Circuits with Capacitors and Inductors (LC Circuits) |...

SECTION 1

Historical Context & Motivation

The story of LC circuits is inseparable from the broader quest to understand how electric and magnetic energy interact dynamically. While static capacitors and steady-state inductors were studied independently throughout the early nineteenth century, physicists gradually realized that connecting these two elements could produce something entirely new: self-sustaining electrical oscillations. These oscillations ultimately became the foundation of radio communication, radar, and modern wireless technology.

1831

Faraday's Induction

Michael Faraday demonstrated that a changing magnetic flux induces an EMF, establishing the principle that underlies every inductor.

1853

Thomson's Oscillation Analysis

William Thomson (Lord Kelvin) predicted that a capacitor discharging through an inductor would produce oscillations with a frequency dependent on L and C, deriving f = 1/(2π√(LC)).

1857

Feddersen's Experimental Confirmation

Berend Feddersen used a rotating mirror to photograph the oscillatory spark discharge of a Leyden jar through a coil, providing the first experimental proof of LC oscillations.

1887

Hertz Generates Radio Waves

Heinrich Hertz used an LC oscillator to produce and detect electromagnetic waves, confirming Maxwell's theory and launching the era of wireless communication.

The central question that drove this physics is deceptively simple: what happens when a charged capacitor is connected to an inductor with no resistance? The answer—a perpetual, lossless exchange of energy between electric and magnetic forms—reveals deep structural parallels between electrical circuits and mechanical oscillators. Understanding this ideal LC circuit is essential for analyzing real RLC circuits, resonant filters, and the electromagnetic radiation that Hertz first demonstrated.

SECTION 2

Core Principles of LC Oscillation

An ideal LC circuit contains only two passive components—a capacitor (capacitance C) and an inductor (inductance L)—connected in a closed loop with no resistance. The capacitor stores energy in its electric field, while the inductor stores energy in its magnetic field. When the circuit is set into motion (e.g., by initially charging the capacitor), these two energy reservoirs exchange energy back and forth indefinitely at a characteristic natural frequency determined solely by L and C.

Energy Storage Duality

The capacitor stores energy U_C = q²/(2C) in its electric field; the inductor stores U_L = Li²/2 in its magnetic field. Total energy is conserved.

Kirchhoff's Voltage Law

Around the single loop: q/C + L(di/dt) = 0. This constraint, combined with i = dq/dt, produces a second-order ODE identical in form to simple harmonic motion.

Simple Harmonic Oscillation

The charge on the capacitor oscillates sinusoidally: q(t) = Q₀ cos(ωt + φ), where ω = 1/√(LC). Current leads charge by 90°.

Mechanical Analogy

L corresponds to mass (inertia), 1/C to a spring constant, charge q to displacement, and current i to velocity. The LC circuit is the electrical twin of a mass–spring system.

✦ KEY TAKEAWAY

KEY TAKEAWAY

SECTION 3

Visual Explanation of LC Oscillation

Top row: the four quarter-cycle phases showing charge on the capacitor plates and magnetic field in the inductor. Bottom: the complementary oscillation of electric energy U_C (cyan) and magnetic energy U_L (violet), summing to the constant total energy (dashed amber line).

The diagram above captures the essential rhythm of LC oscillation. In Phase 1 (t = 0), the capacitor holds its maximum charge Q₀ and all energy resides in the electric field. No current flows. As the capacitor begins to discharge, current rises and the inductor's magnetic field grows. By Phase 2 (t = T/4), the capacitor is fully discharged, current reaches its maximum, and all energy has transferred to the magnetic field. The inductor's self-induced EMF now drives current to continue flowing, recharging the capacitor with reversed polarity in Phase 3 (t = T/2). The cycle completes through Phase 4 (t = 3T/4), and the system returns to its initial state at t = T. The bottom graph confirms that U_C and U_L are always 90° out of phase, and their sum remains constant—a direct consequence of energy conservation in the absence of dissipation.

SECTION 4

Mathematical Framework

We derive the governing differential equation by applying Kirchhoff's voltage law to the single LC loop. Traversing the loop in the direction of positive current i, the voltage drop across the capacitor is q/C and the voltage drop across the inductor is L(di/dt). Setting their sum to zero and substituting i = dq/dt yields the fundamental equation of motion.

