SCIENCE · INVESTIGATE AND EXPLAIN

Fit two hypotheses before revealing a third location

Two calibrated models can fit the same two records perfectly and still make different predictions elsewhere. At exact perpendicular distances y=1,2,3 m, compare a known-half-length 1 m straight-segment shape f(y)=1/[y√(L²+y²)] with L=1 m with an infinite-line shape g(y)=1/y. Each has its own unknown amplitude and constant indicated baseline. The finite-shape amplitude has units µT m², while the infinite-line amplitude has units µT m; these coefficients are not interchangeable.

Learning outcome: Fit both amplitudes and baselines from only distances 1 and 2. Freeze both third-distance predictions and propagate ±0.005 µT calibration bounds. Then compare the unused third observation, retaining any additional observation uncertainty. State an independent measurement needed to separate background from instrument offset.

Starting recap

Review cross products, source integration measures, line integrals, differential initial states and signed work. Distinguish symmetry premises from integral laws.

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

Take generating current 3 A and baseline 0.1 µT. The first two indicated fields are 0.1+0.6/√2 and 0.1+0.6/(2√5) µT. Both candidates fit these exactly. The finite-segment third prediction is 0.1+0.6/(3√10)≈0.163246 µT; the infinite-line fit gives a different third value because its shape decays differently after the baseline is fitted.

Open the experiment

Predict

Why can the third reading compare these two calibrated hypotheses?

Your prediction and reason

Experiment

Fit both amplitudes and baselines from only distances 1 and 2. Freeze both third-distance predictions and propagate ±0.005 µT calibration bounds. Then compare the unused third observation, retaining any additional observation uncertainty. State an independent measurement needed to separate background from instrument offset.

Fit two spatial hypotheses and reserve a third location

Worked teaching model · synthetic values · Perpendicular readings at exact distances 1,2,3 m. Generator is the known-half-length 1 m segment contribution plus constant baseline; real return/background fields are not uniquely identified. Both candidate models fit amplitude and baseline using only distances 1,2. Distance 3 is withheld. Field gain is stipulated calibrated; baseline may represent sensor offset or an approximately uniform background.

Generating segment current: 3 A Constant indicated baseline: 0.1 microtesla

Reset keeps your prediction, captures and practice history.

Two calibrated hypotheses and one unused location

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: finite known-length fit (solid).2: infinite-line fit (dashed).3: fresh third-distance observation (open-circle marked location). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Two calibrated hypotheses and one unused location

finite known-length fit

perpendicular distance (m)	indicated field (microtesla)
1	0.5242641
1.025	0.5087749
1.05	0.4940887
1.075	0.480151
1.1	0.4669124
1.125	0.4543274
1.15	0.4423541
1.175	0.4309542
1.2	0.4200922
1.225	0.4097354
1.25	0.3998536
1.275	0.3904188
1.3	0.381405
1.325	0.372788
1.35	0.3645456
1.375	0.3566567
1.4	0.3491021
1.425	0.3418635
1.45	0.334924
1.475	0.3282678
1.5	0.3218801
1.525	0.315747
1.55	0.3098556
1.575	0.3041936
1.6	0.2987496
1.625	0.2935129
1.65	0.2884734
1.675	0.2836215
1.7	0.2789483
1.725	0.2744454
1.75	0.2701048
1.775	0.265919
1.8	0.261881
1.825	0.257984
1.85	0.2542217
1.875	0.2505882
1.9	0.2470778
1.925	0.2436851
1.95	0.2404051
1.975	0.237233
2	0.2341641
2.025	0.2311942
2.05	0.2283191
2.075	0.225535
2.1	0.2228382
2.125	0.220225
2.15	0.2176923
2.175	0.2152367
2.2	0.2128553
2.225	0.2105452
2.25	0.2083036
2.275	0.2061279
2.3	0.2040155
2.325	0.2019642
2.35	0.1999715
2.375	0.1980355
2.4	0.1961538
2.425	0.1943247
2.45	0.1925462
2.475	0.1908165
2.5	0.1891338
2.525	0.1874965
2.55	0.1859029
2.575	0.1843517
2.6	0.1828413
2.625	0.1813704
2.65	0.1799375
2.675	0.1785414
2.7	0.177181
2.725	0.1758549
2.75	0.1745621
2.775	0.1733015
2.8	0.1720721
2.825	0.1708728
2.85	0.1697027
2.875	0.1685608
2.9	0.1674463
2.925	0.1663584
2.95	0.1652961
2.975	0.1642588
3	0.1632456

