Circuits with Capacitors and Inductors (LC Circuits)
An ideal LC circuit contains a capacitor network, represented by its equivalent capacitance C, and a single inductor but no resistance. Energy therefore oscillates between the capacitor electric field and the inductor magnetic field instead of being dissipated. Kirchhoff’s loop rule gives a second-order differential equation, d²q/dt²=−q/(LC), mathematically identical to simple harmonic motion. The natural angular frequency is ω=1/√(LC), so larger L or C produces slower oscillation. At maximum capacitor charge the current is zero; when the capacitor is momentarily uncharged, the current and magnetic energy are maximum.
Topic sequence · 6 of 6
Learning objectives
Objective 1
Use conservation of energy to connect equivalent capacitance, capacitor charge, circuit current, electric-field energy, magnetic-field energy, and the maximum current in an ideal LC circuit.
Objective 2
Derive the ideal LC differential equation from Kirchhoff’s loop rule and identify its simple-harmonic-motion structure, including ω=1/√(LC), T=2π√(LC), and f=1/(2π√(LC)).
Objective 3
Interpret and create charge, current, voltage, and energy-versus-time graphs, including quarter-cycle phase relationships and experimental methods for determining L or C from the oscillation period.
- Topic 10.3 capacitor relations, series/parallel equivalent capacitance, and electric energy U_C=q²/(2C_eq)=½C_eqV².
- Topic 13.4 inductance and magnetic energy U_L=½LI².
- Topic 13.5 inductor current continuity and Kirchhoff loop reasoning; unlike LR circuits, the ideal LC model contains no dissipative resistance.
- Mechanics simple harmonic motion and second-order differential equations, especially the analogy between d²x/dt²=−(k/m)x and d²q/dt²=−q/(LC).
Definitions first
Representations you must read and create
Representation 1
Charge-versus-time graph: q is sinusoidal. If q starts at +Q_max with zero current, it follows a cosine curve and first crosses zero at T/4.
Representation 2
Current-versus-time graph: I=dq/dt is shifted by one quarter cycle relative to q. With q=Q_max cosωt, current is −ωQ_max sinωt and has maximum magnitude when q=0.
Representation 3
Energy-versus-time graph: U_C∝q² and U_L∝I² are always nonnegative. They exchange with period T/2, while their sum remains a horizontal constant line in the ideal model.
Representation 4
Circuit snapshots at 0, T/4, T/2, and 3T/4: energy progresses electric → magnetic → electric with reversed capacitor polarity → magnetic with reversed current direction.
Representation 5
Phase-plane representation: plotting I against q gives an ellipse described by q²/Q_max² + I²/I_max² = 1, a compact graphical statement of energy conservation.
Representation 6
Factor-change representation: because T∝√L√C, quadrupling L with C fixed doubles T; multiplying both L and C by four makes T four times larger.
Core relationships and calculus
Variables: q C; C_eq F; L H; dI/dt A/s
Use: Kirchhoff loop equation for an ideal source-free LC loop after any capacitor network is replaced by its equivalent capacitance.
Variables: I A; q C; t s
Use: Current is the time rate of change of the capacitor charge variable for the chosen orientation. A different orientation changes signs consistently.
Variables: q C; L H; C F; t s
Use: Required ideal LC differential equation; same mathematical form as simple harmonic motion.
Variables: ω rad/s; L H; C F
Use: Natural angular frequency of the ideal LC oscillator.
Variables: T s
Use: Oscillation period, derived from T=2π/ω.
Variables: f Hz
Use: Ordinary frequency f=1/T.
Variables: q C; Q_max C; φ rad
Use: General sinusoidal charge solution of the ideal LC differential equation.
Variables: I A
Use: Current obtained by differentiating q(t); charge and current extrema are separated by a quarter cycle.
Variables: U J
Use: Conservation of stored electric plus magnetic energy in the ideal zero-resistance model.
Variables: I A; Q C
Use: Maximum-current result obtained by equating maximum capacitor energy to maximum inductor energy.
Start with Kirchhoff’s loop rule for the ideal source-free loop. With V_C=q/C and the inductor term L dI/dt, choose a consistent loop orientation so q/C+L dI/dt=0. Since I=dq/dt for the chosen charge coordinate, dI/dt=d²q/dt². Therefore L d²q/dt²+q/C=0, or d²q/dt²=−q/(LC). Comparing with the SHM form d²x/dt²=−ω²x identifies ω²=1/(LC). This route explains the oscillation instead of treating the frequency formula as isolated memorization.