GOVERNING DIFFERENTIAL EQUATION

L (d²q/dt²) + q/C = 0 → d²q/dt² = −(1/LC) q

This is identical in form to d²x/dt² = −(k/m)x for a mass–spring system, with L ↔ m and 1/C ↔ k.

GENERAL SOLUTION

q(t) = Q₀ cos(ωt + φ), i(t) = −ωQ₀ sin(ωt + φ)

Q₀ is the maximum charge, φ is the phase constant determined by initial conditions, and ω = 1/√(LC) is the angular frequency.

ANGULAR FREQUENCY & PERIOD

ω = 1/√(LC), f = 1/(2π√(LC)), T = 2π√(LC)

ω has units rad/s, f in Hz, and T in seconds. Larger L or C means slower oscillations.

ENERGY CONSERVATION

U_total = q²/(2C) + Li²/2 = Q₀²/(2C) = constant

At any instant, U_C + U_L equals the initial energy stored on the capacitor. The maximum current is i_max = Q₀/√(LC) = Q₀ω.

Derivation from Energy

SECTION 5

LC–Mechanical Analogy in Detail

The structural isomorphism between the LC circuit and the simple harmonic oscillator is one of the most powerful conceptual tools in physics. Every variable, parameter, and energy expression in one system maps directly onto a counterpart in the other. The table below makes these correspondences explicit, and the diagram that follows illustrates the parallel side by side.

Complete mapping between mass–spring and LC circuit oscillators

Mechanical (Mass–Spring)	Electrical (LC Circuit)	Role
Displacement x	Charge q	Oscillating variable
Velocity v = dx/dt	Current i = dq/dt	Rate of change
Mass m	Inductance L	Inertia
Spring constant k	1/C (inverse capacitance)	Restoring force per unit displacement
½kx²	q²/(2C)	Potential / electric energy
½mv²	½Li²	Kinetic / magnetic energy
ω = √(k/m)	ω = 1/√(LC)	Natural angular frequency
Friction b	Resistance R	Damping (absent in ideal LC)

Left: a mass m connected to a spring of constant k oscillates with ω = √(k/m). Right: an LC circuit oscillates with ω = 1/√(LC). The mathematical structure is identical—only the physical interpretation of the variables changes.

✦ KEY TAKEAWAY

KEY TAKEAWAY

SECTION 6

Worked Example

Step 1 — Identify Given ValuesA 4.00 mH inductor is connected to a 10.0 µF capacitor. The capacitor is initially charged to 12.0 V. Find (a) the angular frequency, (b) the period, (c) the maximum current, and (d) the total energy of the oscillation.

Step 2 — Compute Angular Frequencyω = 1/√(LC) = 1/√((4.00 × 10⁻³ H)(10.0 × 10⁻⁶ F)) = 1/√(4.00 × 10⁻⁸) = 1/(2.00 × 10⁻⁴)

ω = 5.00 × 10³ rad/s

Step 3 — Compute PeriodT = 2π/ω = 2π/(5.00 × 10³)

T ≈ 1.26 × 10⁻³ s = 1.26 ms

Step 4 — Find Initial Charge and Maximum CurrentThe initial charge is Q₀ = CV₀ = (10.0 × 10⁻⁶ F)(12.0 V) = 1.20 × 10⁻⁴ C. Maximum current occurs when all capacitor energy has transferred to the inductor: i_max = Q₀ω = (1.20 × 10⁻⁴ C)(5.00 × 10³ rad/s)

i_max = 0.600 A

Step 5 — Compute Total EnergyU_total = Q₀²/(2C) = (1.20 × 10⁻⁴)²/(2 × 10.0 × 10⁻⁶) = 1.44 × 10⁻⁸/(2.00 × 10⁻⁵)

U_total = 7.20 × 10⁻⁴ J = 0.720 mJ

Step 6 — Verify via Magnetic EnergyAs a check, U_L,max = ½Li²_max = ½(4.00 × 10⁻³)(0.600)² = ½(4.00 × 10⁻³)(0.360) = 7.20 × 10⁻⁴ J. ✓ This matches U_total, confirming energy conservation.

