SCIENCE · INVESTIGATE AND EXPLAIN

Solve the RC differential equation with a signed initial condition

For a source emf E, series resistor R and capacitor C, loop balance is E − IR − V = 0.

Learning outcome: Compare V0 = −2, 0, 6 and 8 V at the same R. Predict current sign, final voltage and source-work sign. Derive the solution and independently differentiate it. Retain the finite-time energy residual and explain why 63% applies to the zero-initial charging special case rather than every arbitrary V0.

Starting recap

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

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Try

For R = 2 kohm, C = 2 microfarad, E = 6 V and V0 = 0, tau = 4 ms. At 8 ms, V ≈ 5.18799 V and I ≈ 0.406006 mA. Source work is about 62.2559 microjoule, heat 35.3406 microjoule and stored energy 26.9152 microjoule.

Open the experiment

Predict

If V0 exceeds the ideal source voltage, what is initial current in the source-to-capacitor convention?

Your prediction and reason

Experiment

Compare V0 = −2, 0, 6 and 8 V at the same R. Predict current sign, final voltage and source-work sign. Derive the solution and independently differentiate it. Retain the finite-time energy residual and explain why 63% applies to the zero-initial charging special case rather than every arbitrary V0.

Solve an RC initial-value problem

Worked teaching model · synthetic values · 6 V ideal source, selected resistance, capacitor 2 microfarad. Positive current flows from source toward positive capacitor plate. Arbitrary supplied initial capacitor voltage; no leakage or inductance. Finite source work and resistor heating are included in the model energy account.

Series resistance: 2 kohm Initial capacitor voltage: 0 V

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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	2 kΩ
P	B	2 μF
B	S	6 V source

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.

Capacitor voltage initial-value solution

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.

Capacitor voltage initial-value solution: numerical coordinates

time (ms)	voltage (V)
0	0
0.333333	0.479734
0.666667	0.92111
1	1.3272
1.33333	1.70081
1.66667	2.04456
2	2.36082
2.33333	2.65179
2.66667	2.9195
3	3.1658
3.33333	3.39241
3.66667	3.6009
4	3.79272
4.33333	3.96921
4.66667	4.13158
5	4.28097
5.33333	4.41842
5.66667	4.54487
6	4.66122
6.33333	4.76826
6.66667	4.86675
7	4.95736
7.33333	5.04072
7.66667	5.11742
8	5.18799
8.33333	5.25291
8.66667	5.31265
9	5.3676
9.33333	5.41817
9.66667	5.46469
10	5.50749
10.3333	5.54687
10.6667	5.5831
11	5.61643
11.3333	5.6471
11.6667	5.67532
12	5.70128
12.3333	5.72516
12.6667	5.74714
13	5.76735
13.3333	5.78596
13.6667	5.80307
14	5.81882
14.3333	5.8333
14.6667	5.84663
15	5.85889
15.3333	5.87018
15.6667	5.88056
16	5.89011
16.3333	5.89889
16.6667	5.90698
17	5.91441
17.3333	5.92126
17.6667	5.92755
18	5.93335
18.3333	5.93868
18.6667	5.94358
19	5.94809
19.3333	5.95224
19.6667	5.95606
20	5.95957

Signed branch current

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.

Signed branch current: numerical coordinates

time (ms)	current (mA)
0	3
0.333333	2.76013
0.666667	2.53945
1	2.3364
1.33333	2.14959
1.66667	1.97772
2	1.81959
2.33333	1.67411
2.66667	1.54025
3	1.4171
3.33333	1.30379
3.66667	1.19955
4	1.10364
4.33333	1.0154
4.66667	0.93421
5	0.859514
5.33333	0.790791
5.66667	0.727563
6	0.66939
6.33333	0.615869
6.66667	0.566627
7	0.521322
7.33333	0.479639
7.66667	0.441289
8	0.406006
8.33333	0.373543
8.66667	0.343677
9	0.316198
9.33333	0.290916
9.66667	0.267656
10	0.246255
10.3333	0.226566
10.6667	0.20845
11	0.191784
11.3333	0.176449
11.6667	0.162341
12	0.149361
12.3333	0.137419
12.6667	0.126432
13	0.116323
13.3333	0.107022
13.6667	0.098465
14	0.0905922
14.3333	0.0833488
14.6667	0.0766846
15	0.0705532
15.3333	0.0649121
15.6667	0.059722
16	0.0549469
16.3333	0.0505536
16.6667	0.0465116
17	0.0427927
17.3333	0.0393712
17.6667	0.0362232
18	0.033327
18.3333	0.0306623
18.6667	0.0282107
19	0.0259551
19.3333	0.0238798
19.6667	0.0219705
20	0.0202138

Calculated values under the declared model
Quantity	Value	Unit
Time constant	4	ms
Selected observation time	8	ms
Capacitor voltage at two time constants	5.18799	V
Signed current at two time constants	0.406006	mA
Source work to observation time	62.2559	microjoule
Resistor heat to observation time	35.3406	microjoule
Stored-energy change	26.9152	microjoule
Energy residual	3.38813e-15	microjoule

The initial capacitor voltage fixes both the amplitude and sign of the transient.

Kirchhoff gives R C dV/dt + V = E. Separation for V−E yields the exponential with V(0) supplied. For initially zero V, one time constant reaches 1−exp(−1) of final voltage; pure discharge retains exp(−1). A precharged capacitor above the source voltage initially returns energy to the ideal source. Source work, heating and stored-energy change obey one signed balance.

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

The initial capacitor voltage fixes both the amplitude and sign of the transient.

Observation to attach to a capture

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

Understand

Derive the differential equation

For a source emf E, series resistor R and capacitor C, loop balance is E − IR − V = 0. Since charge on the positive capacitor plate is CV and current into that plate is C dV/dt, RC dV/dt + V = E. Set w = V − E; then dw/w = −dt/(RC). Integrating and applying V(0) = V0 gives V = E + (V0 − E) exp[−t/(RC)].

Differentiate to recover current

Current is I = (E − V0)exp[−t/(RC)]/R. Initially uncharged means V0 = 0: the capacitor’s initial voltage drop is zero, so it acts like a zero-drop branch at that instant, not an actual conducting path through the dielectric. At long times its voltage approaches E and branch current approaches zero. A precharged capacitor above E gives negative current in the chosen source-to-capacitor convention.

Interpret one time constant

Tau = RC has units seconds. Starting from zero, voltage and charge reach 1 − exp(−1), approximately 63.2% of their final values after tau. For pure discharge with E = 0, they retain exp(−1), approximately 36.8%, after tau. No finite multiple makes the exponential exactly zero. A long-time approximation needs an explicitly acceptable residual tolerance.

Close the finite-time energy account

Source work through time t is E C[V(t) − V0]. Integrating I²R gives resistor heat C(E − V0)²[1 − exp(−2t/tau)]/2. Stored energy change is C[V(t)² − V0²]/2. Source work equals heat plus stored-energy change, including negative source work when a precharged capacitor returns energy. Capacitor voltage is continuous for finite current; changing initial voltage without an impulse would violate charge continuity.

Worked original case

Investigation

Changed-case practice

Try a new practice case

For a source of 6 V, initial capacitor voltage 8 V and R = 4 kohm, what is initial current in mA?

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

Your explanation and limitations

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Previous: Derive network capacitance from a floating-node inventory Next: Carry capacitor state across a topology change

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