Challenge an exact two-gap capacitance fit

Write the indicated rather than ideal quantity

For a vacuum parallel-plate source, geometric C = epsilon0 A/d. The synthetic instrument reports Cm = g epsilon0 A/d + Cp, with common gain g and additive parasitic capacitance Cp. A model that assumes g = 1 can solve area and offset from two gap readings, but that does not prove the assumed gain.

Eliminate one nuisance parameter

Subtracting two readings removes Cp and gives Afit = (C1 − C2)/[epsilon0(1/d1 − 1/d2)] = gA. Substituting back returns Cp for this exact packet. A third same-instrument gap still follows the same inverse-gap curve with biased Afit, so exact held-out agreement tests the curve shape but not common gain independently.

Choose an independent constraint

A separately justified optical area record can expose the fitted-area bias, conditional on known permittivity, parallel plates and calibrated gaps. A wrong gap zero, fringing, material polarization or time-dependent stray capacitance could create other discrepancies. Distinguish rejecting one joint model from uniquely diagnosing a gain error.

Bound a two-reading inference

For a separate supplied pair of finite reading bounds, let C1 be 105 to 107 pF and C2 be 60 to 62 pF at gaps 1 and 2 mm. The independent rectangular bound on the difference is 43 to 47 pF, not 45 pF exactly. Dividing by epsilon0×500 per metre gives an inferred area interval of approximately 0.009713 to 0.010616 m². These are stipulated hard bounds, not confidence intervals; common gain remains outside that two-reading bound unless explicitly bounded too.

Worked original case

For true A = 0.01 m², g = 1.05 and Cp = 20 pF, the indications at 1 and 2 mm are approximately 112.969 and 66.4845 pF. The unit-gain fit returns Afit = 0.0105 m² and Cp = 20 pF. The independent supplied optical area is 0.01 m², revealing a 5% conditional discrepancy.

Investigation

Vary gain and parasitic offset separately, retaining which fit parameter changes. Predict a third-gap indication and explain why it does not resolve common gain. Then calculate the supplied finite-bound area interval and name an additional uncertainty that it omits.

Changed-case practice

Sources and boundaries

Original cases checked against the official framework on 30 October 2026, with OpenStax verification on 31 October 2026 against the current official framework, Unit 10 and OpenStax capacitance. These are virtual supplied models, not instructions to charge, ground, wire or handle real equipment. Public answers are practice; retained plans and self-reports do not establish independent mastery, actual laboratory participation or AP authorization.