SCIENCE · INVESTIGATE AND EXPLAIN

Require both curl equations for a vacuum wave

Perpendicular sinusoidal arrows do not automatically form a valid electromagnetic wave. Test the candidate Ey=E0 cos(kx−omega t) and Bz=B0 cos(kx−omega t), with E0=2 V/m and k=π per metre. Controls propose phase speed a times c and signed magnetic amplitude bE0/c. These proposals include deliberately inconsistent choices.

Learning outcome: Compare the valid pair with a Faraday-only pair and with reversed magnetic sign. Predict each residual before inspecting the plots. Explain which violation could be interpreted as requiring a current and which cannot be repaired by an electric-current term.

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 and b=1, both residuals are zero and cB0=E0. At a=0.5 and b=2, their product is one, so Faraday passes, but b−a=1.5 leaves a nonzero Ampère–Maxwell residual. Matching one derivative relation is therefore an insufficient validation test.

Open the experiment

Predict

For a=0.5 and b=2, which statement follows?

Your prediction and reason

Experiment

Compare the valid pair with a Faraday-only pair and with reversed magnetic sign. Predict each residual before inspecting the plots. Explain which violation could be interpreted as requiring a current and which cannot be repaired by an electric-current term.

Test wave amplitude and speed against both curl laws

Worked teaching model · synthetic values · Candidate Ey=2 cos(kx−omega t) V/m,Bz=(b*2/c)cos(kx−omega t) T,k=pi per m,omega=a c k. Source-free vacuum claim must satisfy both curl laws, not just perpendicular arrows. Teaching c3e8,mu0=4pi e−7,epsilon0=1/(mu0 c²) are mutually consistent.

Proposed phase speed divided by c: 1 ratio Proposed signed cB amplitude divided by E: 1 ratio

Reset keeps your prediction, captures and practice history.

Candidate fields at zero time, with explicit constraint audit

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: candidate Ey (solid).2: candidate cBz (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Candidate fields at zero time, with explicit constraint audit

candidate Ey

x (m)	E and cB on common electric-field scale (V/m)
0	2
0.025	1.993835
0.05	1.975377
0.075	1.94474
0.1	1.902113
0.125	1.847759
0.15	1.782013
0.175	1.70528
0.2	1.618034
0.225	1.520812
0.25	1.414214
0.275	1.298896
0.3	1.175571
0.325	1.044997
0.35	0.907981
0.375	0.7653669
0.4	0.618034
0.425	0.4668907
0.45	0.3128689
0.475	0.1569182
0.5	1.22465e-16
0.525	-0.1569182
0.55	-0.3128689
0.575	-0.4668907
0.6	-0.618034
0.625	-0.7653669
0.65	-0.907981
0.675	-1.044997
0.7	-1.175571
0.725	-1.298896
0.75	-1.414214
0.775	-1.520812
0.8	-1.618034
0.825	-1.70528
0.85	-1.782013
0.875	-1.847759
0.9	-1.902113
0.925	-1.94474
0.95	-1.975377
0.975	-1.993835
1	-2
1.025	-1.993835
1.05	-1.975377
1.075	-1.94474
1.1	-1.902113
1.125	-1.847759
1.15	-1.782013
1.175	-1.70528
1.2	-1.618034
1.225	-1.520812
1.25	-1.414214
1.275	-1.298896
1.3	-1.175571
1.325	-1.044997
1.35	-0.907981
1.375	-0.7653669
1.4	-0.618034
1.425	-0.4668907
1.45	-0.3128689
1.475	-0.1569182
1.5	-3.67394e-16
1.525	0.1569182
1.55	0.3128689
1.575	0.4668907
1.6	0.618034
1.625	0.7653669
1.65	0.907981
1.675	1.044997
1.7	1.175571
1.725	1.298896
1.75	1.414214
1.775	1.520812
1.8	1.618034
1.825	1.70528
1.85	1.782013
1.875	1.847759
1.9	1.902113
1.925	1.94474
1.95	1.975377
1.975	1.993835
2	2

