Use an independent field to challenge a potential fit

Write the measurement model

A known-radius ring has V(z) = kQ/sqrt(R² + z²). The synthetic instrument reports m(z) = gV(z) + C. A fit that silently fixes g = 1 while solving Q and C can absorb the true gain into Q. Two observations exactly fit two parameters, but exact interpolation is not evidence that the assumed gain is correct.

Eliminate offset, not gain

Subtracting the observations at z = 1 and 2 m removes C. It yields Qfit = gQ because the geometric factors are known. Substituting back gives the same offset C for this exact packet. A third observation from the same instrument follows the same biased model, so it need not reveal the gain error.

State what makes the new record independent

The stipulated separate field sensor has known calibration and observes Ez at z = 1 m. The potential-derived charge predicts g times the true field, revealing disagreement when g differs from one. This inference still depends on the shared ring geometry and static-source assumption. A geometric mismatch, time-varying source or correlated calibration could provide alternative explanations; actual independence must be justified, not declared by naming two instruments.

Worked original case

With true Q = 2 nC, gain g = 1.05 and offset C = 1 V, the two-point potential fit returns Qfit = 2.10 nC and Cfit = 1 V. It predicts approximately 6.67292 N/C at z = 1 m, while the independent supplied field is 6.35516 N/C. The residual is approximately 0.317758 N/C.

Investigation

Vary gain and offset separately. Retain fitted charge, fitted offset and the independent field residual. Construct a same-instrument third-potential prediction and explain why it cannot by itself identify gain. Propose a controlled alternative test for a wrong source radius.

Changed-case practice

Sources and evidence boundary

Original teaching and cases checked 30 October 2026 against the current official framework, Unit 9 and OpenStax electric potential. All records are supplied or generated, not measured by a learner. No physical charging, battery connection, high-voltage equipment or particle experiment is requested. Retain predictions, derivations and critique in the course evidence workspace; public feedback is not independent mastery or performed laboratory time.