Infer a source location without assuming an instrument zero

Define the source and measurement boundary

An inverse problem asks which source parameters could have produced a set of observations. Here the stipulated family is a single positive point charge at position s left of x=1 m, with indicated potential R(x)=K/(x−s)+o and K=kQ. Three readings at x=1,2,3 m contain information about source position, strength and a common offset. Assuming o=0 without evidence can produce a misleading source estimate.

Derive the quantitative comparison

Subtract neighboring readings: D12=K/[(1−s)(2−s)] and D23=K/[(2−s)(3−s)]. Their ratio R=D12/D23=(3−s)/(1−s) eliminates both strength and offset. Solving gives s=(3−R)/(1−R), after which K=D12(1−s)(2−s) and o=R(1)−K/(1−s). These numerical coordinate formulas use metres throughout; a change of coordinate units must be carried consistently rather than substituted into bare constants.

Test an alternative before claiming identification

Three exact readings can fit the three parameters of this family, but do not prove that the source really is a single point charge. A different arrangement can agree at a finite set of sample locations. Predicting the field at x=2, E=K/(2−s)², supplies a different observable for an independent check. Measurement noise matters: when the difference ratio is close to one, the denominator in the position formula is small and inference is poorly conditioned. A precise displayed fit is not automatically accurate or uniquely identified.

Worked original case

For Q=3 nC, s=−.5 m and offset +2 V, the differences ratio is (3.5)/(1.5)=7/3. The inverse formula recovers s=−.5 m and K=k×3 nC. The predicted independent field at x=2 is k×3 nC/(2.5 m)²≈4.31402486 N/C.

Investigation

Change only the voltage offset and show that both neighboring differences remain unchanged. Then change source position and recover all parameters. Freeze a field prediction before revealing it, and explain why another voltage used in fitting is weaker evidence than a truly independent measurement.

Changed-case practice

Sources and evidence boundary

Original teaching, numerical cases and diagrams checked 30 October 2026 against the relevant primary textbook explanation. No textbook exercise, image or passage is reproduced. These are bounded synthetic records, not observed apparatus. No charging, wiring, capacitor handling, medical application or safety certification is requested. Preserve first predictions, contrasting settings and critiques in the course evidence workspace; public feedback is not independent mastery or performed laboratory time.