SCIENCE · INVESTIGATE AND EXPLAIN

Carry capacitor state across a topology change

The capacitor begins uncharged and charges through 2 kohm from a 6 V source.

Learning outcome: Predict the event voltage for several charging durations, retain continuity and both one-sided currents, then compare voltage and energy decay ratios. Explain why restarting the discharge at the source voltage would invent stored charge after a finite charging time.

Starting recap

Integrate current or resistance, preserve node topology and derive transient state with a signed energy account.

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With C = 2 microfarad and ts = 2 ms, tauc = 4 ms and Vs ≈ 2.36082 V. Current changes from about +1.81959 mA to −0.590204 mA at the event. The discharge time constant is 8 ms; one such interval later V ≈ 0.868496 V.

Open the experiment

Predict

Which quantity must remain continuous at this finite-resistance switch event?

Your prediction and reason

Experiment

Predict the event voltage for several charging durations, retain continuity and both one-sided currents, then compare voltage and energy decay ratios. Explain why restarting the discharge at the source voltage would invent stored charge after a finite charging time.

Match two RC states at a switching event

Worked teaching model · synthetic values · Initially uncharged capacitor charges from 6 V through 2 kohm, then the source branch is disconnected in the model and the capacitor discharges through 4 kohm. Positive current always means increasing positive-plate charge; discharge current is negative. No physical switching instructions.

Charging duration: 2 ms Capacitance: 2 microfarad

Reset keeps your prediction, captures and practice history.

Read the node connections before calculating

Generic source–resistor–capacitor series topology. Capacitor voltage is P relative to B. Positive current enters plate P. For the switching lesson, the second state removes the source and replaces the branch by the stated discharge resistor; it is not simultaneous with this charging state. Component orientation and drawing lengths are schematic; source polarities are specified in the caption and node labels.

First node	Second node	Element
S	P	R
P	B	C
B	S	source branch

Common schematic symbols and conventions

The symbol key is not a wiring recommendation. An open switch removes a conducting path; an ideal closed switch supplies a zero-drop connection. A current arrow is a chosen conventional-charge reference, not electron drift. A junction dot signifies connection; a crossing without a dot need not connect.

Piecewise capacitor voltage

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.

Piecewise capacitor voltage: numerical coordinates

time (ms)	voltage (V)
0	0
0.566667	0.792537
1.13333	1.48039
1.7	2.07738
2.26667	2.28342
2.83333	2.12727
3.4	1.9818
3.96667	1.84628
4.53333	1.72003
5.1	1.60241
5.66667	1.49283
6.23333	1.39075
6.8	1.29564
7.36667	1.20704
7.93333	1.1245
8.5	1.04761
9.06667	0.975967
9.63333	0.909228
10.2	0.847052
10.7667	0.789129
11.3333	0.735166
11.9	0.684893
12.4667	0.638058
13.0333	0.594426
13.6	0.553777
14.1667	0.515908
14.7333	0.480629
15.3	0.447762
15.8667	0.417143
16.4333	0.388618
17	0.362043
17.5667	0.337285
18.1333	0.314221
18.7	0.292733
19.2667	0.272716
19.8333	0.254066
20.4	0.236693
20.9667	0.220507
21.5333	0.205428
22.1	0.19138
22.6667	0.178293
23.2333	0.166101
23.8	0.154743
24.3667	0.144161
24.9333	0.134303
25.5	0.125119
26.0667	0.116563
26.6333	0.108592
27.2	0.101166
27.7667	0.094248
28.3333	0.087803
28.9	0.0817988
29.4667	0.0762052
30.0333	0.070994
30.6	0.0661393
31.1667	0.0616165
31.7333	0.057403
32.3	0.0534776
32.8667	0.0498206
33.4333	0.0464138
34	0.0432399

Discharge current after switching

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.

