SCIENCE · INVESTIGATE AND EXPLAIN

Distinguish required electric sources from a forbidden magnetic gradient

Maxwell equations can be used to audit a proposed field instead of merely generating a familiar wave. In a local Cartesian region take static E=(a x,0,0), where a has units V/m². Compare a transverse magnetic gradient B=(0,gx,0) with a longitudinal one B=(gx,0,0); the numerical coefficient g is a times 10⁻⁶ T/m under the explicitly linked teaching controls.

Learning outcome: Compare transverse and longitudinal gradients at positive, negative and zero coefficient. Compute divergences and curls before assigning sources. Keep electric Gauss, magnetic Gauss, Faraday and Ampère as separate checks rather than claiming that one passing equation validates the whole field.

Starting recap

Review flux integrals, linear differential equations, initial conditions, complex numbers and vector derivatives. Keep source constraints, RMS conventions and calibration roles explicit.

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Try

At a=1 V/m² with the transverse magnetic option, required charge density is epsilon0 C/m³ and current density is 10⁻⁶/mu0=1/(0.4π)≈0.795775 A/m² along positive z. Switching to the longitudinal magnetic gradient leaves div B=10⁻⁶ T/m, so the ordinary Maxwell model rejects it.

Open the experiment

Predict

Can a suitable electric current repair a nonzero divergence of B in ordinary Maxwell theory?

Your prediction and reason

Experiment

Compare transverse and longitudinal gradients at positive, negative and zero coefficient. Compute divergences and curls before assigning sources. Keep electric Gauss, magnetic Gauss, Faraday and Ampère as separate checks rather than claiming that one passing equation validates the whole field.

Identify required sources and reject nonzero magnetic divergence

Worked teaching model · synthetic values · Static ansatz E=(a x,0,0) V/m. B is either (0,a*1e−6 x,0) T or (a*1e−6 x,0,0) T, with x in metres. Evaluate derivatives, not a global finite apparatus. Ordinary Maxwell theory has no magnetic monopoles; nonzero div B is rejected rather than assigned a fictional magnetic charge.

Electric-gradient coefficient: 1 V/m² Magnetic direction:0 transverse,1 longitudinal: 0 code

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Candidate electric field

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

Equivalent numerical coordinates for Candidate electric field

static electric gradient

x (m)	Ex (V/m)
-1	-1
-0.975	-0.975
-0.95	-0.95
-0.925	-0.925
-0.9	-0.9
-0.875	-0.875
-0.85	-0.85
-0.825	-0.825
-0.8	-0.8
-0.775	-0.775
-0.75	-0.75
-0.725	-0.725
-0.7	-0.7
-0.675	-0.675
-0.65	-0.65
-0.625	-0.625
-0.6	-0.6
-0.575	-0.575
-0.55	-0.55
-0.525	-0.525
-0.5	-0.5
-0.475	-0.475
-0.45	-0.45
-0.425	-0.425
-0.4	-0.4
-0.375	-0.375
-0.35	-0.35
-0.325	-0.325
-0.3	-0.3
-0.275	-0.275
-0.25	-0.25
-0.225	-0.225
-0.2	-0.2
-0.175	-0.175
-0.15	-0.15
-0.125	-0.125
-0.1	-0.1
-0.075	-0.075
-0.05	-0.05
-0.025	-0.025
0	0
0.025	0.025
0.05	0.05
0.075	0.075
0.1	0.1
0.125	0.125
0.15	0.15
0.175	0.175
0.2	0.2
0.225	0.225
0.25	0.25
0.275	0.275
0.3	0.3
0.325	0.325
0.35	0.35
0.375	0.375
0.4	0.4
0.425	0.425
0.45	0.45
0.475	0.475
0.5	0.5
0.525	0.525
0.55	0.55
0.575	0.575
0.6	0.6
0.625	0.625
0.65	0.65
0.675	0.675
0.7	0.7
0.725	0.725
0.75	0.75
0.775	0.775
0.8	0.8
0.825	0.825
0.85	0.85
0.875	0.875
0.9	0.9
0.925	0.925
0.95	0.95
0.975	0.975
1	1

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.

