Reconcile current surfaces without inventing gap conduction
A charging capacitor exposes why an electric-flux derivative belongs in Ampère’s law. A spanning surface through the lead sees conduction current, while another surface through the vacuum gap sees no material charge crossing it. Both surfaces have the same boundary loop and must predict the same magnetic circulation.
Learning outcome: Compare full and half-radius flux derivatives, and explain how both spanning-surface accounts agree. Change charge amplitude and frequency independently. Predict the field at t=0 before confusing zero charge with zero charging current.
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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For Q0=2 nC and omega=3 rad/s, wire and full-gap displacement currents at t=0 are 6 nA. The half-radius contour encloses 1.5 nA. Its field is 6×10⁻¹⁶ T, equal here to the field at radius 2 m outside the plate: different enclosed currents and circumferences can give the same local B.
Open the experimentPredict
At t=0 in this sinusoidal charge model, why can B be nonzero even though Q=0?
Experiment
Compare full and half-radius flux derivatives, and explain how both spanning-surface accounts agree. Change charge amplitude and frequency independently. Predict the field at t=0 before confusing zero charge with zero charging current.
Reconcile two surfaces through a charging capacitor
Worked teaching model · synthetic values · Ideal circular parallel plates radius1 m,uniform gap field,charge Q(t)=Q0 sin(omega t). Evaluate at t=0. No material current crosses the vacuum gap; displacement-current integral equals wire current under the uniform no-fringe premise. Rounded mu0 and c define epsilon0 consistently; no high-voltage apparatus.
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Surface-independent magnetic circulation under the ideal geometry
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: Ampere-Maxwell result (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Surface-independent magnetic circulation under the ideal geometry
Ampere-Maxwell result
| centered path radius (m) | azimuthal magnetic field (T) |
|---|---|
| 0 | 0 |
| 0.025 | 3.00000e-17 |
| 0.05 | 6.00000e-17 |
| 0.075 | 9.00000e-17 |
| 0.1 | 1.20000e-16 |
| 0.125 | 1.50000e-16 |
| 0.15 | 1.80000e-16 |
| 0.175 | 2.10000e-16 |
| 0.2 | 2.40000e-16 |
| 0.225 | 2.70000e-16 |
| 0.25 | 3.00000e-16 |
| 0.275 | 3.30000e-16 |
| 0.3 | 3.60000e-16 |
| 0.325 | 3.90000e-16 |
| 0.35 | 4.20000e-16 |
| 0.375 | 4.50000e-16 |
| 0.4 | 4.80000e-16 |
| 0.425 | 5.10000e-16 |
| 0.45 | 5.40000e-16 |
| 0.475 | 5.70000e-16 |
| 0.5 | 6.00000e-16 |
| 0.525 | 6.30000e-16 |
| 0.55 | 6.60000e-16 |
| 0.575 | 6.90000e-16 |
| 0.6 | 7.20000e-16 |
| 0.625 | 7.50000e-16 |
| 0.65 | 7.80000e-16 |
| 0.675 | 8.10000e-16 |
| 0.7 | 8.40000e-16 |
| 0.725 | 8.70000e-16 |
| 0.75 | 9.00000e-16 |
| 0.775 | 9.30000e-16 |
| 0.8 | 9.60000e-16 |
| 0.825 | 9.90000e-16 |
| 0.85 | 1.02000e-15 |
| 0.875 | 1.05000e-15 |
| 0.9 | 1.08000e-15 |
| 0.925 | 1.11000e-15 |
| 0.95 | 1.14000e-15 |
| 0.975 | 1.17000e-15 |
| 1 | 1.20000e-15 |
| 1.025 | 1.17073e-15 |
| 1.05 | 1.14286e-15 |
| 1.075 | 1.11628e-15 |
| 1.1 | 1.09091e-15 |
| 1.125 | 1.06667e-15 |
| 1.15 | 1.04348e-15 |
| 1.175 | 1.02128e-15 |
| 1.2 | 1.00000e-15 |
| 1.225 | 9.79592e-16 |
| 1.25 | 9.60000e-16 |
| 1.275 | 9.41176e-16 |
| 1.3 | 9.23077e-16 |
| 1.325 | 9.05660e-16 |
| 1.35 | 8.88889e-16 |
| 1.375 | 8.72727e-16 |
| 1.4 | 8.57143e-16 |
| 1.425 | 8.42105e-16 |
| 1.45 | 8.27586e-16 |
| 1.475 | 8.13559e-16 |
| 1.5 | 8.00000e-16 |
| 1.525 | 7.86885e-16 |
| 1.55 | 7.74194e-16 |
| 1.575 | 7.61905e-16 |
| 1.6 | 7.50000e-16 |
| 1.625 | 7.38462e-16 |
| 1.65 | 7.27273e-16 |
| 1.675 | 7.16418e-16 |
| 1.7 | 7.05882e-16 |
| 1.725 | 6.95652e-16 |
| 1.75 | 6.85714e-16 |
| 1.775 | 6.76056e-16 |
| 1.8 | 6.66667e-16 |
| 1.825 | 6.57534e-16 |
| 1.85 | 6.48649e-16 |
| 1.875 | 6.40000e-16 |
| 1.9 | 6.31579e-16 |
| 1.925 | 6.23377e-16 |
| 1.95 | 6.15385e-16 |
| 1.975 | 6.07595e-16 |
| 2 | 6.00000e-16 |
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.
| Quantity | Value | Unit |
|---|---|---|
| Wire current at zero time | 6.00000e-9 | A |
| Gap electric-field time derivative | 216 | V/(m s) |
| Full-plate displacement current | 6.00000e-9 | A |
| Displacement current inside half radius | 1.50000e-9 | A |
| Azimuthal magnetic field at half radius | 6.00000e-16 | T |
| Azimuthal magnetic field at two metres | 6.00000e-16 | T |
| Material conduction current through gap | 0 | A |
The gap term is an electric-flux derivative, not electrons physically crossing the vacuum.
Uniform gap E=Q/(epsilon0 pi R²) gives epsilon0 dPhiE/dt=dQ/dt for the full plate area. A radius r inside R encloses fraction r²/R² of that displacement current, so B grows linearly with r; outside the plate region the enclosed flux derivative saturates and B falls as 1/r in the stipulated approximation. A spanning surface cutting the lead and one crossing the gap must give the same circulation once the displacement term is included.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The gap term is an electric-flux derivative, not electrons physically crossing the vacuum.
Uniform gap E=Q/(epsilon0 pi R²) gives epsilon0 dPhiE/dt=dQ/dt for the full plate area. A radius r inside R encloses fraction r²/R² of that displacement current, so B grows linearly with r; outside the plate region the enclosed flux derivative saturates and B falls as 1/r in the stipulated approximation. A spanning surface cutting the lead and one crossing the gap must give the same circulation once the displacement term is included.
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