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
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
\( \mathrm{f = \dfrac{1}{2\pi \sqrt{LC}} = \dfrac{1}{2\pi \sqrt{(5.0 \times 10^{-3})(2.0 \times 10^{-6})}}} \).
\( \mathrm{f = \dfrac{1}{2\pi \sqrt{1.0 \times 10^{-8}}} = \dfrac{1}{2\pi (1.0 \times 10^{-4})}} \).
\( \mathrm{f \approx 1590 \, Hz} \).