SECTION 7

Ideal LC vs. Real RLC: Strengths & Limitations

The ideal LC model assumes zero resistance, which is never perfectly realized in practice. Every real inductor has some wire resistance and every real circuit has contact and lead resistance. When resistance R is present, the circuit becomes an RLC circuit, and the oscillations decay exponentially—a phenomenon called damping. Despite this, the ideal LC analysis is far from useless: it provides the natural frequency, the energy scale, and the phase relationships that persist even in damped systems.

Comparison of ideal LC and real RLC circuit behavior

Feature	Ideal LC (R = 0)	Real RLC (R > 0)
Oscillation amplitude	Constant forever	Decays as e^(−Rt/(2L))
Frequency	ω₀ = 1/√(LC)	ω' = √(ω₀² − (R/(2L))²), slightly lower
Energy	Conserved: U_C + U_L = const	Dissipated as i²R; total energy decreases
Differential equation	L d²q/dt² + q/C = 0	L d²q/dt² + R dq/dt + q/C = 0
Mechanical analog	Frictionless mass–spring	Damped mass–spring (friction = R)

✦ KEY TAKEAWAY

KEY TAKEAWAY

SECTION 8

Connection to Advanced Theory: Driven Oscillations & Resonance

The free LC oscillation you have studied is the starting point for the richer topic of driven RLC circuits, where an AC voltage source of angular frequency ω_d is placed in the loop. In steady state the system responds at the driving frequency, but the amplitude peaks sharply when ω_d = ω₀ = 1/√(LC)—this phenomenon is called resonance. Resonance is exploited in radio tuners (selecting a station by adjusting C to match the broadcast frequency), MRI machines, and wireless power transfer.

Free LC oscillation vs. driven RLC resonance

Concept	Free LC (this lesson)	Driven RLC (next topic)
Energy source	Initial charge on capacitor	External AC EMF: ε₀ cos(ω_d t)
Oscillation frequency	Always ω₀ = 1/√(LC)	Responds at ω_d; amplitude depends on \|ω_d − ω₀\|
Key phenomenon	Free oscillation / natural frequency	Resonance when ω_d = ω₀
Phase relationships	Current leads charge by 90°	At resonance, current is in phase with driving EMF

Beyond classical circuits, the LC oscillator concept extends into quantum mechanics, where the quantized LC circuit serves as a model for the quantum harmonic oscillator—a cornerstone of quantum field theory and quantum computing. The superconducting LC resonators used in modern transmon qubits are, at their core, the same LC oscillation studied here, operating at microwave frequencies with effectively zero resistance thanks to superconductivity.

SECTION 9

Practice Problems

In an ideal LC circuit, at the instant the current through the inductor is at its maximum value, which of the following is true?

An LC circuit has L = 50.0 mH and C = 20.0 µF. What is the frequency of oscillation?

An LC circuit oscillates with Q₀ = 6.00 × 10⁻⁴ C, L = 8.00 mH, and C = 5.00 µF. At the instant when the charge on the capacitor is 3.00 × 10⁻⁴ C, what is the current in the circuit?

PROBLEM 4 — APPLIED

A capacitor of capacitance C = 2.00 µF is charged to a potential difference V₀ = 100 V and then connected to an inductor of inductance L = 0.500 H at time t = 0. (a) Write an expression for q(t), the charge on the capacitor as a function of time. (b) Write an expression for i(t). (c) Determine the maximum energy stored in the inductor. (d) At what time does the current first reach its maximum value? (e) Sketch q(t) and i(t) on the same axes for two full periods, clearly labeling the axes and key features.

PROBLEM 5 — CRITICAL THINKING

Two identical LC circuits each oscillate at frequency f₀. They are then combined by connecting their capacitors in parallel and their inductors in series to form a single LC circuit. (a) Derive the new oscillation frequency f_new in terms of f₀. (b) Is the new frequency higher, lower, or equal to f₀? Justify your answer both mathematically and with a physical argument using the mechanical analogy. (c) If instead the capacitors were connected in series and the inductors in parallel, would the result change? Explain.