For an initially charged capacitor, q(0)=Q_max and I(0)=0, so the differential-equation solution is q=Q_max cosωt. Differentiate to obtain I=−ωQ_max sinωt. The total energy becomes q²/(2C)+½LI² = [Q_max²/(2C)]cos²ωt + ½L(ω²Q_max²)sin²ωt. Because Lω²=1/C, the two terms combine to Q_max²[cos²ωt+sin²ωt]/(2C)=Q_max²/(2C), explicitly proving constant total energy in the ideal model.
The required Topic 13.6 model is an ideal circuit containing one or more capacitors and a single inductor with resistance neglected; multiple capacitors are represented by their equivalent capacitance before applying the LC oscillator model. Use conservation of energy, the SHM charge equation, and the natural angular frequency. Detailed RLC damping, driven AC resonance, impedance, phasors, quality factor, transformer coupling, and radiation losses go beyond this ideal topic unless a problem explicitly supplies the needed model. Real laboratory circuits will show some damping, which should be treated as a limitation of the ideal approximation rather than hidden.
Physical examples
Example 1
If two capacitors form the LC storage network, first replace them by C_eq using the usual series/parallel capacitor rules. Then use C_eq everywhere in ω=1/√(LC_eq), T=2π√(LC_eq), and the energy expressions.
Example 2
If C is increased by a factor of 9 while L stays fixed, ω falls by a factor of 3 and T rises by a factor of 3. The oscillation amplitude does not enter the frequency formula.
Example 3
At an instant when q=0 in an ideal LC circuit, capacitor energy is zero and all stored energy is magnetic, so |I|=I_max.
Example 4
At an instant when |q|=Q_max, current is zero because the charge is at a turning point; all stored energy is electric.
Example 5
If the capacitor has one half of its maximum charge magnitude, U_C/U_total=(q/Q_max)²=1/4, so U_L/U_total=3/4 and |I|=(√3/2)I_max.
Example 6
Doubling Q_max with L and C unchanged doubles I_max but leaves ω and T unchanged. Amplitude changes energy and current scale, not the ideal oscillator frequency.
Example 7
Adding appreciable resistance changes the model from ideal LC to damped RLC behavior. The required ideal LC equations should not be used unchanged when dissipation is explicitly significant.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Begin with energy or the differential equation, depending on the question. Use energy for amplitudes such as I_max; use the differential equation or sinusoidal solution for timing and phase.
Reasoning pattern 2
Keep q and I phase-shifted: maximum |q| occurs when I=0, while maximum |I| occurs when q=0. They are not maximum simultaneously.
Reasoning pattern 3
Do not confuse the period of q or I with the period of the energy curves. q and I repeat every T, while q² and I² repeat every T/2.
Reasoning pattern 4
Use the square-root dependence. If L or C changes by a factor r, period changes by √r when the other element is fixed.
Reasoning pattern 5
Check units: LC has units of s², so √(LC) is a time and 1/√(LC) has units s⁻¹, appropriate for angular frequency.
Reasoning pattern 6
If resistance is stated as non-negligible, ideal LC energy is not constant. Identify whether the problem still asks for an ideal approximation or requires a different model.
Deeper calculus and limiting-case check
The energy ellipse q²/Q_max² + I²/I_max² = 1 follows directly from q²/(2C)+½LI²=Q_max²/(2C) and I_max²=Q_max²/(LC). It is the electrical analogue of the position-velocity phase ellipse for a mass-spring oscillator. The analogy is exact at the equation level: q↔x, L↔m, 1/C↔k, and I=dq/dt↔v. This lets you transfer SHM reasoning about turning points, phase, amplitude independence of frequency, and energy exchange into the LC circuit without confusing the physical quantities.
Model-selection workflow: decide what the problem is before choosing an equation
For Circuits with Capacitors and Inductors (LC Circuits), the fastest reliable route is not equation hunting. Start by identifying the loop geometry and changing flux, encode it with the flux/emf/current representation, choose the Faraday-Lenz or inductor equation, and then test the orientation, ideal-element, and transient assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat LC circuit and equivalent capacitance C_eq as physical quantities first and symbols second.
