AP Physics C E&M- 13.6 Circuits with Capacitors and Inductors (LC Circuits)- Study Notes- New Syllabus

AP Physics C E&M- 13.6 Circuits with Capacitors and Inductors (LC Circuits) – Study Notes

AP Physics C E&M- 13.6 Circuits with Capacitors and Inductors (LC Circuits) – Study Notes – per latest Syllabus.

Key Concepts:

LC Circuits: Combination of a Capacitor and an Inductor

An LC circuit consists of a capacitor and an inductor connected together without resistance.

Such a circuit exhibits electrical oscillations due to continuous energy exchange between the capacitor’s electric field and the inductor’s magnetic field.
This oscillatory behavior is mathematically equivalent to simple harmonic motion in mechanics.

Energy in an LC Circuit:

At any time, total energy is the sum of electric potential energy in the capacitor and magnetic energy in the inductor:

\( \mathrm{U = \tfrac{1}{2} C V^2 + \tfrac{1}{2} L I^2} \)

Conservation of energy applies:

\( \mathrm{U_{total} = \text{constant}} \)

Maximum capacitor energy occurs when current is zero; maximum inductor energy occurs when capacitor charge is zero.

Differential Equation for Charge:

From Kirchhoff’s loop rule and Faraday’s law:

\( \mathrm{L \dfrac{d^2 Q}{dt^2} + \dfrac{Q}{C} = 0} \).

This is the equation of simple harmonic motion, with solution:

\( \mathrm{Q(t) = Q_{max} \cos(\omega t + \phi)} \)

where \( \mathrm{\omega} \) is the angular frequency.

Angular Frequency of Oscillation:

\( \mathrm{\omega = \dfrac{1}{\sqrt{LC}}} \)

Oscillation frequency:

\( \mathrm{f = \dfrac{1}{2\pi \sqrt{LC}}} \)

Key Properties:

Charge, current, and voltage oscillate sinusoidally in time.
Energy oscillates between the capacitor’s electric field and the inductor’s magnetic field.
Ideal LC circuits have no resistance; in real circuits, resistance causes the oscillations to decay (damped oscillations).

Example:

A capacitor of \(C = 2.0 \, \mu F\) is charged to a potential difference of \(100 \, V\) and then connected to an inductor of \(L = 5.0 \, mH\). Find

(a) the maximum current in the inductor, and

(b) the oscillation frequency of the circuit.

▶️ Answer/Explanation

Part (a): Maximum Current

Initial energy stored in capacitor: \( \mathrm{U = \tfrac{1}{2} C V^2 = \tfrac{1}{2}(2.0 \times 10^{-6})(100^2) = 0.010 \, J} \).

At maximum current, all energy is in inductor: \( \mathrm{U = \tfrac{1}{2} L I_{max}^2} \).

Solve for \( \mathrm{I_{max}} \): \( \mathrm{I_{max} = \sqrt{\dfrac{2U}{L}} = \sqrt{\dfrac{0.020}{5.0 \times 10^{-3}}} = 2.0 \, A} \).

Part (b): Frequency

\( \mathrm{f = \dfrac{1}{2\pi \sqrt{LC}} = \dfrac{1}{2\pi \sqrt{(5.0 \times 10^{-3})(2.0 \times 10^{-6})}}} \).

\( \mathrm{f = \dfrac{1}{2\pi \sqrt{1.0 \times 10^{-8}}} = \dfrac{1}{2\pi (1.0 \times 10^{-4})}} \).

\( \mathrm{f \approx 1590 \, Hz} \).