Discharge current after switching: numerical coordinates

time since switch (ms)	current (mA)
0	-0.590204
0.533333	-0.55214
1.06667	-0.516531
1.6	-0.483218
2.13333	-0.452054
2.66667	-0.4229
3.2	-0.395626
3.73333	-0.37011
4.26667	-0.346241
4.8	-0.323911
5.33333	-0.303021
5.86667	-0.283478
6.4	-0.265196
6.93333	-0.248092
7.46667	-0.232092
8	-0.217124
8.53333	-0.203121
9.06667	-0.190021
9.6	-0.177766
10.1333	-0.166301
10.6667	-0.155576
11.2	-0.145543
11.7333	-0.136156
12.2667	-0.127375
12.8	-0.11916
13.3333	-0.111475
13.8667	-0.104286
14.4	-0.0975601
14.9333	-0.0912681
15.4667	-0.085382
16	-0.0798754
16.5333	-0.074724
17.0667	-0.0699048
17.6	-0.0653965
18.1333	-0.0611789
18.6667	-0.0572332
19.2	-0.0535421
19.7333	-0.050089
20.2667	-0.0468586
20.8	-0.0438366
21.3333	-0.0410094
21.8667	-0.0383646
22.4	-0.0358903
22.9333	-0.0335757
23.4667	-0.0314103
24	-0.0293845
24.5333	-0.0274894
25.0667	-0.0257166
25.6	-0.024058
26.1333	-0.0225064
26.6667	-0.0210549
27.2	-0.019697
27.7333	-0.0184267
28.2667	-0.0172383
28.8	-0.0161266
29.3333	-0.0150865
29.8667	-0.0141135
30.4	-0.0132033
30.9333	-0.0123518
31.4667	-0.0115552
32	-0.01081

Calculated values under the declared model
Quantity	Value	Unit
Charging time constant	4	ms
Discharge time constant	8	ms
Voltage continuous at switch	2.36082	V
Current just before switch	1.81959	mA
Current just after switch	-0.590204	mA
Voltage one discharge time constant later	0.868496	V
Energy ratio after one discharge time constant	0.135335	ratio
Energy at switch	5.57345	microjoule

Capacitor voltage is continuous at the event, while the circuit change permits an immediate current change.

Solve charging up to the switch, retain its terminal value, then use that value as the discharge initial condition. Restarting the discharge at 6 V would invent charge unless the charging interval were infinite. Energy is proportional to V², so its decay exponent is twice that of voltage. Branch-current arrows keep one charge-increasing convention across both states.

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

Capacitor voltage is continuous at the event, while the circuit change permits an immediate current change.

Observation to attach to a capture

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

Understand

Solve the first interval

The capacitor begins uncharged and charges through 2 kohm from a 6 V source. Until the supplied switch time ts, V(t) = 6[1 − exp(−t/tauc)] with tauc = 2 kohm times C. Calculate Vs = V(ts) before changing the circuit model; it is not generally 6 V.

Use the terminal value as the next initial value

After the modeled source branch is removed and the 4 kohm discharge branch is selected, V(t) = Vs exp[−(t − ts)/taud], where taud = 4 kohm times C. Voltage and charge remain continuous at ts because the finite-resistance model contains no impulsive current. The derivative and branch current can jump because the circuit equation changed.

Keep one current convention

Positive current means increasing positive-plate charge throughout. Just before the event, I = (6 − Vs)/(2 kohm); just after, I = −Vs/(4 kohm). These opposite signs reflect charging and discharging, not inconsistent arithmetic. The source branch and discharge branch are distinct connections, represented as separate states rather than an impossible simultaneous topology.

Track the squared state variable

Stored energy is CV²/2. During discharge it therefore decays as exp[−2(t − ts)/taud]. After one discharge time constant, voltage retains about 36.8% while energy retains about 13.5%. The lost energy is delivered to the resistor within this ideal account. A plot of voltage alone cannot be labeled a plot of energy.

Worked original case

Investigation

Changed-case practice

Try a new practice case

What fraction of the switch-time energy remains after one discharge time constant?

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 evidence boundary

Original teaching checked 31 October 2026 against the current official framework, Unit 11 and OpenStax RC circuits. All diagrams and numerical records are virtual, supplied or generated. No physical wiring, battery connection, short circuit, meter operation or electrical handling is requested. Public feedback and retained self-reports do not certify independent reasoning, actual laboratory participation or institutional credit.

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

Which source, topology, initial condition or instrument assumption would invalidate this inference?

Your explanation and limitations

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Previous: Solve the RC differential equation with a signed initial condition Next: Infer a decay without confusing the meter and leakage

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