SCIENCE · INVESTIGATE AND EXPLAIN

Integrate a signed current-segment contribution

Biot–Savart is a vector source integral. For a straight segment from x=−1 to +1 m carrying 3 A along positive x, an observer at (0,y,0) with positive y sees a contribution along positive z: the sign follows from dl crossed with the source-to-observer vector. We use the explicitly rounded teaching value μ0=4π×10⁻⁷, not a claim that this is an exact modern SI constant.

Learning outcome: At y=0.25,0.5,1 m compare the exact integral with N and 2N midpoint cells. Retain the signed quadrature error separately from the infinite-wire error. Sketch a possible return route and explain why its field would be an additional physical contribution, not numerical error.

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

At L=1 m and y=1 m, the exact segment contribution is 6×10⁻⁷/√2 T≈0.424264 µT along positive z. The infinite-wire approximation gives 0.6 µT, a relative overestimate √2−1≈0.414214. Both expressions concern the specified current direction; reversing current reverses the field.

Open the experiment

Predict

For current along positive x and an observer at positive y, which way does the segment contribution point?

Your prediction and reason

Experiment

At y=0.25,0.5,1 m compare the exact integral with N and 2N midpoint cells. Retain the signed quadrature error separately from the infinite-wire error. Sketch a possible return route and explain why its field would be an additional physical contribution, not numerical error.

Integrate a finite current segment and test resolution

Worked teaching model · synthetic values · Straight segment along x from −1 to +1 m carries +3 A. Observer is (0,y,0), y positive. Calculate only this segment contribution; a steady physical circuit needs a return path whose field is excluded here, not an isolated current endpoint. Rounded teaching mu0=4 pi×10⁻7, not a claim of modern SI exactness.

Perpendicular observer distance: 0.5 m Midpoint segment count: 8 count

Reset keeps your prediction, captures and practice history.

Signed Biot–Savart contribution along the segment

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

Equivalent numerical coordinates for Signed Biot–Savart contribution along the segment

cross-product kernel

source x coordinate (m)	field contribution per length (T/m)
-1	1.07331e-7
-0.975	1.14020e-7
-0.95	1.21236e-7
-0.925	1.29027e-7
-0.9	1.37446e-7
-0.875	1.46552e-7
-0.85	1.56407e-7
-0.825	1.67082e-7
-0.8	1.78651e-7
-0.775	1.91198e-7
-0.75	2.04812e-7
-0.725	2.19591e-7
-0.7	2.35637e-7
-0.675	2.53064e-7
-0.65	2.71990e-7
-0.625	2.92540e-7
-0.6	3.14845e-7
-0.575	3.39037e-7
-0.55	3.65252e-7
-0.525	3.93620e-7
-0.5	4.24264e-7
-0.475	4.57293e-7
-0.45	4.92792e-7
-0.425	5.30814e-7
-0.4	5.71367e-7
-0.375	6.14400e-7
-0.35	6.59784e-7
-0.325	7.07299e-7
-0.3	7.56611e-7
-0.275	8.07262e-7
-0.25	8.58650e-7
-0.225	9.10026e-7
-0.2	9.60493e-7
-0.175	1.00902e-6
-0.15	1.05449e-6
-0.125	1.09569e-6
-0.1	1.13144e-6
-0.075	1.16061e-6
-0.05	1.18222e-6
-0.025	1.19551e-6
0	1.20000e-6
0.025	1.19551e-6
0.05	1.18222e-6
0.075	1.16061e-6
0.1	1.13144e-6
0.125	1.09569e-6
0.15	1.05449e-6
0.175	1.00902e-6
0.2	9.60493e-7
0.225	9.10026e-7
0.25	8.58650e-7
0.275	8.07262e-7
0.3	7.56611e-7
0.325	7.07299e-7
0.35	6.59784e-7
0.375	6.14400e-7
0.4	5.71367e-7
0.425	5.30814e-7
0.45	4.92792e-7
0.475	4.57293e-7
0.5	4.24264e-7
0.525	3.93620e-7
0.55	3.65252e-7
0.575	3.39037e-7
0.6	3.14845e-7
0.625	2.92540e-7
0.65	2.71990e-7
0.675	2.53064e-7
0.7	2.35637e-7
0.725	2.19591e-7
0.75	2.04812e-7
0.775	1.91198e-7
0.8	1.78651e-7
0.825	1.67082e-7
0.85	1.56407e-7
0.875	1.46552e-7
0.9	1.37446e-7
0.925	1.29027e-7
0.95	1.21236e-7
0.975	1.14020e-7
1	1.07331e-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
Exact segment field along +z	1.07331e-6	T
Midpoint field	1.07459e-6	T
Doubled-grid midpoint field	1.07365e-6	T
Midpoint signed error	1.27851e-9	T
Doubled-grid signed error	3.33829e-10	T
Infinite-wire approximation	1.20000e-6	T
Infinite-wire relative overestimate	0.118034	ratio

The source-to-observer cross product fixes the sign before the scalar integral is evaluated.

For current along +x and observer above the segment, dl cross r points +z. Integrate (mu0 I/4pi)y dx/(x²+y²)^(3/2) to obtain mu0 I L/[2pi y sqrt(L²+y²)] with half-length L=1 m. The infinite-wire limit is appropriate only when L is large relative to y, and agreement between two grids tests this prescribed integral rather than the omitted return path.

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

The source-to-observer cross product fixes the sign before the scalar integral is evaluated.

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

The signed scalar kernel is (μ0I/4π)y/(x²+y²)^(3/2). Integrating over a symmetric half-length L gives Bz=μ0IL/[2πy√(L²+y²)]. The infinite-wire expression omits the finite-end factor L/√(L²+y²). Separately approximate the original integral with equal midpoint cells and double the resolution; the finite-source approximation error and discretization error are different quantities.

Check energy, symmetry and evidence limits

A steady current cannot simply begin and end at isolated endpoints. This lesson calculates one segment contribution within a complete circuit whose return-path field is deliberately not included. A real field claim would need that return contribution or evidence that it is negligible. Numerical convergence of the displayed segment integral cannot supply the missing circuit geometry.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

For L=1 m, I=3 A and y=√3 m, find the segment 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: Test numerical decay against passivity and exact time Next: Include the return arc in a closed source geometry

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