Choose a representation
Build Charge-versus-time graph: q is sinusoidal. If q starts at +Q_max with zero current, it follows a cosine curve and first crosses zero at T/4.. If the prompt supplies Current-versus-time graph: I=dq/dt is shifted by one quarter cycle relative to q. With q=Q_max cosωt, current is −ωQ_max sinωt and has maximum magnitude when q=0., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is q/C_eq + L dI/dt = 0. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
The required Topic 13.6 model is an ideal circuit containing one or more capacitors and a single inductor with resistance neglected; multiple capacitors are represented by their equivalent capacitance before applying the LC oscillator model. Use conservation of energy, the SHM charge equation, and the natural angular frequency. Detailed RLC damping, driven AC resonance, impedance, phasors, quality factor, transformer coupling, and radiation losses go beyond this ideal topic unless a problem explicitly supplies the needed model. Real laboratory circuits will show some damping, which should be treated as a limitation of the ideal approximation rather than hidden. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | Charge-versus-time graph: q is sinusoidal. If q starts at +Q_max with zero current, it follows a cosine curve and first crosses zero at T/4. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | q/C_eq + L dI/dt = 0 | Kirchhoff loop equation for an ideal source-free LC loop after any capacitor network is replaced by its equivalent capacitance.. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | I = dq/dt | Current is the time rate of change of the capacitor charge variable for the chosen orientation. A different orientation changes signs consistently.. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | Start with Kirchhoff’s loop rule for the ideal source-free loop. With V_C=q/C and the inductor term L dI/dt, choose a consistent loop orientation so q/C+L dI/dt=0. Since I=dq/dt for the chosen charge coordinate, dI/dt=d²q/dt². Therefore L d²q/dt²+q/C=0, or d²q/dt²=−q/(LC). Comparing with the SHM form d²x/dt²=−ω²x identifies ω²=1/(LC). This route explains the oscillation instead of treating the frequency formula as isolated memorization. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
An ideal LC circuit has C=8.00 μF, L=0.200 H, and begins with capacitor charge Q_max=120 μC and zero current. First, ω=1/√(LC)=1/√[(0.200)(8.00×10⁻⁶)]=790.57 rad/s. The period is T=2π/ω=7.95×10⁻³ s and f≈125.8 Hz. Initial electric energy is U=Q_max²/(2C)=(120×10⁻⁶)²/[2(8.00×10⁻⁶)]=9.00×10⁻⁴ J. At q=0 all of this energy is magnetic, so ½LI_max²=9.00×10⁻⁴ J and I_max=√(2U/L)=9.49×10⁻² A. The same result follows from I_max=ωQ_max=(790.57)(120×10⁻⁶)=0.0949 A. At t=T/8, ωt=π/4, so q=Q_max/√2≈84.9 μC and |I|=I_max/√2≈0.0671 A. Each energy is then one half of the total because q²/Q_max²=I²/I_max²=1/2. This simultaneously checks the sinusoidal solution, phase relationship, and energy conservation. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 13.6
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship q/C_eq + L dI/dt = 0 should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Charge-versus-time graph: q is sinusoidal. If q starts at +Q_max with zero current, it follows a cosine curve and first crosses zero at T/4. and Current-versus-time graph: I=dq/dt is shifted by one quarter cycle relative to q. With q=Q_max cosωt, current is −ωQ_max sinωt and has maximum magnitude when q=0.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Construct a low-resistance LC circuit using a known capacitor, inductor, switch, and voltage/current sensor or oscilloscope. Charge the capacitor from a DC source, isolate the source, then connect the capacitor to the inductor and record V_C(t) or I(t). Determine the oscillation period from repeated peaks and compare the measured value with T=2π√(LC). Repeat with a different capacitor or inductor and test the square-root scaling. Estimate q(t)=CV_C(t) and compare electric and magnetic energy when current data are available. Real traces should decay slowly because coil resistance and other losses make the circuit nonideal; quantify that discrepancy rather than forcing the data to appear perfectly undamped.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Using τ=L/R from LR circuits in an LC oscillator. Ideal LC behavior is oscillatory with ω=1/√(LC), not exponential. | Return to the flux/emf/current representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Assuming capacitor energy and inductor energy can both equal the total energy at the same instant. | Return to the Faraday-Lenz or inductor equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Writing I and q in phase. Because I=dq/dt, their extrema are separated by one quarter cycle. | Return to the flux/emf/current representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Forgetting the square root in T=2π√(LC) or writing ω=1/(LC). | Return to the Faraday-Lenz or inductor equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Using Q_max/C as the maximum current. Q_max/C is a voltage scale; I_max=Q_max/√(LC)=ωQ_max. | Return to the flux/emf/current representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Claiming a real LC circuit oscillates forever. Undamped behavior belongs to the ideal zero-resistance model. | Return to the Faraday-Lenz or inductor equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 10.3 supplies capacitor charge, voltage, and electric-field energy; Topic 13.4 supplies inductor magnetic energy. LC oscillation is the direct exchange between those two storage mechanisms. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 13.5 uses the same inductor but includes resistance, producing first-order exponential behavior. Removing the dissipative resistor and adding a capacitor produces a second-order oscillatory model. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Mechanics simple harmonic motion has the same mathematical structure. Charge q plays the role of displacement, inductance L the role of mass, and 1/C the role of the spring constant. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write q/C_eq + L dI/dt = 0 and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 13.6 mastery ladder
Interactive LC energy-exchange check
Start from maximum capacitor charge with zero current. The tool calculates ω, T, f, q(t), I(t), electric energy, magnetic energy, total energy, and I_max for the ideal undamped LC model.
