Defend independent spatial and temporal calibration in one capacitor dossier

Declare the coupled system and independent premises

A long coaxial capacitor has inner radius a0, outer radius s a0 and radial relative permittivity κ(r)=1+r/a0. Gauss law gives radial displacement D=Q/(2π ell r); integrating E=D/(epsilon0 κ) gives V=Q F(s)/(2π epsilon0 ell), where F(s)=ln[2s/(1+s)]. Thus C=A/F(s), with A=2π epsilon0 ell expressed in nF. This is a prescribed linear dielectric profile and long-device approximation, not a unique material reconstruction.

Derive the state or parameter relation

The capacitance instrument adds an unknown offset B. Define fi=1/F(i). Indications y2=A f2+B and y3=A f3+B determine A=(y2−y3)/(f2−f3) and B=y2−A f2. Freeze the prediction y4=A f4+B before revealing the third geometry. In a separate discharge, the physical ratio-four capacitor is in parallel with independently established stray capacitance 0.4 nF. A readout offset does not store charge; the physical stray does. With independently known R=2 Mohm, physical tau=2(C4+0.4) ms.

Challenge conservation and identification

The discharge has zero voltage baseline, independently zeroed time origin and unknown initial amplitude. Two readings at displayed times 10 and 20 ms give displayed time constant T=10/ln(V10/V20); recover amplitude from either point and freeze a 30 ms prediction. Dividing T by independently predicted physical tau infers timestamp gain. A third same-model voltage tests response shape but cannot establish the separate resistance, stray or dielectric premises. With independent ±0.01 nF bounds on the two calibration indications, A has hard half-width 0.02/(f2−f3). Holding T, R and stray exact gives monotone gain bounds T/[2((A±dA)f4+0.4)], with the plus sign supplying the lower gain. Shared errors or uncertain baselines require a different interval model.

Worked original case

For A=2 nF and readout offset B=0.25 nF, first derive f2=1/ln(4/3), f3=1/ln(3/2) and f4=1/ln(8/5). The two geometry records recover A and B, while the frozen third prediction is 2/ln(8/5)+0.25 nF. Physical total capacitance is instead 2/ln(8/5)+0.4 nF. Its physical time constant is about 9.31057 ms. A timestamp gain 1.05 gives displayed time constant about 9.77610 ms and is recovered only after the independent physical premises are retained.

Cumulative investigation

Fit geometry amplitude and readout offset from exactly two records; withhold the third. Then fit the displayed exponential from exactly two voltage readings and withhold a third. Propagate the two capacitance bounds into clock gain without treating readout offset as stored capacitance. Retain one alternative dielectric profile and one alternative timing explanation that these records alone cannot rule out.

Changed-case practice

Sources and exit evidence

Original synthesis checked 31 October 2026 against the primary component relation, inductive state and energy, RC state equations and capacitance geometry. Inter-stage derivations and evidence packets are original, not copied source experiments. The course workspace preserves first responses, source-bound reasoning, criterion critique, revision and a genuinely new frozen assignment. Actual laboratory technique, browser interaction, independent learner competence and credit require separate evidence; no physical circuit or field apparatus is requested.