Question 130: AC fundamentals

Question

A capacitor C is first charged fully with potential difference of $V_0$ and disconnected from the battery. The charged capacitor is connected across an inductor having inductance L. In $t$ s, 25% of the initial energy in the capacitor is transferred to the inductor. The value of $t$ is ________ s.

Accepted Answer

When a charged capacitor is connected to an inductor, the energy oscillates between the electric field of the capacitor and the magnetic field of the inductor.
The charge q on the capacitor at any time t is given by :
$$ \mathrm{q}(\mathrm{t})=\mathrm{Q}_0 \cos (\omega \mathrm{t}) $$
Where :
$\mathrm{Q}_0$ is the initial maximum charge.
$\omega$ is the angular frequency, defined as $\omega=\frac{1}{\sqrt{\mathrm{LC}}}$.
Initially all energy is in the capacitor.
$$ \mathrm{U}_0=\frac{\mathrm{Q}_0^2}{2 \mathrm{C}} $$
Energy in capacitor at time t is :
$$ U_C(t)=\frac{q(t)^2}{2 C}=\frac{\left(Q_0 \cos (\omega t)\right)^2}{2 C}=U_0 \cos ^2(\omega t) $$
By conservation of energy the energy stored in inductor as magnetic field,
$$\mathrm{U}_{\mathrm{L}}=\mathrm{U}_0-\mathrm{U}_{\mathrm{C}} $$
$$\Rightarrow $$ $$ \mathrm{U}_{\mathrm{L}}(\mathrm{t})=\mathrm{U}_0\left(1-\cos ^2(\omega \mathrm{t})\right)=\mathrm{U}_0 \sin ^2(\omega \mathrm{t})$$
It is that $25 \%$ of the initial energy is transferred to the inductor :
$$ \mathrm{U}_{\mathrm{L}}=\frac{1}{4} \mathrm{U}_0 $$
$$\Rightarrow $$ $U_0 \sin ^2(\omega t)=\frac{1}{4} U_0$
$$\Rightarrow $$  $\sin ^2(\omega \mathrm{t})=\frac{1}{4} $
$$\Rightarrow $$$ \sin (\omega \mathrm{t})=\frac{1}{2} $
$$\Rightarrow $$$ \omega \mathrm{t}=\frac{\pi}{6} $
$$\Rightarrow $$$ \frac{\mathrm{t}}{\sqrt{\mathrm{LC}}}=\frac{\pi}{6} $
$$\Rightarrow $$ $ \mathrm{t}=\frac{\pi \sqrt{\mathrm{LC}}}{6}$

Answer

$\frac{\pi \sqrt{LC}}{3}$

Answer

$\frac{\pi \sqrt{LC}}{2}$

Answer

$\pi \sqrt{\frac{LC}{2}}$

Answer

$\frac{\pi \sqrt{LC}}{6}$

AC fundamentals

A capacitor C is first charged fully with potential difference of V_0 and disconnected from the battery. The charged capacitor is connected across an inductor having inductance L. In t s, 25% of the initial energy in the capacitor is transferred to the inductor. The value of t is ________ s.

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