SCIENCE · INVESTIGATE AND EXPLAIN

Separate LC components with independent amplitude evidence

In an ideal LC circuit, T = 2pi sqrt(LC).

Learning outcome: Construct an equal-period alternative, retain two independent relations and solve both parameters. Add a gain-confounded counterexample, then specify one genuinely independent observation that would resolve it. Separate synthetic record agreement from performed measurement and independent validation.

Starting recap

Differentiate linked flux, preserve circuit state and test energy and independent-calibration constraints.

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Try

L = 2 H and C = 3 F with V0 = 2 V give T = 2pi sqrt(6) ≈ 15.390598 s and Ipeak = 2sqrt(3/2) ≈ 2.449490 A. Substitution recovers L = 2 H and C = 3 F. A current gain of two would instead produce apparent L = 1 H and C = 6 F while preserving their product.

Open the experiment

Predict

An unknown multiplicative current gain changes which inference?

Your prediction and reason

Experiment

Construct an equal-period alternative, retain two independent relations and solve both parameters. Add a gain-confounded counterexample, then specify one genuinely independent observation that would resolve it. Separate synthetic record agreement from performed measurement and independent validation.

Identify two components from independent records

Worked teaching model · synthetic values · Ideal LC circuit starts at capacitor voltage 2 V and zero current. Period and independently calibrated peak-current magnitude are supplied. These are synthetic records, not actual measurements. Unknown sensor gain is a separate counterexample.

True inductance: 2 H True capacitance: 3 F

Reset keeps your prediction, captures and practice history.

State and topology reference

This original schematic defines one closed loop; it is not an apparatus wiring instruction. Exact element connections and event conditions are stated below. Component sizes are schematic.

The capacitor and inductor form one isolated loop; there is no source or resistor.
The q label marks the selected left capacitor plate. The arrow is toward that plate, so i = dq/dt.
L d²q/dt² + q/C = 0. Reversing the chosen plate reverses q and i together, not the nonnegative energy.

Equal-period capacitance alternatives

Axes rescale independently. Curves join sampled coordinates; a joining segment is not additional measured evidence. Read physical units rather than screen angles or apparent slopes.

Equal-period capacitance alternatives: numerical coordinates

candidate inductance (H)	candidate capacitance (F)
1	6
1.05	5.71429
1.1	5.45455
1.15	5.21739
1.2	5
1.25	4.8
1.3	4.61538
1.35	4.44444
1.4	4.28571
1.45	4.13793
1.5	4
1.55	3.87097
1.6	3.75
1.65	3.63636
1.7	3.52941
1.75	3.42857
1.8	3.33333
1.85	3.24324
1.9	3.15789
1.95	3.07692
2	3
2.05	2.92683
2.1	2.85714
2.15	2.7907
2.2	2.72727
2.25	2.66667
2.3	2.6087
2.35	2.55319
2.4	2.5
2.45	2.44898
2.5	2.4
2.55	2.35294
2.6	2.30769
2.65	2.26415
2.7	2.22222
2.75	2.18182
2.8	2.14286
2.85	2.10526
2.9	2.06897
2.95	2.0339
3	2
3.05	1.96721
3.1	1.93548
3.15	1.90476
3.2	1.875
3.25	1.84615
3.3	1.81818
3.35	1.79104
3.4	1.76471
3.45	1.73913
3.5	1.71429
3.55	1.69014
3.6	1.66667
3.65	1.64384
3.7	1.62162
3.75	1.6
3.8	1.57895
3.85	1.55844
3.9	1.53846
3.95	1.51899
4	1.5

Calculated values under the declared model
Quantity	Value	Unit
Period record	15.3906	s
Peak-current record	2.44949	A
Known initial voltage	2	V
Recovered inductance	2	H
Recovered capacitance	3	F
Product constrained by period alone	6	H F

Period fixes a product; an independently calibrated amplitude fixes a ratio and separates the two components.

Period gives LC and energy gives Ipeak/V0 = sqrt(C/L). Combine to get L = T V0/(2pi Ipeak) and C = T Ipeak/(2pi V0). If current gain is unknown, these two inferred components trade off with that gain, so numerical agreement is not independent identification.

Derive before checking. Retain contrasting captures and explain one assumption that could fail; calculated evidence is not actual laboratory work.

Period fixes a product; an independently calibrated amplitude fixes a ratio and separates the two components.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

A fitted period does not identify everything

In an ideal LC circuit, T = 2pi sqrt(LC). A measured period determines the product LC, not each factor. The pairs L = 2 H, C = 3 F and L = 3 H, C = 2 F produce the same period. A good period fit is therefore compatible with distinct components; more digits in the same record do not resolve that structural ambiguity.

Use an independently calibrated second relation

Suppose the circuit begins with known capacitor voltage V0 and zero current. Conservation gives peak current Ipeak = V0 sqrt(C/L). This supplies a ratio while the period supplies a product. Solving gives L = T V0/(2pi Ipeak) and C = T Ipeak/(2pi V0). The voltage, current calibration, initial condition and negligible-loss model must all be independent assumptions or evidence, not quantities inferred from the same desired answer.

Expose sensor-gain confounding

If a current sensor reports g times true current and g is unknown, using that report as Ipeak gives inferred L divided by g and inferred C multiplied by g. Their product still agrees perfectly with the period. A period residual cannot reveal this gain error. An independent current calibration or an independently known component can break the ambiguity; another identical uncalibrated trace may not.

Design a virtual evidence comparison

Choose two component pairs with the same product and predict equal periods but different peak currents from the same initial voltage. Retain the predictions before revealing the model record. Then introduce a supplied current-gain error and explain which reconstructed quantity changes and which remains fixed. A real apparatus plan would require qualified supervision and instrument constraints; this workspace records a conceptual design and supplied-data analysis only.

Worked original case

Investigation

Changed-case practice

Try a new practice case

With V0 = 2 V, T = 4pi s and Ipeak = 1 A, what inductance in henries is inferred?

Numerical result in the requested units

This public, unsaved check is not a secure assessment. Explain the source integral, symmetry or charge boundary in the retained course evidence workspace.

Sources and boundaries

Original teaching checked 31 October 2026 against current official Unit 13 and the relevant OpenStax discussion. No source exercises or diagrams are reproduced. All records and apparatus are virtual, supplied or generated; no real wiring, switching, magnetic braking or electrical measurement is requested. Public feedback and retained work do not establish independent mastery, supervised practical participation or AP authorization.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which orientation, initial state, symmetry or independent calibration assumption would change this induction inference?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Use both initial conditions in an LC oscillation Next: Recover a flux history without inventing its missing anchor

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