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13.6 Circuits with Capacitors and Inductors (LC Circuits)

Part of Electromagnetic Induction in AP® Physics C: Electricity and Magnetism. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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An LC circuit contains a capacitor and an inductor and can oscillate even with no battery present. If a charged capacitor is connected to an inductor, the capacitor begins to discharge, current grows, and energy is transferred into the magnetic field of the inductor. Once the capacitor is fully discharged, the current is maximum, but the inductor keeps the current going, recharging the capacitor with opposite polarity. This produces oscillation between electric-field energy and magnetic-field energy.

The loop rule for an ideal LC circuit is

qC+LdIdt=0\frac{q}{C} + L\frac{dI}{dt} = 0

with I=dq/dt{I = dq/dt}, so

Ld2qdt2+qC=0L\frac{d^2 q}{dt^2} + \frac{q}{C} = 0

or

d2qdt2+1LCq=0\frac{d^2 q}{dt^2} + \frac{1}{LC}q = 0

This is the differential equation of simple harmonic motion. Therefore,

q(t)=Qmax⁡cos⁡(ωt+ϕ)q(t) = Q_{\max}\cos(\omega t + \phi)

with

ω=1LC\omega = \frac{1}{\sqrt{LC}}

and

T=2πLCT = 2\pi\sqrt{LC}

So LC circuits are the electrical analogue of mass-spring oscillators.

In this unit

6 lessons

  1. 13.1Magnetic FluxPreview lesson
  2. 13.2Electromagnetic InductionPreview lesson
  3. 13.3Induced Currents and Magnetic ForcesPreview lesson
  4. 13.4InductancePreview lesson
  5. 13.5Circuits with Resistors and Inductors (LR Circuits)Preview lesson
  6. 13.6Circuits with Capacitors and Inductors (LC Circuits)This lesson

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