Infer an unknown charge from spatial force data

Establish the model and evidence

A single force reading can fit many explanations: a larger source charge, a shorter separation or a sensor offset. To distinguish them, keep the probe charge fixed and compare several known separations. For point charges in vacuum, the magnitude is F=k|Qq|/r², where r is center-to-center separation and k=8.99×10⁹ N m²/C². Like signs repel and opposite signs attract along their line of separation.

The supplied original record holds a positive 2 nC probe outside a positive source. Outward force is positive. Plot force against 1/r² rather than r: an ideal point source gives a straight line whose slope is kQq. A common force-zero error shifts the intercept. Fit both slope and intercept instead of forcing the line through zero or trusting the nearest point.

The controls expose the hidden source charge and a constant sensor bias to let you test your inference. Four fixed small residual errors remain. These are synthetic readings, not completed experimental observations. A fitted charge can differ slightly from the selected charge even with zero bias because of those residuals. A good fit also cannot exclude a mismeasured distance scale or an unwanted field without independent checks.

Both electrostatic and gravitational forces follow an inverse-square distance law for point sources, but electrical force can repel as well as attract. Their ratio for two charged particles is k|Qq|/(Gm₁m₂); the distance cancels. Electrical forces can dominate particle interactions while nearly neutral large objects have little net electrical force. Contact forces such as normal force arise from electromagnetic interactions microscopically, not a new fundamental force.

Worked evidence

Ignoring the stipulated reading errors, a +3 nC source and +2 nC probe at 0.10 m repel with 5.394 µN. At 0.20 m the force is 1.3485 µN. A +1 µN offset changes those readings to 6.394 and 2.3485 µN: their ratio is no longer four, although the inverse-square slope is unchanged.

Predict and investigate

Will adding the same 2 µN to every reading alter the fitted charge when an intercept is allowed?

Capture source=3 at offsets 0 and 2 µN. Compare fitted charge, intercept and the nearest-reading estimate that ignores bias. Propose a separate check that could expose a wrong distance origin.

Transfer to changed conditions

A different probe has charge +4 nC. Its fitted force versus 1/r² slope is 0.07192 µN m². Infer Q and say whether an unknown constant force offset changes it.

Compare the worked transfer

Convert the slope to 7.192×10⁻⁸ N m². Q=slope/(kq)=2.00×10⁻⁹ C=2 nC. A fitted intercept absorbs a constant additive offset; variable bias or distance error would need further investigation.

Explain the boundary

Why does a small residual not prove that the point-charge model or the distance calibration is correct?

Sources and authorship

Physical relationship reference. Current AP Physics 2 framework, checked 30 October 2026. These cases, diagrams and numerical records are original. All experiments are bounded browser calculations or synthetic data, not measured laboratory evidence. Do not construct or handle charged capacitors, high-voltage sources or particle-beam equipment for these activities.