# UY1: L-C Circuit

We shall look at an important circuit – one containing an inductor and a capacitor.

From Kirchhoff’s loop rule,

\begin{aligned} -L\frac{di}{dt} &=\frac{q}{C} \\ L\frac{di}{dt} + \frac{q}{C} &= 0 \\ \frac{d^{2}q}{dt^{2}} + \frac{1}{LC}q &= 0 \end{aligned}

Solving the differential equation above, will give:

\begin{aligned} q &= Q \cos{\left( \omega t + \phi \right)} \\ i &=-\omega Q \sin{\left( \omega t + \phi \right)} \end{aligned}

where the angular frequency, $\omega = \sqrt{\frac{1}{LC}}$.

The constants Q and $\phi$ are determined by the initial conditions.

Recall that the electric field energy in the capacitor is given by $\frac{q^{2}}{2C}$, which is:

$$\frac{q^{2}}{2C} = \frac{Q^{2}}{2C} \cos^{2}{\left( \omega t + \phi \right)}$$

The L-C circuit is a conservative system – the total energy in the L-C circuit is constant. Let’s calculate the energy stored in the inductor at any time:

\begin{aligned} \frac{Q^{2}}{2C}-\frac{q^{2}}{2C} &= \frac{Q^{2}}{2C} \left[ 1-\cos^{2}{\left( \omega t + \phi \right)} \right] \\ &= \frac{Q^{2}}{2C} \sin^{2}{\left( \omega t + \phi \right)} \\ &= \frac{1}{2} L \omega^{2} Q^{2} \sin^{2}{\left( \omega t + \phi \right)} \\ &= \frac{1}{2} L i^{2} \end{aligned}

This is the magnetic-field energy in the inductor at any time. The total energy oscillates between the electric and magnetic forms.

### Step-by-step process in a L-C circuit

1. Capacitor has maximum charge so no current flows.
2. Capacitor discharges, current flows counter-clockwise and increases.
3. The capacitor has zero charge. Current is at its maximum value and flows counter-clockwise
4. Capacitor charges with opposite polarity. Current flow counter-clockwise and decreases.
5. Capacitor has maximum charge so no current flows.
6. Capacitor discharges, current flows clockwise and increases.
7. Capacitor has zero charge. Current is at maximum value and flows clockwise.
8. Capacitor charges with original polarity. Current flows clockwise and decreases.

In an oscillating L-C circuit, energy is transferred between magnetic energy in the inductor $\left( U_{B} \right)$ and electric energy in the capacitor $\left( U_{E} \right)$ – electrical oscillation.

The oscillating L-C circuit is very similar to a mass-spring system which undergoes simple harmonic motion. In the table below, we shall compare the oscillation of a mass-spring system with electrical oscillation in an L-C circuit.

Mass-Spring System Inductor-Capacitor Circuit
Kinetic energy = $\frac{1}{2} mv^{2}_{x}$ Magnetic energy = $\frac{1}{2} L i^{2}$
Potential energy = $\frac{1}{2} k x^{2}$ Electric energy = $\frac{q^{2}}{2C}$
$\frac{1}{2} mv_{x}^{2} + \frac{1}{2} kx^{2} = \frac{1}{2} kA^{2}$ $\frac{1}{2} Li^{2} + \frac{q^{2}}{2C} = \frac{Q^{2}}{2C}$
$v_{x} = \pm \sqrt{\frac{k}{m}} \sqrt{A^{2} – x^{2}}$ $i = \pm \sqrt{\frac{1}{LC}}\sqrt{Q^{2}-q^{2}}$
$v_{x} = \frac{dx}{dt}$  $i = \frac{dq}{dt}$
$\omega = \sqrt{\frac{k}{m}}$ $\omega = \sqrt{\frac{1}{LC}}$
$x = A \cos{\left(\omega t + \phi \right)}$  $q = Q \cos{\left(\omega t + \phi \right)}$

Next: Magnetic-Field Energy In Inductor

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