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 cBz

x (m)	E and cB on common electric-field scale (V/m)
0	2
0.025	1.993835
0.05	1.975377
0.075	1.94474
0.1	1.902113
0.125	1.847759
0.15	1.782013
0.175	1.70528
0.2	1.618034
0.225	1.520812
0.25	1.414214
0.275	1.298896
0.3	1.175571
0.325	1.044997
0.35	0.907981
0.375	0.7653669
0.4	0.618034
0.425	0.4668907
0.45	0.3128689
0.475	0.1569182
0.5	1.22465e-16
0.525	-0.1569182
0.55	-0.3128689
0.575	-0.4668907
0.6	-0.618034
0.625	-0.7653669
0.65	-0.907981
0.675	-1.044997
0.7	-1.175571
0.725	-1.298896
0.75	-1.414214
0.775	-1.520812
0.8	-1.618034
0.825	-1.70528
0.85	-1.782013
0.875	-1.847759
0.9	-1.902113
0.925	-1.94474
0.95	-1.975377
0.975	-1.993835
1	-2
1.025	-1.993835
1.05	-1.975377
1.075	-1.94474
1.1	-1.902113
1.125	-1.847759
1.15	-1.782013
1.175	-1.70528
1.2	-1.618034
1.225	-1.520812
1.25	-1.414214
1.275	-1.298896
1.3	-1.175571
1.325	-1.044997
1.35	-0.907981
1.375	-0.7653669
1.4	-0.618034
1.425	-0.4668907
1.45	-0.3128689
1.475	-0.1569182
1.5	-3.67394e-16
1.525	0.1569182
1.55	0.3128689
1.575	0.4668907
1.6	0.618034
1.625	0.7653669
1.65	0.907981
1.675	1.044997
1.7	1.175571
1.725	1.298896
1.75	1.414214
1.775	1.520812
1.8	1.618034
1.825	1.70528
1.85	1.782013
1.875	1.847759
1.9	1.902113
1.925	1.94474
1.95	1.975377
1.975	1.993835
2	2

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
Proposed angular frequency	9.42478e+8	rad/s
Proposed signed magnetic amplitude	6.66667e-9	T
Faraday sine-coefficient residual	0	V/m²
Ampere-Maxwell sine-coefficient residual	0	T/m
Source-free vacuum curl constraints satisfied	1	flag
Electric divergence	0	V/m²
Magnetic divergence	0	T/m

Both curl constraints hold with E/B=c and phase speed c for this traveling vacuum wave.

For phase kx−omega t, curl E has z component −kE0 sin(phase), while −dB/dt is −omega B0 sin(phase). Their equality requires kE0=omega B0. The source-free Ampere-Maxwell equation independently requires kB0=mu0 epsilon0 omega E0. With consistent constants both imply speed c and B0=E0/c for +x propagation. A current could modify Ampere, but cannot repair a Faraday violation of this candidate field pair.

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

Both curl constraints hold with E/B=c and phase speed c for this traveling vacuum 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

Faraday requires kE0=omega B0, while source-free Ampère–Maxwell independently requires kB0=mu0epsilon0omega E0. With epsilon0=1/(mu0c²), the two imply positive propagation speed c and B0=E0/c for this positive-x phase. Satisfying only one relation does not establish the source-free wave claim.

Audit energy, uncertainty and domain

The ledger reports the actual sine-coefficient residuals: curl E plus dB/dt has coefficient omega B0−kE0, and curl B minus mu0epsilon0 dE/dt has coefficient kB0−mu0epsilon0omega E0. A current source can modify the second law, but cannot repair a Faraday violation for the same proposed fields. Their divergences vanish separately because the field components are transverse.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

For the valid positive-x vacuum wave with E0=3 V/m and c=3×10⁸ m/s, find B0 in nanotesla.

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

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.

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