Experimental connection
Construct a low-resistance LC circuit using a known capacitor, inductor, switch, and voltage/current sensor or oscilloscope. Charge the capacitor from a DC source, isolate the source, then connect the capacitor to the inductor and record V_C(t) or I(t). Determine the oscillation period from repeated peaks and compare the measured value with T=2π√(LC). Repeat with a different capacitor or inductor and test the square-root scaling. Estimate q(t)=CV_C(t) and compare electric and magnetic energy when current data are available. Real traces should decay slowly because coil resistance and other losses make the circuit nonideal; quantify that discrepancy rather than forcing the data to appear perfectly undamped.
Connections across Physics C
Connection 1
Topic 10.3 supplies capacitor charge, voltage, and electric-field energy; Topic 13.4 supplies inductor magnetic energy. LC oscillation is the direct exchange between those two storage mechanisms.
Connection 2
Topic 13.5 uses the same inductor but includes resistance, producing first-order exponential behavior. Removing the dissipative resistor and adding a capacitor produces a second-order oscillatory model.
Connection 3
Mechanics simple harmonic motion has the same mathematical structure. Charge q plays the role of displacement, inductance L the role of mass, and 1/C the role of the spring constant.
Connection 4
Kirchhoff’s loop rule from Unit 11 supplies the circuit equation whose derivative structure generates the LC oscillator.
Connection 5
Energy conservation provides an independent check of the differential-equation solution and gives the fastest route to maximum current.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Using τ=L/R from LR circuits in an LC oscillator. Ideal LC behavior is oscillatory with ω=1/√(LC), not exponential.
Avoid this
Assuming capacitor energy and inductor energy can both equal the total energy at the same instant.
Avoid this
Writing I and q in phase. Because I=dq/dt, their extrema are separated by one quarter cycle.
Avoid this
Forgetting the square root in T=2π√(LC) or writing ω=1/(LC).
Avoid this
Using Q_max/C as the maximum current. Q_max/C is a voltage scale; I_max=Q_max/√(LC)=ωQ_max.
Avoid this
Claiming a real LC circuit oscillates forever. Undamped behavior belongs to the ideal zero-resistance model.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
Why does an LC circuit oscillate?
A charged capacitor drives current through the inductor. As capacitor energy falls, magnetic energy rises; the inductor then sustains current long enough to recharge the capacitor with opposite polarity, repeating the exchange.
What is the angular frequency of an ideal LC circuit?
ω=1/√(LC). The period is T=2π√(LC) and the ordinary frequency is f=1/(2π√(LC)).
When is current maximum?
Current magnitude is maximum when the capacitor is momentarily uncharged, so all stored energy is magnetic.
Does increasing the initial capacitor charge change the frequency?
No in the ideal linear model. It increases charge amplitude, current amplitude, and total energy, but ω depends only on L and C.
Why are charge and current one quarter cycle out of phase?
Current is the derivative of charge. The derivative of a cosine is a negative sine, so current extrema occur when charge crosses zero.
What happens in a real LC circuit?
Real resistance and other losses dissipate energy, so the amplitude gradually decreases. Topic 13.6 uses the ideal undamped approximation unless another model is specified.