SCIENCE · INVESTIGATE AND EXPLAIN

Match the state instead of restarting the transient

A finite resistor limits current even when an ideal source changes abruptly. The same RC=1 s element starts at zero voltage, sees 6 V until time ts, and then sees −2 V. The resistor remains connected. Capacitor voltage is continuous because a finite current cannot transfer finite charge in zero time.

Learning outcome: Compare observations just before, exactly at and after a chosen switch. Retain which one-sided convention is used for current. Predict how delaying the switch changes the second initial state without changing the −2 V final asymptote.

Starting recap

Review charge continuity, definite integrals, first-order differential equations, initial conditions and signed source work. Distinguish displayed time from physical time.

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Try

If ts=1 s, boundary voltage is 6(1−1/e)≈3.792723 V. Just before switching, current is 3/e mA≈1.103638 mA. Just after, it is −2.896362 mA. Their difference is −4 mA while capacitor voltage has not jumped.

Open the experiment

Predict

What changes discontinuously at the ideal switch in this bounded model?

Your prediction and reason

Experiment

Compare observations just before, exactly at and after a chosen switch. Retain which one-sided convention is used for current. Predict how delaying the switch changes the second initial state without changing the −2 V final asymptote.

Match capacitor state across a source change

Worked teaching model · synthetic values · Same 2000 ohm and 500 microfarad element, initially zero voltage. Source is 6 V before the switch and −2 V afterward; switch does not disconnect the resistor. At the exact switch instant display the right-hand source/current and continuous capacitor voltage. No impulsive capacitor voltage jump.

Switch time: 1 s Observation time: 1.5 s

Reset keeps your prediction, captures and practice history.

Continuous state with a changed derivative

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: piecewise analytic voltage (solid).2: actual switch coordinate (open-circle marked location). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Continuous state with a changed derivative

piecewise analytic voltage

time (s)	capacitor voltage (V)
0	0
0.05	0.2926235
0.1	0.5709755
0.15	0.8357521
0.2	1.087615
0.25	1.327195
0.3	1.555091
0.35	1.771871
0.4	1.97808
0.45	2.174231
0.5	2.360816
0.55	2.538301
0.6	2.70713
0.65	2.867725
0.7	3.020488
0.75	3.165801
0.8	3.304026
0.85	3.43551
0.9	3.560582
0.95	3.679554
1	3.792723
1.05	3.510209
1.1	3.241473
1.15	2.985843
1.2	2.742681
1.25	2.511377
1.3	2.291355
1.35	2.082063
1.4	1.882979
1.45	1.693603
1.5	1.513464
1.55	1.342111
1.6	1.179114
1.65	1.024067
1.7	0.8765813
1.75	0.7362888
1.8	0.6028384
1.85	0.4758965
1.9	0.3551456
1.95	0.2402838
2	0.1310238
2.05	0.02709257
2.1	-0.0717699
2.15	-0.1658108
2.2	-0.2552653
2.25	-0.340357
2.3	-0.4212987
2.35	-0.4982929
2.4	-0.571532
2.45	-0.6411992
2.5	-0.7074687
2.55	-0.7705062
2.6	-0.8304693
2.65	-0.887508
2.7	-0.9417649
2.75	-0.9933756
2.8	-1.042469
2.85	-1.089169
2.9	-1.13359
2.95	-1.175846
3	-1.21604
3.05	-1.254274
3.1	-1.290644
3.15	-1.325239
3.2	-1.358148
3.25	-1.389451
3.3	-1.419228
3.35	-1.447553
3.4	-1.474496
3.45	-1.500125
3.5	-1.524504
3.55	-1.547695
3.6	-1.569754
3.65	-1.590737
3.7	-1.610697
3.75	-1.629684
3.8	-1.647744
3.85	-1.664924
3.9	-1.681266
3.95	-1.696811
4	-1.711597

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.

actual switch coordinate

time (s)	capacitor voltage (V)
1	3.792723

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
Voltage at switching boundary	3.792723	V
Observed capacitor voltage	1.513464	V
Right-hand selected source voltage	-2	V
Observed signed current	-0.001756732	A
Current immediately before switch	0.001103638	A
Current immediately after switch	-0.002896362	A
Current jump	-0.004	A

Finite resistor current allows the derivative to jump, not the capacitor voltage itself.

The first interval ends at Vs=6(1−e^(−ts)). Use that voltage as the second initial condition: V=−2+(Vs+2)e^(−(t−ts)). The source changes by −8 V, so resistor current jumps by −8/2000 A. At the exact switch the display uses the right-hand current. Resetting the capacitor to zero or restarting the original formula discards stored charge.

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

Finite resistor current allows the derivative to jump, not the capacitor voltage itself.

Observation to attach to a capture

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

Understand

State the current, source and measurement model

Derive the integral or initial-value solution

First integrate V=6(1−e^(−t)) up to ts and call that boundary value Vs. The second interval is V=−2+(Vs+2)e^(−(t−ts)). The new source fixes the new asymptote, while the old history supplies the new initial state. Resetting voltage to zero at the switch loses stored charge.

Test conservation, uncertainty and limits

At the switching instant the source changes by −8 V, so current changes by −8/2000=−0.004 A. A derivative jump is compatible with continuous state. The display uses the right-hand source and current at the exact switch, and marks the actual switch coordinate rather than inventing two nearby observation locations.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

For any allowed switch time, find the current jump in mA.

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. No live circuit construction, energized switching, component rating or electrical handling instruction is requested. Preserve first predictions, critique, revision and fresh assigned work separately from public practice. Software agreement does not certify measurement technique or learning.

Apply

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

Which initial state, calibration or constitutive assumption is needed for this conclusion, and what fresh evidence could test 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.

Skipping retains experiments and awards no mastery.

Previous: Solve an RC initial-value problem with energy return Next: Retain initial current and the inductive energy cross term

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