Circuits with Capacitors and Inductors (LC Circuits) - AP Physics C: Electricity and Magnetism Study Guide

Getting Started

We consider an idealized electrical circuit, an isolated system containing only a capacitor and an inductor. If the capacitor is initially charged and then connected to the inductor, the circuit exhibits a remarkable behavior. The central question is: how do the charge, current, and energy within this system evolve over time, and what physical principles govern this evolution?

What You Should Be Able to Do

• Derive the second-order linear differential equation that governs the charge on the capacitor as a function of time using Kirchhoff's Loop Rule.

• Model the time-dependence of charge and current in an LC circuit as simple harmonic motion, and from the governing differential equation, derive the circuit's natural angular frequency of oscillation.

• Apply the principle of conservation of energy to relate the maximum energy stored in the capacitor's electric field to the maximum energy stored in the inductor's magnetic field, thereby determining the maximum current.

Key Concepts & Mechanisms

System & Preconditions

The system is an ideal LC circuit, consisting of a single capacitor with capacitance C (in Farads, F) and a single inductor with inductance L (in Henrys, H). We make several key idealizations:

• The circuit has zero resistance, meaning there is no mechanism for energy dissipation as heat.

• The system is closed and does not lose energy by radiating electromagnetic waves.

• The components themselves are ideal; the inductor has no internal capacitance, and the capacitor has no internal inductance.

The process begins with a specific set of initial conditions. Typically, at time t = 0, the capacitor holds its maximum charge, $Q_{max}$, and the switch is closed, so the initial current in the circuit is zero, i(0) = 0.

Key Steps / Relations

The behavior of the circuit is governed by the principle of conservation of energy, expressed for circuits as Kirchhoff's Loop Rule.

1. Governing Law: Kirchhoff's Loop Rule states that the sum of the potential differences ($\Delta V$) around any closed loop must be zero. For our LC circuit, this gives:

   $\Delta V_C+\Delta V_L=0$

2. Component Relations: We substitute the defining relations for the potential difference across a capacitor and an inductor. The potential difference across the capacitor is $\Delta V_C=\frac{q}{C}$, where q is the charge on the capacitor at time t. The potential difference (or self-induced EMF) across the inductor is $\Delta V_L=L\frac{di}{dt}$, where i is the current at time t. The loop rule becomes:

   $\frac{q}{C}+L\frac{di}{dt}=0$

3. Differential Form: To express this equation in terms of a single variable, charge q, we use the definition of current: $i=\frac{dq}{dt}$. Substituting this into our loop equation yields:

   $\frac{q}{C}+L\frac{d}{dt}\left(\frac{dq}{dt}\right)=0$

4. Standard Form: Rearranging the terms gives the final differential equation for charge in an LC circuit:

   $\frac{d^{2}q}{d{t}^2}=-\frac{1}{LC}q$

Outputs & Effects

This equation is the hallmark of Simple Harmonic Motion (SHM). It states that the second time derivative of the charge is proportional to the negative of the charge itself.

• Oscillatory Solution: The general solution to this differential equation is a sinusoidal function: $q(t)=A\cos (\omega t+\varphi )$. Applying our initial conditions ($q(0)=Q_{max}$ and $i(0)=0$), we find the amplitude $A=Q_{max}$ and the phase constant $\varphi =0$. The specific solution is:

  $q(t)=Q_{max}\cos (\omega t)$

• Angular Frequency: By comparing the derived differential equation to the general equation for SHM ($\frac{d^{2}x}{d{t}^2}=-{\omega }^{2}x$), we can identify the angular frequency of oscillation, $\omega$ (in rad/s):

  $\omega =\frac{1}{\sqrt{LC}}$

• Current: The current as a function of time is found by differentiating the charge function:

  $i(t)=\frac{dq}{dt}=-\omega Q_{max}\sin (\omega t)$

  The current also oscillates, but it is 90° out of phase with the charge. The maximum current is $I_{max}=\omega Q_{max}$.

Regulation & Limits

The ideal nature of the circuit (zero resistance) implies that the total energy is conserved. Energy continuously transfers between the capacitor's electric field and the inductor's magnetic field.

• Energy Conservation: The total energy $U_{total}$ in the circuit is the sum of the electric energy $U_E$ stored in the capacitor and the magnetic energy $U_B$ stored in the inductor:

  $U_{total}=U_E+U_B=\frac{1}{2}\frac{q(t{)}^2}{C}+\frac{1}{2}Li(t{)}^2=\text{constant}$

• Maximum Values:

  • When the capacitor is fully charged ($q=Q_{max}$), the current is momentarily zero ($i=0$). All energy is electric: $U_{total}=\frac{Q_{max}^2}{2C}$.

  • When the capacitor is fully discharged ($q=0$), the current is at its maximum ($i=I_{max}$). All energy is magnetic: $U_{total}=\frac{1}{2}L{I}_{max}^2$.

• Relating Maxima: By equating these two expressions for the total energy, we can find the maximum current in the inductor:

  $\frac{Q_{max}^2}{2C}=\frac{1}{2}L{I}_{max}^2⟹I_{max}=Q_{max}\sqrt{\frac{1}{LC}}=\frac{Q_{max}}{\sqrt{LC}}$

  This confirms our earlier result, $I_{max}=\omega Q_{max}$.

Key Models & Diagrams

The derivation of the oscillatory behavior of an LC circuit can be visualized as a direct causal chain from physical principles to observable predictions.

