LC Oscillations — Energy Exchange

LC Oscillations — A Complete Guide

LC oscillations are one of the most elegant topics in Class 12 physics. A capacitor and an inductor, wired together, trade energy back and forth indefinitely (in the ideal case) — exactly like a mass on a spring trades kinetic and potential energy. Once we see this analogy, the entire chapter collapses into a familiar story.

JEE Main, JEE Advanced, and NEET all reliably draw 1–2 questions from this section per year. The questions test two ideas: the natural frequency formula, and the energy distribution between $L$ and $C$ at a given instant. Both reduce to a handful of equations.

We’ll set up the differential equation, solve it physically (rather than mathematically), look at the energy bookkeeping, and finish with PYQ-style worked examples.

Key Terms & Definitions

Inductor (L). Stores energy in its magnetic field. Energy stored: $U_L = \tfrac{1}{2}LI^2$ . Resists changes in current (Lenz’s law).

Capacitor (C). Stores energy in its electric field. Energy stored: $U_C = \tfrac{Q^2}{2C}$ . Resists changes in voltage.

LC Oscillation. When an initially charged capacitor is connected across an inductor, the charge sloshes back and forth, producing a sinusoidal current. The frequency of this oscillation depends only on $L$ and $C$ .

Natural (Angular) Frequency.

\omega_0 = \frac{1}{\sqrt{LC}}, \quad f_0 = \frac{1}{2\pi\sqrt{LC}}

Methods & Concepts

Setting Up the Equation

Apply Kirchhoff’s voltage law around the LC loop. The capacitor voltage $V_C = Q/C$ equals the inductor’s back-EMF $L\, dI/dt$ . With $I = -dQ/dt$ (current flows out of the positive plate as it discharges):

\frac{Q}{C} = -L\frac{d^2 Q}{dt^2}

Rearranging:

\frac{d^2 Q}{dt^2} + \frac{1}{LC}Q = 0

This is the simple harmonic motion equation with $\omega^2 = 1/LC$ . So $Q(t) = Q_0 \cos(\omega_0 t)$ if the capacitor starts fully charged at rest.

The Mechanical Analogy

Mechanical (mass-spring)	Electrical (LC)
Mass $m$	Inductance $L$
Spring constant $k$	Inverse capacitance $1/C$
Position $x$	Charge $Q$
Velocity $v = dx/dt$	Current $I = dQ/dt$
KE $= \tfrac{1}{2}mv^2$	$U_L = \tfrac{1}{2}LI^2$
PE $= \tfrac{1}{2}kx^2$	$U_C = \tfrac{Q^2}{2C}$

If you’ve internalised SHM from Class 11, LC oscillations are the same chapter in different clothes.

Energy Conservation

In an ideal lossless LC circuit:

U_{\text{total}} = \frac{Q^2}{2C} + \frac{1}{2}LI^2 = \frac{Q_0^2}{2C}

When all charge sits on the capacitor, current is zero — all energy is in $C$ . A quarter-period later, the capacitor is uncharged and current is maximum — all energy is in $L$ . This swap repeats forever.

Solved Examples

Example 1 — Frequency Calculation (NEET pattern)

An LC circuit has $L = 4\ \text{mH}$ and $C = 25\ \mu\text{F}$ . Find the frequency of oscillation.

\omega_0 = \frac{1}{\sqrt{4 \times 10^{-3} \times 25 \times 10^{-6}}} = \frac{1}{\sqrt{10^{-7}}} \approx 3162\ \text{rad/s}

$f_0 = \omega_0/2\pi \approx 503\ \text{Hz}$ .

Example 2 — Maximum Current (JEE Main pattern)

The capacitor in the previous problem is initially charged to $Q_0 = 10\ \mu\text{C}$ . Find the maximum current through the inductor.

By energy conservation:

\frac{Q_0^2}{2C} = \frac{1}{2}LI_{\max}^2 \implies I_{\max} = Q_0 \sqrt{\frac{1}{LC}} = Q_0 \omega_0

$I_{\max} = 10 \times 10^{-6} \times 3162 \approx 31.6\ \text{mA}$ .

Example 3 — Energy at an Instant (JEE Advanced pattern)

In the same circuit, find the charge on the capacitor when the current is half its maximum value.