Candidate magnetic field with separate physical units

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

Equivalent numerical coordinates for Candidate magnetic field with separate physical units

transverse magnetic gradient

x (m)	By (microtesla)
-1	-1
-0.975	-0.975
-0.95	-0.95
-0.925	-0.925
-0.9	-0.9
-0.875	-0.875
-0.85	-0.85
-0.825	-0.825
-0.8	-0.8
-0.775	-0.775
-0.75	-0.75
-0.725	-0.725
-0.7	-0.7
-0.675	-0.675
-0.65	-0.65
-0.625	-0.625
-0.6	-0.6
-0.575	-0.575
-0.55	-0.55
-0.525	-0.525
-0.5	-0.5
-0.475	-0.475
-0.45	-0.45
-0.425	-0.425
-0.4	-0.4
-0.375	-0.375
-0.35	-0.35
-0.325	-0.325
-0.3	-0.3
-0.275	-0.275
-0.25	-0.25
-0.225	-0.225
-0.2	-0.2
-0.175	-0.175
-0.15	-0.15
-0.125	-0.125
-0.1	-0.1
-0.075	-0.075
-0.05	-0.05
-0.025	-0.025
0	0
0.025	0.025
0.05	0.05
0.075	0.075
0.1	0.1
0.125	0.125
0.15	0.15
0.175	0.175
0.2	0.2
0.225	0.225
0.25	0.25
0.275	0.275
0.3	0.3
0.325	0.325
0.35	0.35
0.375	0.375
0.4	0.4
0.425	0.425
0.45	0.45
0.475	0.475
0.5	0.5
0.525	0.525
0.55	0.55
0.575	0.575
0.6	0.6
0.625	0.625
0.65	0.65
0.675	0.675
0.7	0.7
0.725	0.725
0.75	0.75
0.775	0.775
0.8	0.8
0.825	0.825
0.85	0.85
0.875	0.875
0.9	0.9
0.925	0.925
0.95	0.95
0.975	0.975
1	1

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
Electric divergence	1	V/m²
Required charge density	8.84194e-12	C/m³
Magnetic divergence residual	0	T/m
Faraday residual for static electric field	0	V/m²
Ordinary Maxwell magnetic constraint satisfied	1	flag
Required current density along z	0.7957747	A/m²

These static fields require electric charge and current sources; they are not a source-free wave.

Gauss gives rho=epsilon0 div E. For static fields Ampere gives J=curl B/mu0 and Faraday requires curl E=0. The transverse magnetic gradient has curl B along z and zero divergence, compatible locally with the stated charge/current sources. The longitudinal alternative has nonzero div B when a is nonzero; introducing an electric current cannot fix that magnetic Gauss-law violation. A local compatible ansatz does not specify a finite global apparatus or boundary conditions.

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

These static fields require electric charge and current sources; they are not a source-free wave.

Observation to attach to a capture

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

Understand

State geometry, sources and initial data

Derive the field or circuit relation

Gauss requires charge density rho=epsilon0a. For the transverse magnetic option, divergence is zero and curl B has z component g, so static Ampère requires Jz=g/mu0. The electric curl is zero, satisfying static Faraday. These nonzero fields therefore require sources; they are not source-free traveling waves simply because their local formulas are smooth.

Audit energy, uncertainty and domain

The longitudinal magnetic option has divergence g. When g is nonzero it violates ordinary magnetic Gauss law and is rejected rather than assigned an invented magnetic charge. An electric current term cannot repair this violation. At a=0 both fields vanish and the constraint passes. Even the locally compatible transverse option does not specify a finite global apparatus or its boundary conditions.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

For a=−2 V/m² with the transverse option, find required charge density divided by epsilon0.

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. Constants are rounded teaching values and are mutually consistent where Maxwell relations require it. No live circuit, high-voltage, energized-inductor, magnet, beam or optical exposure activity is requested. Retain first work, criterion-linked critique, revision and newly assigned evidence; public numerical feedback is not practical or learner certification.

Apply

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

Which source, initial-state, calibration or field-law premise makes the result conditional, and what unused evidence would challenge it?

Your explanation and limitations

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