KAHIBARO
Table of Contents
Energy Exchange Between Electric and Magnetic Forms
An LC circuit is an ideal circuit made from a capacitor, with capacitance $C$, and an inductor, with inductance $L$. When connected together with no resistance, the circuit can oscillate on its own. Charge flows back and forth, current reverses direction repeatedly, and energy moves between the electric field of the capacitor and the magnetic field of the inductor.
This is the electrical analog of a mass on a spring. In a spring system, energy moves between kinetic energy and elastic potential energy. In an LC circuit, energy moves between magnetic energy in the inductor and electric energy in the capacitor.
Suppose the capacitor is initially charged to charge $Q_0$, and the current is zero. The capacitor begins to discharge through the inductor. As charge leaves the capacitor, current grows in the circuit and a magnetic field builds in the inductor. Later, the capacitor becomes uncharged, but the current is then at its maximum. Because the inductor resists changes in current, the current does not stop immediately. It continues, charging the capacitor again with opposite polarity. The process repeats periodically.
Differential Equation of an LC Circuit
For an ideal LC loop, Kirchhoff's loop rule gives the sum of potential differences as zero:
$$
V_L + V_C = 0
$$
Using
$$
V_L = L\frac{dI}{dt}, \qquad V_C = \frac{Q}{C}
$$
and the relation
$$
I = \frac{dQ}{dt}
$$
we get
$$
L\frac{d^2Q}{dt^2} + \frac{Q}{C} = 0
$$
This is the equation of simple harmonic motion.
For an ideal LC circuit, the charge satisfies
$$
\boxed{L\frac{d^2Q}{dt^2} + \frac{Q}{C} = 0}
$$
This means the circuit oscillates naturally.
The equation can also be written as
$$
\frac{d^2Q}{dt^2} + \frac{1}{LC}Q = 0
$$
Comparing with the standard SHM form,
$$
\frac{d^2x}{dt^2} + \omega^2 x = 0
$$
we identify the angular frequency
$$
\omega = \frac{1}{\sqrt{LC}}
$$
Charge and Current as Functions of Time
A general solution for the charge on the capacitor is
$$
Q(t) = Q_0 \cos(\omega t + \phi)
$$
where $Q_0$ is the maximum charge and $\phi$ is the phase constant, determined by initial conditions.
Since current is the time derivative of charge,
$$
I(t) = \frac{dQ}{dt} = -\omega Q_0 \sin(\omega t + \phi)
$$
The maximum current is therefore
$$
I_{\max} = \omega Q_0 = \frac{Q_0}{\sqrt{LC}}
$$
Charge and current are out of phase by one quarter of a cycle. When charge is maximum, current is zero. When current is maximum, charge is zero.
Important LC formulas:
$$
\boxed{\omega = \frac{1}{\sqrt{LC}}}
$$
$$
\boxed{Q(t) = Q_0 \cos(\omega t + \phi)}
$$
$$
\boxed{I(t) = -\omega Q_0 \sin(\omega t + \phi)}
$$
$$
\boxed{I_{\max} = \omega Q_0}
$$
The period of oscillation is
$$
T = \frac{2\pi}{\omega} = 2\pi\sqrt{LC}
$$
and the ordinary frequency is
$$
f = \frac{1}{T} = \frac{1}{2\pi\sqrt{LC}}
$$
The natural period and frequency of an ideal LC oscillator are
$$
\boxed{T = 2\pi\sqrt{LC}}
$$
$$
\boxed{f = \frac{1}{2\pi\sqrt{LC}}}
$$
Energy in an LC Circuit
The energy stored in the capacitor is
$$
U_C = \frac{Q^2}{2C}
$$
The energy stored in the inductor is
$$
U_L = \frac{1}{2}LI^2
$$
In an ideal LC circuit, total energy remains constant:
$$
U = U_C + U_L = \text{constant}
$$
If the capacitor initially has charge $Q_0$ and current is zero, then the total energy is
$$
U = \frac{Q_0^2}{2C}
$$
At any later time,
$$
\frac{Q^2}{2C} + \frac{1}{2}LI^2 = \frac{Q_0^2}{2C}
$$
This expresses continuous conversion between electric and magnetic energy.
