SCIENCE · INVESTIGATE AND EXPLAIN

Include the return arc in a closed source geometry

A closed annular-sector loop makes the return-path issue explicit. The inner arc has radius 0.5 m and carries 2 A counterclockwise; the outer arc carries the same current clockwise, connected by radial segments. Both arcs subtend the same positive sector angle. Their current directions are opposite because they are parts of one closed path.

Learning outcome: Compare 60°,90°,180° at the same radii, converting each angle to radians before calculating. Then change only outer radius and retain both arc terms. Explain why the zero connector contribution must not be reused for an off-center observer.

Starting recap

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

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Try

For a 90° sector, inner radius 0.5 m and outer radius 1 m, inner field is 0.2π µT and outer field −0.1π µT. Their sum is 0.1π µT≈0.314159 µT. Adding their magnitudes would incorrectly discard the return direction.

Open the experiment

Predict

Why do the radial connectors contribute zero at the common center?

Your prediction and reason

Experiment

Compare 60°,90°,180° at the same radii, converting each angle to radians before calculating. Then change only outer radius and retain both arc terms. Explain why the zero connector contribution must not be reused for an off-center observer.

Add opposite arcs in a closed current loop

Worked teaching model · synthetic values · Closed annular-sector loop: inner radius .5 m, outer radius ratio times .5 m, current 2 A counterclockwise on inner arc and clockwise on outer arc, joined by radial segments. Field at their common center only. Thin prescribed current path, no finite wire cross section or apparatus.

Outer-to-inner radius ratio: 2 ratio Sector angle: 90 degrees

Reset keeps your prediction, captures and practice history.

Closed-loop center field as outer radius changes

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: opposed arc contributions (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Closed-loop center field as outer radius changes

opposed arc contributions

outer to inner radius ratio	center axial field (T)
1.5	2.09440e-7
1.51875	2.14611e-7
1.5375	2.19656e-7
1.55625	2.24580e-7
1.575	2.29386e-7
1.59375	2.34079e-7
1.6125	2.38664e-7
1.63125	2.43142e-7
1.65	2.47519e-7
1.66875	2.51798e-7
1.6875	2.55982e-7
1.70625	2.60073e-7
1.725	2.64076e-7
1.74375	2.67992e-7
1.7625	2.71826e-7
1.78125	2.75578e-7
1.8	2.79253e-7
1.81875	2.82851e-7
1.8375	2.86376e-7
1.85625	2.89830e-7
1.875	2.93215e-7
1.89375	2.96533e-7
1.9125	2.99786e-7
1.93125	3.02976e-7
1.95	3.06104e-7
1.96875	3.09173e-7
1.9875	3.12183e-7
2.00625	3.15138e-7
2.025	3.18038e-7
2.04375	3.20884e-7
2.0625	3.23679e-7
2.08125	3.26424e-7
2.1	3.29119e-7
2.11875	3.31767e-7
2.1375	3.34368e-7
2.15625	3.36924e-7
2.175	3.39436e-7
2.19375	3.41906e-7
2.2125	3.44333e-7
2.23125	3.46719e-7
2.25	3.49066e-7
2.26875	3.51374e-7
2.2875	3.53644e-7
2.30625	3.55877e-7
2.325	3.58074e-7
2.34375	3.60236e-7
2.3625	3.62364e-7
2.38125	3.64458e-7
2.4	3.66519e-7
2.41875	3.68549e-7
2.4375	3.70547e-7
2.45625	3.72515e-7
2.475	3.74452e-7
2.49375	3.76361e-7
2.5125	3.78242e-7
2.53125	3.80094e-7
2.55	3.81919e-7
2.56875	3.83718e-7
2.5875	3.85490e-7
2.60625	3.87237e-7
2.625	3.88959e-7
2.64375	3.90657e-7
2.6625	3.92330e-7
2.68125	3.93981e-7
2.7	3.95608e-7
2.71875	3.97213e-7
2.7375	3.98796e-7
2.75625	4.00357e-7
2.775	4.01897e-7
2.79375	4.03417e-7
2.8125	4.04916e-7
2.83125	4.06396e-7
2.85	4.07856e-7
2.86875	4.09297e-7
2.8875	4.10719e-7
2.90625	4.12123e-7
2.925	4.13509e-7
2.94375	4.14877e-7
2.9625	4.16228e-7
2.98125	4.17562e-7
3	4.18879e-7

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
Sector angle	1.570796	rad
Inner arc contribution	6.28319e-7	T
Outer arc contribution	-3.14159e-7	T
Radial connector contribution at center	0	T
Total central axial field	3.14159e-7	T
Signed total arc factor	1.570796	m⁻¹

The return arc is part of the field model and opposes, rather than duplicates, the inner arc.

At the center of an arc, dl is perpendicular to the source-to-center vector, so its contribution is mu0 I theta/(4pi R), with theta in radians. The closed loop traverses the two arcs in opposite directions. Each radial connector has dl parallel or antiparallel to that vector and contributes zero at this particular point. Zero connector contribution at the center does not mean those segments have zero field everywhere.

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

The return arc is part of the field model and opposes, rather than duplicates, the inner arc.

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

At the common center, an arc gives signed B=μ0Iθ/(4πR), with θ in radians. Inner and outer contributions therefore subtract. Along each radial connector, dl is parallel or antiparallel to the vector toward the center, so its cross product vanishes at that point. That geometric cancellation is point-specific, not a claim that radial wires never make a magnetic field.

Check energy, symmetry and evidence limits

The net center field is μ0Iθ(1/Rinner−1/Router)/(4π). Holding current fixed while changing arc length does not mean holding a total charge on the wire fixed; a current is a rate of transport. The source is an ideal thin current path, with no statement about actual wire heating, source compliance or field near a finite wire cross section.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

For a 180° sector, inner radius 0.5 m, outer radius 1.5 m and current 2 A, find net center field in microtesla.

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

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Previous: Integrate a signed current-segment contribution Next: Derive local uniformity instead of matching one amplitude

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