The charge in an $LC$ circuit with negligible resistance oscillates as given by the equation $\frac{d^2q}{dt^2} + 16\pi^2q = 0$. If the charge is maximum equal to $24\,\mu C$ at $t = 0$,find the charge at $t = \frac{1}{12}\,s$ in $\mu C$.

$A$ $1 \, \mu F$ capacitor is charged to $50 \, V$. The charging battery is then disconnected and a $10 \, mH$ coil is connected across the capacitor so that $LC$ oscillations occur. What is the maximum current in the coil (in $, A$)? Assume that the circuit contains no resistance.

$A$ $4 \mu F$ capacitor is charged to $10 \ V$. The battery is then disconnected and a pure $10 \ mH$ coil is connected across the capacitor so that $LC$ oscillations are set up. The maximum current in the coil is (in $A$)

In an $LC$ circuit,the capacitor has a maximum charge $q_0$. The value of $\left| \frac{di}{dt} \right|_{\max }$ is

In an oscillating $LC$-circuit,$L = 3 \ mH$ and $C = 2.7 \ \mu F$. At $t = 0$,the charge on the capacitor is zero and the current is $2 \ A$. The maximum charge that will appear on the capacitor will be

An $LC$ circuit contains a $20 \; mH$ inductor and a $50 \; \mu F$ capacitor with an initial charge of $10 \; mC$. The resistance of the circuit is negligible. Let the instant the circuit is closed be $t=0$.
$(a)$ What is the total energy stored initially? Is it conserved during $LC$ oscillations?
$(b)$ What is the natural frequency of the circuit?
$(c)$ At what time is the energy stored $(i)$ completely electrical (i.e.,stored in the capacitor)? $(ii)$ completely magnetic (i.e.,stored in the inductor)?
$(d)$ At what times is the total energy shared equally between the inductor and the capacitor?
$(e)$ If a resistor is inserted in the circuit,how much energy is eventually dissipated as heat?