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

infinite-line fit

perpendicular distance (m)	indicated field (microtesla)
1	0.5242641
1.025	0.5101128
1.05	0.4966355
1.075	0.483785
1.1	0.4715186
1.125	0.4597974
1.15	0.4485858
1.175	0.4378513
1.2	0.4275641
1.225	0.4176967
1.25	0.4082241
1.275	0.3991229
1.3	0.3903718
1.325	0.3819509
1.35	0.3738419
1.375	0.3660277
1.4	0.3584926
1.425	0.351222
1.45	0.344202
1.475	0.33742
1.5	0.3308641
1.525	0.3245231
1.55	0.3183867
1.575	0.312445
1.6	0.3066891
1.625	0.3011102
1.65	0.2957004
1.675	0.2904521
1.7	0.2853582
1.725	0.2804119
1.75	0.2756069
1.775	0.2709373
1.8	0.2663974
1.825	0.2619819
1.85	0.2576857
1.875	0.2535041
1.9	0.2494325
1.925	0.2454667
1.95	0.2416025
1.975	0.2378362
2	0.2341641
2.025	0.2305826
2.05	0.2270885
2.075	0.2236785
2.1	0.2203498
2.125	0.2170994
2.15	0.2139245
2.175	0.2108227
2.2	0.2077914
2.225	0.2048281
2.25	0.2019307
2.275	0.199097
2.3	0.1963249
2.325	0.1936125
2.35	0.1909577
2.375	0.1883588
2.4	0.1858141
2.425	0.1833218
2.45	0.1808804
2.475	0.1784883
2.5	0.1761441
2.525	0.1738463
2.55	0.1715935
2.575	0.1693845
2.6	0.1672179
2.625	0.1650927
2.65	0.1630075
2.675	0.1609613
2.7	0.158953
2.725	0.1569815
2.75	0.1550459
2.775	0.1531452
2.8	0.1512784
2.825	0.1494446
2.85	0.147643
2.875	0.1458728
2.9	0.144133
2.925	0.1424231
2.95	0.140742
2.975	0.1390893
3	0.1374641

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

fresh third-distance observation

perpendicular distance (m)	indicated field (microtesla)
3	0.1632456

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
First calibration field	0.5242641	microtesla
Second calibration field	0.2341641	microtesla
Fitted finite-segment current	3	A
Fitted finite-segment baseline	0.1	microtesla
Fitted infinite-line amplitude	0.5802	microtesla m
Fitted infinite-line baseline	-0.05593591	microtesla
Fresh third observation	0.1632456	microtesla
Finite-segment third prediction	0.1632456	microtesla
Infinite-line third prediction	0.1374641	microtesla
Infinite-line fresh residual	0.02578147	microtesla
Finite prediction calibration-error bound	0.007444624	microtesla
Infinite prediction calibration-error bound	0.008333333	microtesla

Both two-parameter hypotheses fit the first two records exactly; only the unused spatial record compares their predictions.

Each candidate has shape f(y), amplitude A and constant baseline B0. Solve A=(Y1−Y2)/(f1−f2), B0=Y1−A f1, then freeze Y3pred=wY1+(1−w)Y2 where w=(f3−f2)/(f1−f2). Independent ±0.005 microtesla calibration bounds propagate as epsilon(|w|+|1−w|); a fresh observation error would be additional. Segment half-length is stipulated independently, not inferred from these same two readings. A constant fitted baseline does not distinguish sensor offset from uniform background or prove omitted return fields are constant.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Both two-parameter hypotheses fit the first two records exactly; only the unused spatial record compares their predictions.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Specify sources, directions and independent premises

Derive the integral or vector evolution

For either shape h, fit amplitude A=(Y1−Y2)/(h1−h2) and baseline B0=Y1−Ah1 using only the first two fields. Then freeze the third prediction Ah3+B0 before revealing Y3. The finite segment half-length is stipulated independently; fitting that length too would add an unknown and consume additional calibration information.

Check energy, symmetry and evidence limits

The third prediction can be written wY1+(1−w)Y2, where w=(h3−h2)/(h1−h2). Independent calibration-reading bounds ε therefore contribute ε(|w|+|1−w|), and a fresh observation error must be added separately when comparing a residual. A constant fitted baseline cannot distinguish a sensor zero from uniform background, nor prove an omitted return-path field is actually constant.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

For the infinite-line candidate with exact fields Y1=0.5 µT at 1 m and Y2=0.3 µT at 2 m, predict the indicated field at 3 m in µT.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original derivations and synthetic numerical evidence checked 31 October 2026 against the primary explanation. The rounded teaching permeability is not asserted to be an exact modern SI constant; NIST CODATA 2022 lists its measured uncertainty. No live electrical construction, beam, strong-magnet handling, medical or physical certification claim is requested. Public feedback remains separate from retained first work, critique, revision and fresh assigned evidence.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which source geometry, symmetry, initial state or calibration premise is necessary, and what independent observation could reject it?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Separate magnetic guiding drift from electric work Next: Integrate changing flux before inferring an induced field

Save a science backup or import one

Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.

A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.

Choose a science backup (maximum 36 MB)

Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.

Complete file sets and course backups

A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.

Export scope

The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.

Choose the manifest and all parts together

0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.

Science storage: loading. No account or remote computation is required.