Key Components & Evidence

• Charge (q): The quantity of electric charge stored on the capacitor plates. Its oscillation is the primary dynamic variable of the system. Unit: Coulomb (C).

• Current (i): The rate of flow of charge, $i=dq/dt$. It represents the kinetic aspect of the electrical oscillation. Unit: Ampere (A).

• Capacitance (C): A measure of the capacitor's ability to store energy in an electric field. It acts as the "electrical inertia" for potential. Unit: Farad (F).

• Inductance (L): A measure of the inductor's ability to store energy in a magnetic field, resisting changes in current. It acts as the "electrical inertia" for current. Unit: Henry (H).

• Kirchhoff's Loop Rule: A manifestation of energy conservation in a circuit, stating that the net change in electric potential around a closed loop is zero.

• Electric Energy ($U_E$): The potential energy stored in the capacitor's electric field, given by $U_E=q^2/(2C)$. Unit: Joule (J).

• Magnetic Energy ($U_B$): The energy stored in the inductor's magnetic field, given by $U_B=\frac{1}{2}L{i}^2$. Unit: Joule (J).

• Angular Frequency ($\omega$): The natural rate of oscillation of the circuit, determined solely by its physical components: $\omega =1/\sqrt{LC}$. Unit: radians per second (rad/s).

Skill Snapshots

Causation

• Driver → Change: The potential difference across the charged capacitor ($V_C=q/C$) → drives a current to flow through the inductor.

• Driver → Change: A changing current ($di/dt$) through the inductor → causes a self-induced EMF ($V_L=L(di/dt)$) that opposes the change in current, as dictated by Lenz's Law.

• Driver → Change: The combined properties of inductance and capacitance ($L$ and $C$) → determine the natural frequency of oscillation ($\omega =1/\sqrt{LC}$) at which energy is exchanged between the components.

Comparison

• LC Circuit vs. Mass-Spring System: The differential equation $\frac{d^{2}q}{d{t}^2}=-(\frac{1}{LC})q$ is mathematically identical to that of a mass-spring system, $\frac{d^{2}x}{d{t}^2}=-(\frac{k}{m})x$. Inductance (L) is analogous to mass (m), and the inverse of capacitance (1/C) is analogous to the spring constant (k).

• Electric Energy vs. Magnetic Energy: Electric energy ($U_E\propto q^2$) is maximum when the capacitor charge is maximum, analogous to a spring's potential energy at maximum displacement. Magnetic energy ($U_B\propto i^2$) is maximum when the current is maximum, analogous to a mass's kinetic energy at the equilibrium position.

• LC Oscillator vs. RC Circuit: An ideal LC circuit oscillates indefinitely with a constant total energy. In contrast, an RC circuit exhibits exponential decay, where the resistor continuously dissipates energy, preventing any oscillation.

Change and Continuity Over Time

• Baseline: At $t=0$, charge is maximum ($q=Q_{max}$), current is zero ($i=0$), and all energy is stored in the capacitor's electric field ($U_E=max,U_B=0$).

• Change (0 → T/4): The capacitor discharges, creating a growing current. Electric energy transforms into magnetic energy. At $t=T/4$, charge is zero, current is maximum, and all energy is magnetic ($U_E=0,U_B=max$).

• Change (T/4 → T/2): The inductor's collapsing magnetic field maintains the current, which now recharges the capacitor with opposite polarity. Magnetic energy transforms back into electric energy.

• Continuity: Throughout the entire cycle, the total energy of the system, $U_{total}=U_E+U_B$, remains constant due to the absence of a dissipative element like a resistor.

Common Misconceptions & Clarifications

1. Misconception: The current is maximum when the charge on the capacitor is maximum.

   • Clarification: Current is the rate of change of charge ($i=dq/dt$). In any sinusoidal oscillation, the rate of change is zero at the maxima and minima (the turning points) and is greatest when the value itself is passing through zero. Therefore, the current is maximum when the capacitor's charge is zero.

2. Misconception: The oscillation frequency depends on the initial amount of charge.

   • Clarification: The angular frequency, $\omega =1/\sqrt{LC}$, is an intrinsic property of the circuit, determined only by the physical values of L and C. The initial charge $Q_{max}$ determines the amplitude of the charge and current oscillations and the total energy in the system, but not the frequency.

3. Misconception: When the capacitor is fully discharged, the circuit activity stops.

   • Clarification: At the moment the capacitor is fully discharged ($q=0$), the current in the inductor is at its maximum. The energy that was in the capacitor's electric field is now fully stored in the inductor's magnetic field. This stored magnetic energy drives the current to continue flowing, which then recharges the capacitor with the opposite polarity, continuing the oscillation.

One-Paragraph Summary

An ideal LC circuit, containing only a capacitor and an inductor with no resistance, serves as a fundamental model for an electrical oscillator. When initiated with a charged capacitor, the circuit's energy oscillates perpetually between the electric field of the capacitor and the magnetic field of the inductor. Applying Kirchhoff's Loop Rule reveals that the charge on the capacitor follows the differential equation for simple harmonic motion, $\frac{d^{2}q}{d{t}^2}=-\frac{1}{LC}q$. The solution shows that both charge and current vary sinusoidally over time, with a natural angular frequency of $\omega =1/\sqrt{LC}$ that depends only on the circuit's physical components. This principle of energy conservation allows for the direct calculation of the maximum current from the maximum charge, making the LC circuit a perfect electrical analogue to the mechanical mass-spring oscillator.