In an ideal LC circuit, energy is conserved:
$$
\boxed{\frac{Q^2}{2C} + \frac{1}{2}LI^2 = \text{constant}}
$$
Electric energy is maximum when charge is maximum.
Magnetic energy is maximum when current is maximum.
Oscillation Stages
The oscillation can be understood by following one cycle.
| Stage | Capacitor charge | Current | Main stored energy |
|---|---|---|---|
| 1 | Maximum | 0 | Electric |
| 2 | Decreasing | Increasing | Mixed |
| 3 | 0 | Maximum | Magnetic |
| 4 | Opposite maximum | 0 | Electric |
| 5 | Increasing toward original sign | Increasing opposite direction | Mixed |
At the start, the capacitor is fully charged and the inductor has no magnetic energy. As discharge begins, current grows. When the capacitor becomes uncharged, all energy is in the magnetic field. Then current continues and charges the capacitor with reversed polarity. After half a period, the capacitor is fully charged again, but with opposite sign.
Phase Relationship
The charge, current, and rate of current change are not all maximum at the same time. This is a key feature of oscillatory motion.
If
$$
Q(t) = Q_0\cos(\omega t)
$$
then
$$
I(t) = -\omega Q_0\sin(\omega t)
$$
So current is shifted by $\pi/2$ radians relative to charge.
When $Q = \pm Q_0$, the capacitor voltage is maximum and $I = 0$.
When $Q = 0$, current is at its maximum magnitude.
This phase difference explains why energy keeps transferring instead of remaining in one form.
Mechanical Analogy
The LC circuit behaves mathematically like a mass-spring oscillator.
| Mechanical oscillator | LC circuit |
|---|---|
| Displacement $x$ | Charge $Q$ |
| Velocity $v$ | Current $I$ |
| Mass $m$ | Inductance $L$ |
| Spring constant $k$ | $\frac{1}{C}$ |
| Equation $m\frac{d^2x}{dt^2}+kx=0$ | $L\frac{d^2Q}{dt^2}+\frac{Q}{C}=0$ |
This analogy helps show why the circuit oscillates naturally. The capacitor plays a role similar to the spring, and the inductor plays a role similar to inertia.
Example of Natural Frequency
Consider a circuit with
$$
L = 2.0 \times 10^{-3}\ \text{H}, \qquad C = 8.0 \times 10^{-6}\ \text{F}
$$
Then
$$
f = \frac{1}{2\pi\sqrt{LC}}
$$
Substitute the values:
$$
f = \frac{1}{2\pi\sqrt{(2.0 \times 10^{-3})(8.0 \times 10^{-6})}}
$$
$$
LC = 1.6 \times 10^{-8}
$$
$$
\sqrt{LC} = 1.26 \times 10^{-4}
$$
$$
f \approx \frac{1}{2\pi(1.26 \times 10^{-4})} \approx 1.26 \times 10^3\ \text{Hz}
$$
So the circuit oscillates at about
$$
f \approx 1.3\ \text{kHz}
$$
Visualizing the Oscillation
The figure below shows the basic idea of energy exchange in one cycle.
The blue curve represents charge, and the red curve represents current. They reach maxima at different times, showing the phase shift.
Ideal Nature of LC Oscillations
Real circuits usually have some resistance, so real oscillations gradually decrease. That behavior belongs to a more general circuit with resistance included. In the ideal LC case studied here, there is no energy loss, so the oscillations continue forever with constant amplitude.
An ideal LC circuit has no resistance, so its oscillations are undamped and continue with constant amplitude.
Summary
An LC circuit is a natural electrical oscillator. The capacitor stores electric energy, the inductor stores magnetic energy, and energy moves back and forth between them. The charge obeys the simple harmonic motion equation
$$
L\frac{d^2Q}{dt^2} + \frac{Q}{C} = 0
$$
with angular frequency
$$
\omega = \frac{1}{\sqrt{LC}}
$$
and period
$$
T = 2\pi\sqrt{LC}
$$
The charge and current vary sinusoidally, with current shifted by one quarter cycle relative to charge. In the ideal case, total energy remains constant throughout the oscillation.