SCIENCE · INVESTIGATE AND EXPLAIN

Integrate a nonuniform current density

A measured total current is an integral across an oriented surface, not a point value of current density. Consider an imposed axial profile J(r)=J0[1+α(r/R)²] in a circular wire of radius R=1 mm, carrying total conventional current 2 A in the positive axial direction. The coefficient α changes the shape while total current stays fixed.

Learning outcome: Keep total current fixed and compare α=0,1,2,3. Record central density and half-radius current before revealing them. Explain why the profile is renormalized and why the partial-current discrepancy is not lost charge.

Starting recap

Review charge continuity, definite integrals, first-order differential equations, initial conditions and signed source work. Distinguish displayed time from physical time.

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For α=2, J0=10⁶/π A/m². The inner half-radius disk carries fraction (1/4+1/16)/2=5/32 of the total, or 0.3125 A. An area-only estimate would give 0.5 A. At that radius J=1.5×10⁶/π A/m², and electron drift is negative along the chosen axis.

Open the experiment

Predict

For α=2, what current passes inside half the wire radius?

Your prediction and reason

Experiment

Keep total current fixed and compare α=0,1,2,3. Record central density and half-radius current before revealing them. Explain why the profile is renormalized and why the partial-current discrepancy is not lost charge.

Normalize a nonuniform current density

Worked teaching model · synthetic values · Circular wire radius 1 mm, total conventional current 2 A in +z. Stipulated axial J(r)=J0[1+alpha(r/R)²], no radial current. This profile is imposed; it is not inferred from a uniform conductivity assumption. Electron density 8e28 per m³, elementary charge 1.602176634e−19 C. No physical wiring.

Radial profile coefficient: 2 ratio Inner sampled radius fraction: 0.5 ratio

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Normalized radial current profile

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

Equivalent numerical coordinates for Normalized radial current profile

imposed profile

radius fraction	axial current density (A/m²)
0	318309.9
0.0125	318409.4
0.025	318707.8
0.0375	319205.1
0.05	319901.4
0.0625	320796.7
0.075	321890.9
0.0875	323184
0.1	324676.1
0.1125	326367.1
0.125	328257.1
0.1375	330346
0.15	332633.8
0.1625	335120.6
0.175	337806.4
0.1875	340691.1
0.2	343774.7
0.2125	347057.2
0.225	350538.8
0.2375	354219.2
0.25	358098.6
0.2625	362177
0.275	366454.3
0.2875	370930.5
0.3	375605.7
0.3125	380479.8
0.325	385552.8
0.3375	390824.9
0.35	396295.8
0.3625	401965.7
0.375	407834.5
0.3875	413902.3
0.4	420169
0.4125	426634.7
0.425	433299.3
0.4375	440162.9
0.45	447225.4
0.4625	454486.8
0.475	461947.2
0.4875	469606.6
0.5	477464.8
0.5125	485522
0.525	493778.2
0.5375	502233.3
0.55	510887.4
0.5625	519740.4
0.575	528792.3
0.5875	538043.2
0.6	547493
0.6125	557141.8
0.625	566989.5
0.6375	577036.1
0.65	587281.7
0.6625	597726.3
0.675	608369.8
0.6875	619212.2
0.7	630253.6
0.7125	641493.9
0.725	652933.2
0.7375	664571.4
0.75	676408.5
0.7625	688444.6
0.775	700679.6
0.7875	713113.6
0.8	725746.5
0.8125	738578.4
0.825	751609.2
0.8375	764839
0.85	778267.7
0.8625	791895.3
0.875	805721.9
0.8875	819747.4
0.9	833971.9
0.9125	848395.3
0.925	863017.7
0.9375	877839
0.95	892859.2
0.9625	908078.4
0.975	923496.6
0.9875	939113.6
1	954929.7

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
Central current density	318309.9	A/m²
Sampled-radius current density	477464.8	A/m²
Current inside sampled radius	0.3125	A
Area-only current estimate	0.5	A
Local electron drift velocity along z	-3.72513e-5	m/s
Total conventional current	2	A

The area element is 2 pi r dr; averaging endpoint densities is not the current integral.

Integrating J0[1+alpha(r/R)²] over the disk gives I=pi R² J0(1+alpha/2). A partial radius fR contains fraction (f²+alpha f⁴/2)/(1+alpha/2), not generally f². Conventional current points with positive J; electron drift is opposite because J=n q vd and q is negative. This local drift is not the propagation speed of an electrical signal.

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

The area element is 2 pi r dr; averaging endpoint densities is not the current integral.

Observation to attach to a capture

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

Understand

State the current, source and measurement model

Derive the integral or initial-value solution

Use annular area dA=2πr dr. Integrating from zero to R gives I=πR²J0(1+α/2), so J0 must decrease as the outer-weighted profile increases. A smaller disk of radius fR carries fraction (f²+αf⁴/2)/(1+α/2). Multiplying total current by f² assumes uniform current density and fails for this supplied nonuniform profile.

Test conservation, uncertainty and limits

With independently stipulated electron number density n, local electron drift is vd=−J/(ne). Its direction is opposite conventional current. This drift speed does not set signal propagation speed. The profile is imposed for an integration experiment; a uniform scalar conductivity and uniform axial field would not independently produce the same radial profile.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

For α=0, find the current inside radius R/4 in amperes.

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 evidence checked 31 October 2026 against the primary explanation. No live circuit construction, energized switching, component rating or electrical handling instruction is requested. Preserve first predictions, critique, revision and fresh assigned work separately from public practice. Software agreement does not certify measurement technique or learning.

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