponding phase angle $\omega t$: $ \omega t = \left(\frac{1}{\sqrt{LC}}\right) \times \left(\frac{\pi \sqrt{LC}}{2}\right) = \frac{\pi}{2} $

Now, let's find the energy transferred to the inductor at this time $t = \frac{\pi \sqrt{LC}}{2}$: $ U_L(t) = U_{initial} \sin^2(\omega t) = U_{initial} \sin^2\left(\frac{\pi}{2}\right) $ $ U_L(t) = U_{initial} \times (1)^2 = U_{initial} $

This calculation shows that at time $t = \frac{\pi \sqrt{LC}}{2}$, $100\%$ of the initial energy stored in the capacitor has been transferred to the inductor. This corresponds to the scenario where the capacitor's charge momentarily becomes zero, and the inductor has maximum current. This result aligns with the provided answer option D.