SCIENCE · INVESTIGATE AND EXPLAIN

Infer a decay without confusing the meter and leakage

A capacitor discharging through leakage resistance Rl and parallel meter resistance Rm has equivalent resistance Req = RlRm/(Rl + Rm).

Learning outcome: Compare inferred leakage with and without the known meter. Predict a second meter’s time constant while holding the leakage fixed, and identify the shared parameter. Write a baseline-offset counterexample showing why a straight two-point log fit is not always a valid physical inference.

Starting recap

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

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Try

For C = 2 microfarad, Rl = 3 Mohm and Rm = 2 Mohm, Req = 1.2 Mohm and tau = 2.4 s. Starting at 4 V, supplied readings at 1 and 3 s are about 2.63696 and 1.14602 V. Their ratio recovers tau; the known meter then gives Rl = 3 Mohm.

Open the experiment

Predict

What does one ideal exponential decay identify when both parallel resistances are unknown?

Your prediction and reason

Experiment

Compare inferred leakage with and without the known meter. Predict a second meter’s time constant while holding the leakage fixed, and identify the shared parameter. Write a baseline-offset counterexample showing why a straight two-point log fit is not always a valid physical inference.

Distinguish leakage from meter loading

Worked teaching model · synthetic values · Supplied capacitor 2 microfarad initially at 4 V, discharging through leakage and a parallel meter. Exact synthetic observations have no additive voltage baseline. One measured time constant identifies equivalent resistance, not two unknown resistances separately; meter resistance is stipulated known for the inverse step.

True leakage resistance: 3 Mohm Known meter resistance: 2 Mohm

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Meter-loaded discharge

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.

Meter-loaded discharge: numerical coordinates

time (s)	voltage (V)
0	4
0.166667	3.73165
0.333333	3.4813
0.5	3.24775
0.666667	3.02986
0.833333	2.82659
1	2.63696
1.16667	2.46005
1.33333	2.29501
1.5	2.14105
1.66667	1.99741
1.83333	1.86341
2	1.73839
2.16667	1.62177
2.33333	1.51297
2.5	1.41146
2.66667	1.31677
2.83333	1.22843
3	1.14602
3.16667	1.06914
3.33333	0.997409
3.5	0.930495
3.66667	0.86807
3.83333	0.809832
4	0.755502
4.16667	0.704817
4.33333	0.657532
4.5	0.61342
4.66667	0.572267
4.83333	0.533874
5	0.498058
5.16667	0.464644
5.33333	0.433472
5.5	0.404391
5.66667	0.377261
5.83333	0.351952
6	0.32834
6.16667	0.306312
6.33333	0.285762
6.5	0.266591
6.66667	0.248706
6.83333	0.232021
7	0.216455
7.16667	0.201934
7.33333	0.188386
7.5	0.175748
7.66667	0.163957
7.83333	0.152958
8	0.142696
8.16667	0.133123
8.33333	0.124192
8.5	0.11586
8.66667	0.108087
8.83333	0.100836
9	0.094071
9.16667	0.0877599
9.33333	0.0818723
9.5	0.0763797
9.66667	0.0712555
9.83333	0.0664751
10	0.0620154

Calculated values under the declared model
Quantity	Value	Unit
True equivalent resistance	1.2	Mohm
Measured exponential time constant	2.4	s
Supplied voltage at 1 s	2.63696	V
Supplied voltage at 3 s	1.14602	V
Two-point fitted time constant	2.4	s
Leakage inferred with known meter	3	Mohm
Naive leakage if meter ignored	1.2	Mohm

One decay identifies the parallel combination; a known meter resistance is additional information, not something learned from the same slope.

With zero additive baseline, ln[V(t1)/V(t2)] = (t2−t1)/tau. Then 1/Rleak = C/tau − 1/Rmeter must be positive for a finite passive leakage model. A zero result gives an infinite-leakage-resistance limit; a negative result rejects the assumed parameters. Unknown voltage baseline invalidates this simple two-reading logarithm.

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

One decay identifies the parallel combination; a known meter resistance is additional information, not something learned from the same slope.

Observation to attach to a capture

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

Understand

Identify what a slope determines

A capacitor discharging through leakage resistance Rl and parallel meter resistance Rm has equivalent resistance Req = RlRm/(Rl + Rm). Its voltage is V0 exp[−t/(CReq)]. A decay record identifies tau = CReq if C is known; it does not separately identify two unknown branch resistances.

Derive the two-reading fit

With zero additive baseline, ln[V(t1)/V(t2)] = (t2 − t1)/tau. The unknown multiplicative amplitude cancels. An unknown additive voltage offset does not cancel inside the logarithm; two readings alone cannot determine amplitude, baseline and tau simultaneously. The supplied packet fixes baseline at zero rather than pretending to estimate it.

Use independent meter information

If Rm is known independently, 1/Rl = C/tau − 1/Rm. A positive result gives a finite passive leakage resistance; zero corresponds to an infinite-resistance limit, while a negative result rejects the joint assumptions. Ignoring the meter returns Req and therefore underestimates the actual leakage resistance in this positive-resistance family.

Design a discriminating changed record

A second supplied decay with a different known meter can test the same leakage hypothesis: inferred 1/Rl should agree after subtracting the new meter conductance. That is not the same as collecting more samples with one unchanged unknown meter. Leakage may depend on voltage or temperature in real materials, so this constant-resistance model is conditional.

Worked original case

Investigation

Changed-case practice

Try a new practice case

If C = 2 microfarad and both leakage and meter resistances are 4 Mohm, what is tau in seconds?

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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