SCIENCE · INVESTIGATE AND EXPLAIN

Preserve both initial states of an ideal LC oscillation

An ideal LC oscillator needs both initial charge and initial current. Here L=1 H and C=0.25 F, giving angular frequency 2 rad/s. Signed current is dq/dt. Neither initial state is forced to vanish, and the zero-state case is included rather than being assigned a fictitious oscillation amplitude.

Learning outcome: Compare initial states (1,0),(0,2),(1,2) and (0,0). Predict the quarter-period state before inspecting the phase curve. Explain which initial states have equal energy but different phase and why one observed charge cannot identify both states.

Starting recap

Review flux integrals, linear differential equations, initial conditions, complex numbers and vector derivatives. Keep source constraints, RMS conventions and calibration roles explicit.

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Try

For q0=1 C and I0=2 A, the quarter-period state is q=1 C and I=−2 A. Electric and magnetic energies are each 2 J, totaling 4 J. Despite the same charge as initially, current has reversed: charge alone does not specify the full oscillator state.

Open the experiment

Predict

What additional datum is needed besides initial charge to fix this LC solution?

Your prediction and reason

Experiment

Compare initial states (1,0),(0,2),(1,2) and (0,0). Predict the quarter-period state before inspecting the phase curve. Explain which initial states have equal energy but different phase and why one observed charge cannot identify both states.

Solve an LC state with initial charge and current

Worked teaching model · synthetic values · Ideal L1 H,C.25 F,zero resistance and no source. Initial charge/current independently stipulated; omega0=2 rad/s. Observe t=pi/4 s, one quarter period. Fictional large-scale teaching values, not charged-component instructions.

Initial capacitor charge: 1 C Initial signed current dq/dt: 1 A

Reset keeps your prediction, captures and practice history.

Energy-normalized phase plane

Spatial axes use the same pixels per physical unit, preserving geometry. 1: analytic state over one period (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Energy-normalized phase plane

analytic state over one period

charge q (C)	current divided by omega0 (C)
1	0.5
1.036147	0.4199996
1.065906	0.3374097
1.089093	0.2527396
1.105565	0.1665113
1.115221	0.07925633
1.118002	-0.008487238
1.113889	-0.09617848
1.10291	-0.1832768
1.08513	-0.2692451
1.06066	-0.3535534
1.029651	-0.4356819
0.9922937	-0.5151244
0.9488186	-0.5913909
0.8994938	-0.6640113
0.8446232	-0.7325378
0.7845453	-0.796548
0.7196303	-0.8556472
0.6502786	-0.9094711
0.5769178	-0.9576878
0.5	-1
0.4199996	-1.036147
0.3374097	-1.065906
0.2527396	-1.089093
0.1665113	-1.105565
0.07925633	-1.115221
-0.008487238	-1.118002
-0.09617848	-1.113889
-0.1832768	-1.10291
-0.2692451	-1.08513
-0.3535534	-1.06066
-0.4356819	-1.029651
-0.5151244	-0.9922937
-0.5913909	-0.9488186
-0.6640113	-0.8994938
-0.7325378	-0.8446232
-0.796548	-0.7845453
-0.8556472	-0.7196303
-0.9094711	-0.6502786
-0.9576878	-0.5769178
-1	-0.5
-1.036147	-0.4199996
-1.065906	-0.3374097
-1.089093	-0.2527396
-1.105565	-0.1665113
-1.115221	-0.07925633
-1.118002	0.008487238
-1.113889	0.09617848
-1.10291	0.1832768
-1.08513	0.2692451
-1.06066	0.3535534
-1.029651	0.4356819
-0.9922937	0.5151244
-0.9488186	0.5913909
-0.8994938	0.6640113
-0.8446232	0.7325378
-0.7845453	0.796548
-0.7196303	0.8556472
-0.6502786	0.9094711
-0.5769178	0.9576878
-0.5	1
-0.4199996	1.036147
-0.3374097	1.065906
-0.2527396	1.089093
-0.1665113	1.105565
-0.07925633	1.115221
0.008487238	1.118002
0.09617848	1.113889
0.1832768	1.10291
0.2692451	1.08513
0.3535534	1.06066
0.4356819	1.029651
0.5151244	0.9922937
0.5913909	0.9488186
0.6640113	0.8994938
0.7325378	0.8446232
0.796548	0.7845453
0.8556472	0.7196303
0.9094711	0.6502786
0.9576878	0.5769178
1	0.5

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Natural angular frequency	2	rad/s
Charge at quarter period	0.5	C
Current at quarter period	-2	A
Electric energy at quarter period	0.5	J
Magnetic energy at quarter period	2	J
Total conserved energy	2.5	J
Oscillation period	3.141593	s

Both initial charge and initial current set the phase; neither can be silently reset.

qsecond+q/(LC)=0 with omega0=2 has q=q0 cos(2t)+(I0/2)sin(2t) and I=−2q0 sin(2t)+I0 cos(2t). At t=pi/4 the trigonometric boundary gives q=I0/2 and I=−2q0 exactly. In the normalized phase plane q versus I/omega0, a circle expresses constant energy. Its coordinates are not physical wire positions.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Both initial charge and initial current set the phase; neither can be silently reset.

Observation to attach to a capture

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

Understand

State geometry, sources and initial data

Derive the field or circuit relation

The differential equation qsecond+4q=0 has solution q=q0 cos(2t)+(I0/2)sin(2t), with I=−2q0 sin(2t)+I0 cos(2t). At one quarter period, t=π/4 s, charge is exactly I0/2 and current exactly −2q0. The familiar initially charged, zero-current picture is only one member of this family.

Audit energy, uncertainty and domain

Energy is q²/(2C)+LI²/2=2q0²+I0²/2. Plotting q against I/omega0 gives a circle with equal coordinate units and scale; it is a phase-state diagram, not the physical shape of a wire or a charge path. Resistance, radiation or a connected source would change the energy equation and require a different model.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

For q0=−1 C and I0=0, find current at t=π/4 s in amperes.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original derivations and synthetic evidence checked 31 October 2026 against the primary explanation. Constants are rounded teaching values and are mutually consistent where Maxwell relations require it. No live circuit, high-voltage, energized-inductor, magnet, beam or optical exposure activity is requested. Retain first work, criterion-linked critique, revision and newly assigned evidence; public numerical feedback is not practical or learner certification.

Apply

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

Which source, initial-state, calibration or field-law premise makes the result conditional, and what unused evidence would challenge it?

Your explanation and limitations

Continue

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

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Previous: Reject a coupling matrix that permits negative field energy Next: Match three damping regimes without a singular formula

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