By energy conservation:

\frac{Q^2}{2C} + \frac{1}{2}L\left(\frac{I_{\max}}{2}\right)^2 = \frac{Q_0^2}{2C}

Using $Q_0^2/2C = LI_{\max}^2/2$ :

\frac{Q^2}{2C} = \frac{Q_0^2}{2C}\left(1 - \frac{1}{4}\right) \implies Q = \frac{\sqrt{3}}{2}Q_0

So $Q \approx 8.66\ \mu\text{C}$ .

Exam-Specific Tips

JEE Main weightage: Reliably 1 question on $\omega_0$ or energy split per shift. JEE Advanced: Often combines LC with damping or an external EMF (driven oscillator). NEET: Direct frequency-formula plug-ins.

For “maximum current” PYQs, jump straight to $I_{\max} = Q_0/\sqrt{LC}$ . Saves the algebra of solving $Q(t)$ and differentiating.

Common Mistakes to Avoid

Forgetting that $\omega_0$ is in rad/s, not Hz. $f_0 = \omega_0/2\pi$ — divide by $2\pi$ when the question asks for frequency in hertz.

Mixing energy formulas. $U_C = Q^2/2C$ uses charge; $U_C = \tfrac{1}{2}CV^2$ uses voltage. Pick one and stick with it for the whole problem.

Assuming current is in phase with charge. Current is the time-derivative of charge — they’re $90°$ out of phase. When $Q$ is maximum, $I = 0$ , and vice versa.

Ignoring resistance. Real circuits have $R$ , which damps the oscillation. The chapter assumes ideal $R = 0$ unless stated otherwise.

Unit slip on $L$ and $C$ . Inductance in millihenries, capacitance in microfarads. Convert to SI before plugging in.

Practice Questions

Q1. Find $f_0$ for $L = 1\ \text{mH}$ , $C = 1\ \mu\text{F}$ .

$\omega_0 = 1/\sqrt{10^{-9}} = 31623\ \text{rad/s}$ . $f_0 \approx 5033\ \text{Hz}$ .

Q2. If $L$ is doubled and $C$ is halved, how does $\omega_0$ change?

$\omega_0 \propto 1/\sqrt{LC}$ . Product $LC$ stays same, so $\omega_0$ unchanged.

Q3. At what $Q$ is energy equally split between $L$ and $C$ ?

$Q = Q_0/\sqrt{2}$ .

Q4. Time for energy to fully transfer from $C$ to $L$ ?

Quarter period: $T/4 = \pi/(2\omega_0) = \tfrac{\pi}{2}\sqrt{LC}$ .

Q5. Maximum voltage across $C$ if $Q_0 = 5\ \mu\text{C}$ and $C = 10\ \mu\text{F}$ ?

$V_{\max} = Q_0/C = 0.5\ \text{V}$ .

Q6. What happens to oscillations if a small resistance is added?

Oscillations decay exponentially (damped LC, equivalent to RLC with small $R$ ).

Q7. Show $\omega_0 = 1/\sqrt{LC}$ has units of rad/s.

Units: $1/\sqrt{\text{H} \cdot \text{F}} = 1/\sqrt{\text{V·s/A} \cdot \text{C/V}} = 1/\sqrt{\text{s}^2} = 1/\text{s}$ . Correct.

Q8. Energy stored when $I = I_{\max}/2$ ?

$U_L = \tfrac{1}{4} \cdot \tfrac{1}{2}LI_{\max}^2 = U_{\text{total}}/4$ . Three-quarters is in capacitor.

FAQs

Why does the energy oscillate forever in an ideal LC circuit? No resistance means no dissipation. In the SHM analogy, this is a frictionless mass-spring.

What’s the role of phase between $Q$ and $I$ ? $I = -dQ/dt$ , so $I$ leads or lags $Q$ by $90°$ depending on sign convention. This phase difference is why energy keeps swapping rather than damping.

Can LC oscillations exist in reality? Practically, every real inductor has some resistance, so true LC oscillations are an idealisation. But low-loss superconducting circuits come very close.

How is LC related to LC tank circuits in radio? Same physics — radio tuners select a station by matching the LC natural frequency to the broadcast frequency.

Does the energy in an LC circuit depend on temperature? Not in the ideal model. In real components, $L$ and $C$ have small temperature coefficients, but this is a precision concern, not a JEE concern.