Recover a flux history without inventing its missing anchor

Differentiate a history and preserve orientation

A fixed ten-turn coil of area 0.2 m² has normal +z and a supplied uniform field history B(t) = 3 T + a t + b t² for 0 to 2 s. The signed emf is −N A(a + 2bt). The constant field offset disappears on differentiation. Reversing the selected normal changes the sign convention for both linked flux and boundary circulation, not the measured physical phenomenon.

Integrate the record to obtain only a change

From emf(t), Delta B = −integral emf dt/(N A). Recovering B(t) additionally requires B at one time. Two histories differing only by a constant produce identical induced-emf records. A magnetometer anchor would be a different observation; assuming an anchor because it fits the preferred history is not independent evidence.

Do not confuse a secant with an instantaneous rate

For the quadratic packet, the average field-change rate over 0 to 2 s is a + 2b, while the endpoint instantaneous rates are a and a + 4b. The interval average equals the rate at the midpoint for this quadratic history, not for every arbitrary history. If two endpoint field readings each have a stated absolute error bound delta, the worst-case secant-rate error over duration T is 2delta/T; this finite bound is not a probabilistic confidence interval.

Zero net flux change does not mean zero energy transfer

In a separate original waveform, emf is +2t V for 0 ≤ t ≤ 1 s and −2(2 − t) V for 1 < t ≤ 2 s. The two signed triangular areas cancel, so net linked-flux change is zero. For a stipulated 2 ohm resistive loop with negligible self-inductance, heat is integral emf²/R dt = 4/3 J. The finite jump in the ideal emf record at 1 s is supplied; endpoint values do not change its integral. A return to the original flux is not a return of dissipated heat.

Retain an alternative model and a discriminating action

A constant sensor voltage offset would integrate into a false linear field drift; an unknown turn-area product would rescale the recovered change. The shape of one trace alone may not distinguish these errors. Propose an independent zero-change control, geometry record or field anchor, state which ambiguity it addresses and preserve the remaining limitations.

Worked original case

a = 1 T/s and b = 0.5 T/s² give emf from −2 V initially to −6 V at 2 s. Its integral is −8 V s, so Delta B = 4 T and the supplied 3 T anchor yields 7 T finally. The secant rate is 2 T/s, different from the final instantaneous rate of 3 T/s.

Investigation

Differentiate and integrate the same history independently, retaining the missing-constant argument. Compare instantaneous and interval rates, compute the signed-area versus squared-area waveform packet and specify an independent observation that resolves one calibration ambiguity.

Changed-case practice

Try a new practice case

For N A = 2 m² turn, an emf record integrates to −10 V s. What is the field change in tesla?

Numerical result in the requested units Check this practice result

This public, unsaved check is not a secure assessment. Explain the source integral, symmetry or charge boundary in the retained course evidence workspace.

Sources and boundaries

Original teaching checked 31 October 2026 against current official Unit 13 and the relevant OpenStax discussion. No source exercises or diagrams are reproduced. All records and apparatus are virtual, supplied or generated; no real wiring, switching, magnetic braking or electrical measurement is requested. Public feedback and retained work do not establish independent mastery, supervised practical participation or AP authorization.