Q.Show that the total energy of LC circuit is $V = V_E = \dfrac{1}{2}\dfrac{q_m^2}{C}$.

LC Oscillations: The Electrical Pendulum

Imagine a pendulum. You pull it to one side, storing gravitational potential energy. Release it, and that energy smoothly converts to kinetic energy as it swings through the bottom, then back to potential energy as it rises on the other side. The energy sloshes back and forth at a natural frequency determined by the pendulum's length.

An LC circuit does exactly the same thing — but with electric and magnetic fields instead of height and speed.

The Physical Setup

Take a capacitor (stores energy in an electric field) and an inductor (stores energy in a magnetic field). Connect them in a loop. Initially, charge the capacitor fully — say the top plate is positive, bottom negative. There is no current yet; all the energy is in the capacitor's electric field.

Now close the switch. What happens?

The Oscillation, Step by Step

1. Capacitor discharges through the inductor. The voltage across the capacitor pushes current through the inductor. As current builds up, the inductor resists the change (Lenz's law) — it doesn't let the current jump instantly. The capacitor's electric field collapses gradually, feeding energy into the inductor's magnetic field.

2. Capacitor fully discharged, current at maximum. At this instant, the capacitor has zero voltage. All the energy now resides in the inductor's magnetic field. But the inductor wants to keep current flowing (it opposes a decrease just as it opposed an increase). So the current continues in the same direction.

3. Current charges the capacitor in the opposite polarity. The continuing current pushes charge onto the capacitor plates — now the top becomes negative, bottom positive. The inductor's magnetic field collapses, transferring energy back to the capacitor's electric field. Current decreases until it reaches zero.

4. Capacitor fully charged (opposite polarity), current zero. We're back to the starting situation, but with reversed polarity. The capacitor now discharges again, current flows the opposite way, and the cycle repeats.

The process repeats indefinitely (in an ideal circuit with zero resistance) — energy oscillates between the capacitor and the inductor at a natural frequency.

The Precise Statement

This is the natural frequency of free electrical oscillations in an LC circuit. $L$ is inductance in henries, $C$ is capacitance in farads. The period $T = 2\pi\sqrt{LC}$.

Why That Formula?

The derivation comes from equating the two forms of energy and applying Kirchhoff's voltage law. The voltage across the capacitor is $V = q/C$. The voltage across the inductor is $V = -L\,di/dt$. In a closed loop:

Since $i = dq/dt$, this becomes:

This is the exact same differential equation as a mass on a spring ($m\,d^2x/dt^2 + kx = 0$). The solution is sinusoidal oscillation with angular frequency $\omega = 1/\sqrt{LC}$, so $f = \omega/2\pi = 1/(2\pi\sqrt{LC})$.

Energy Exchange

At any instant, the total energy is constant (in an ideal circuit):

$U_{\text{total}} = \frac{1}{2}CV^2 + \frac{1}{2}LI^2 = \frac{1}{2}CV_0^2 = \frac{1}{2}LI_0^2$ …