SCIENCE · INVESTIGATE AND EXPLAIN

Separate geometric slope from a parallel stray capacitance

Capacitance measured at one gap cannot by itself separate geometric capacitance from an added parallel stray term. The synthetic family is C=S/d+Cs, with d in independently exact millimetres, C in picofarads and S in pF mm. Gain is stipulated calibrated. Two distinct gaps fit its two parameters; a third gap is reserved for validation.

Learning outcome: Freeze the third-gap prediction after fitting only the first two records. Compare several stray values without changing S. Propagate reading bounds, then derive the direct third-prediction formula in terms of C1,C2 to avoid treating correlated parameter intervals as independent.

Starting recap

Review charge conservation, conductor equipotential conditions, definite field integrals and energy derivatives under a stated electrical constraint. Convert capacitance and charge units before interpreting small energies.

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Try

For S=20 pF mm and Cs=4 pF, the first two readings are 24 and14 pF. Their fitted parameters predict the third reading 32/3≈10.666667 pF at3 mm. The separate compatible intervals are S=20±0.4 pF mm and Cs=4±0.3 pF, not probabilistic confidence intervals.

Open the experiment

Predict

Which record is fresh validation of the stated two-parameter fit?

Your prediction and reason

Experiment

Freeze the third-gap prediction after fitting only the first two records. Compare several stray values without changing S. Propagate reading bounds, then derive the direct third-prediction formula in terms of C1,C2 to avoid treating correlated parameter intervals as independent.

Separate geometric capacitance from a parallel stray term

Worked teaching model · synthetic values · Synthetic indicated capacitance C=S/d+Cstray at independently exact gaps1 and2 mm; reserve3 mm for validation. Gain is stipulated calibrated. Two readings identify S and stray within this model, not a gap-zero error or arbitrary dielectric geometry. Independent reading-error bounds ±.1 pF; all units are explicit.

True geometric coefficient epsilon0 A: 20 pF mm Constant stray capacitance: 4 pF

Reset keeps your prediction, captures and practice history.

Inverse-gap calibration with a withheld third gap

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: model fitted only to first two gaps (solid).2: fresh third observation (open-circle marked location). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Inverse-gap calibration with a withheld third gap

model fitted only to first two gaps

gap (mm)	indicated capacitance (pF)
1	24
1.025	23.5122
1.05	23.04762
1.075	22.60465
1.1	22.18182
1.125	21.77778
1.15	21.3913
1.175	21.02128
1.2	20.66667
1.225	20.32653
1.25	20
1.275	19.68627
1.3	19.38462
1.325	19.09434
1.35	18.81481
1.375	18.54545
1.4	18.28571
1.425	18.03509
1.45	17.7931
1.475	17.55932
1.5	17.33333
1.525	17.11475
1.55	16.90323
1.575	16.69841
1.6	16.5
1.625	16.30769
1.65	16.12121
1.675	15.9403
1.7	15.76471
1.725	15.5942
1.75	15.42857
1.775	15.26761
1.8	15.11111
1.825	14.9589
1.85	14.81081
1.875	14.66667
1.9	14.52632
1.925	14.38961
1.95	14.25641
1.975	14.12658
2	14
2.025	13.87654
2.05	13.7561
2.075	13.63855
2.1	13.52381
2.125	13.41176
2.15	13.30233
2.175	13.1954
2.2	13.09091
2.225	12.98876
2.25	12.88889
2.275	12.79121
2.3	12.69565
2.325	12.60215
2.35	12.51064
2.375	12.42105
2.4	12.33333
2.425	12.24742
2.45	12.16327
2.475	12.08081
2.5	12
2.525	11.92079
2.55	11.84314
2.575	11.76699
2.6	11.69231
2.625	11.61905
2.65	11.54717
2.675	11.47664
2.7	11.40741
2.725	11.33945
2.75	11.27273
2.775	11.20721
2.8	11.14286
2.825	11.07965
2.85	11.01754
2.875	10.95652
2.9	10.89655
2.925	10.83761
2.95	10.77966
2.975	10.72269
3	10.66667

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.

fresh third observation

gap (mm)	indicated capacitance (pF)
3	10.66667

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
First capacitance reading at 1 mm	24	pF
Second capacitance reading at 2 mm	14	pF
Inferred geometry coefficient	20	pF mm
Inferred constant stray	4	pF
Fresh withheld reading at 3 mm	10.66667	pF
Prediction from the first two readings	10.66667	pF
Withheld observed minus predicted	0	pF
Geometry coefficient lower bound	19.6	pF mm
Geometry coefficient upper bound	20.4	pF mm
Stray lower bound	3.7	pF
Stray upper bound	4.3	pF
Inferred area	0.002261947	m²

Two exact-gap readings separate a geometric slope from a constant parallel stray term; a third gap tests that fitted family.

At gaps1 and2 mm, C1=S+Cs and C2=S/2+Cs, so S=2(C1−C2), Cs=2C2−C1. Independent ±0.1 pF errors give S bound±0.4 pF mm and Cs bound±0.3 pF by absolute coefficient sums; these individual intervals are correlated and must not be treated as an independent rectangle for further inference. A gap-zero error or gain error requires a new model and independent information, not relabeling Cs as proof.

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

Two exact-gap readings separate a geometric slope from a constant parallel stray term; a third gap tests that fitted family.

Observation to attach to a capture

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

Understand

State conductor and source constraints

Derive charge, voltage and energy

At d=1 and2 mm, C1=S+Cs and C2=S/2+Cs. Therefore S=2(C1−C2) and Cs=2C2−C1. Predict the third record as S/3+Cs before revealing it. To infer area from S, convert pF mm to F m by multiplying by 10⁻¹⁵, then use A=S_SI/ε0. The unit conversion is part of the inference.

Check evidence and excluded mechanisms

Independent reading bounds ±0.1 pF give S bound ±0.4 pF mm and Cs bound ±0.3 pF by summing absolute linear coefficients. These parameter errors share the same readings and are correlated; their separate intervals are not an independent rectangle. A gap-zero error would change the reciprocal relation, while an unknown gain would confound amplitude. Neither alternative is ruled out merely by a neat two-point fit.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

Given C1=35 pF at1 mm and C2=20 pF at2 mm, predict capacitance at3 mm in pF.

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 numerical packets checked 30 October 2026 against the primary explanation. No charged-capacitor handling, high-voltage apparatus, material safety limit or practical certification is requested. Preserve first predictions, independent critique, revision and newly assigned work separately from public practice feedback.

Apply

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

Which geometry, calibration or source assumption could fail while an exact arithmetic fit remains possible?

Your explanation and limitations

Continue

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