SCIENCE · INVESTIGATE AND EXPLAIN

Match three damping regimes without a singular formula

The free circuit keeps L=1 H and C=0.25 F but adds a nonnegative series resistance R. Starting from q=1 C and I=0, the equation is qsecond+R qprime+4q=0. Its roots are −R/2 plus or minus the square root of (R/2)²−4, so the critical resistance is 4 Ω.

Learning outcome: Compare R=0,2,4,6 at the same elapsed times. Retain the branch label, charge, current and energy loss; explain why the critical expression needs a separate repeated-root limit and why the zero-resistance history cannot lose energy.

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

At R=4 Ω and t=1 s, q=3e^(−2)≈0.406006 C and I=−4e^(−2)≈−0.541341 A. Stored energy is 26e^(−4)≈0.476207 J, and resistor conversion is 2 minus that value. The critical formula satisfies q(0)=1 and I(0)=0 directly.

Open the experiment

Predict

Why is the critical branch evaluated separately?

Your prediction and reason

Experiment

Compare R=0,2,4,6 at the same elapsed times. Retain the branch label, charge, current and energy loss; explain why the critical expression needs a separate repeated-root limit and why the zero-resistance history cannot lose energy.

Match underdamped, critical and overdamped solutions

Worked teaching model · synthetic values · Free L1 H,C.25 F circuit with q(0)=1 C,I(0)=0. Equation qsecond+R qprime+4q=0; critical R=4 ohm. Ideal positive passive elements, no source or parasitics. Stored energy and resistor conversion, not a physical circuit trial.

Series resistance: 4 ohm Elapsed time: 1 s

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Charge history across damping regimes

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: critical (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Charge history across damping regimes

critical

time (s)	charge (C)
0	1
0.05	0.9953212
0.1	0.9824769
0.15	0.9630637
0.2	0.9384481
0.25	0.909796
0.3	0.8780986
0.35	0.844195
0.4	0.8087921
0.45	0.7724824
0.5	0.7357589
0.55	0.6990293
0.6	0.6626273
0.65	0.6268231
0.7	0.5918327
0.75	0.5578254
0.8	0.5249309
0.85	0.4932455
0.9	0.4628369
0.95	0.433749
1	0.4060058
1.05	0.3796149
1.1	0.3545701
1.15	0.3308542
1.2	0.308441
1.25	0.2872975
1.3	0.2673849
1.35	0.2486604
1.4	0.2310782
1.45	0.2145906
1.5	0.1991483
1.55	0.1847017
1.6	0.1712013
1.65	0.1585976
1.7	0.1468424
1.75	0.1358882
1.8	0.1256891
1.85	0.1162006
1.9	0.1073797
1.95	0.09918537
2	0.09157819
2.05	0.08452064
2.1	0.077977
2.15	0.07191336
2.2	0.06629764
2.25	0.06109948
2.3	0.05629028
2.35	0.05184308
2.4	0.04773253
2.45	0.04393484
2.5	0.04042768
2.55	0.03719015
2.6	0.0342027
2.65	0.03144704
2.7	0.02890612
2.75	0.02656401
2.8	0.0244059
2.85	0.02241797
2.9	0.02058737
2.95	0.01890217
3	0.01735127
3.05	0.01592436
3.1	0.0146119
3.15	0.01340502
3.2	0.01229552
3.25	0.01127579
3.3	0.0103388
3.35	0.009478022
3.4	0.008687446
3.45	0.007961505
3.5	0.007295056
3.55	0.00668335
3.6	0.006122004
3.65	0.005606972
3.7	0.005134523
3.75	0.004701217
3.8	0.004303882
3.85	0.003939596
3.9	0.003605668
3.95	0.003299618
4	0.003019164

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
Damping coefficient	2	s⁻¹
Charge	0.4060058	C
Current	-0.5413411	A
Stored energy	0.4762066	J
Resistor conversion since release	1.523793	J
Instantaneous energy-loss rate	1.172201	W
Critical branch	1	flag

The repeated-root solution is handled directly rather than dividing by a vanishing damped frequency.

The characteristic roots are −R/2 plus or minus sqrt((R/2)²−4). For R below 4 use sine/cosine, at 4 use (1+2t)e^(−2t), and above 4 use the equivalent hyperbolic form. The initial conditions remain q=1,I=0 in every branch. Multiplying the ODE by I gives dU/dt=−R I². Zero resistance preserves exactly the initial 2 J, whereas larger resistance does not necessarily mean faster charge return in the overdamped regime.

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

The repeated-root solution is handled directly rather than dividing by a vanishing damped frequency.

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

Below critical, use a decaying sine/cosine solution with damped frequency sqrt[4−(R/2)²]. At critical, the repeated root gives q=(1+2t)e^(−2t), not a formula that divides by zero frequency. Above critical, two distinct real decay roots produce the corresponding nonoscillatory solution. Every branch must still satisfy the same initial charge and zero initial current.

Audit energy, uncertainty and domain

Multiplying the equation by current gives dU/dt=−RI², with U=2q²+I²/2. Resistance cannot add energy in this model. At R=0 energy remains exactly 2 J. Increasing resistance beyond critical can slow the long-time charge return, so greater dissipation coefficient is not a universal synonym for faster settling.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

At R=4 Ω and t=0.5 s, find charge in coulombs.

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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