Integrate changing flux before inferring an induced field
A changing magnetic field creates an electric circulation even if no wire occupies the contour. Choose a stationary circle with positive z normal and counterclockwise positive traversal. In the local region let Bz=(B0+βt)[1+(r/ell)²], with B0=2 T and ell=1 m. The ramp β has units T/s, while the radial factor is dimensionless.
Learning outcome: Compare radii 0.5,1,2 m and both ramp signs. Derive the area integral before revealing the ledger, reconstruct circulation from Etheta and circumference, and distinguish nonzero flux from nonzero flux change.
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 R=1 m and β=1 T/s, initial flux is 3π Wb, its derivative 1.5π Wb/s, and emf −1.5π V. The azimuthal electric field is −0.75 V/m: clockwise relative to the chosen positive traversal. With β=0, the initial flux remains nonzero but emf is exactly zero.
Open the experimentPredict
At R=1 m and β=1 T/s, what is the signed counterclockwise emf?
Experiment
Compare radii 0.5,1,2 m and both ramp signs. Derive the area integral before revealing the ledger, reconstruct circulation from Etheta and circumference, and distinguish nonzero flux from nonzero flux change.
Integrate a changing nonuniform flux
Worked teaching model · synthetic values · Stationary circular contour, +z normal and counterclockwise positive circulation. Imposed axisymmetric Bz=(2+beta t)(1+r²/(1 m²)) T in the local region. Calculate flux at t=0 and its time derivative. External source currents are unspecified; this is not a source-free global field or apparatus plan.
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Nonconservative azimuthal electric field
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: Faraday integral (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Nonconservative azimuthal electric field
Faraday integral
| radius (m) | counterclockwise field (V/m) |
|---|---|
| 0 | 0 |
| 0.025 | -0.01250391 |
| 0.05 | -0.02503125 |
| 0.075 | -0.03760547 |
| 0.1 | -0.05025 |
| 0.125 | -0.06298828 |
| 0.15 | -0.07584375 |
| 0.175 | -0.08883984 |
| 0.2 | -0.102 |
| 0.225 | -0.1153477 |
| 0.25 | -0.1289063 |
| 0.275 | -0.1426992 |
| 0.3 | -0.15675 |
| 0.325 | -0.171082 |
| 0.35 | -0.1857187 |
| 0.375 | -0.2006836 |
| 0.4 | -0.216 |
| 0.425 | -0.2316914 |
| 0.45 | -0.2477813 |
| 0.475 | -0.264293 |
| 0.5 | -0.28125 |
| 0.525 | -0.2986758 |
| 0.55 | -0.3165938 |
| 0.575 | -0.3350273 |
| 0.6 | -0.354 |
| 0.625 | -0.3735352 |
| 0.65 | -0.3936563 |
| 0.675 | -0.4143867 |
| 0.7 | -0.43575 |
| 0.725 | -0.4577695 |
| 0.75 | -0.4804688 |
| 0.775 | -0.5038711 |
| 0.8 | -0.528 |
| 0.825 | -0.5528789 |
| 0.85 | -0.5785312 |
| 0.875 | -0.6049805 |
| 0.9 | -0.63225 |
| 0.925 | -0.6603633 |
| 0.95 | -0.6893437 |
| 0.975 | -0.7192148 |
| 1 | -0.75 |
| 1.025 | -0.7817227 |
| 1.05 | -0.8144062 |
| 1.075 | -0.8480742 |
| 1.1 | -0.88275 |
| 1.125 | -0.918457 |
| 1.15 | -0.9552187 |
| 1.175 | -0.9930586 |
| 1.2 | -1.032 |
| 1.225 | -1.072066 |
| 1.25 | -1.113281 |
| 1.275 | -1.155668 |
| 1.3 | -1.19925 |
| 1.325 | -1.244051 |
| 1.35 | -1.290094 |
| 1.375 | -1.337402 |
| 1.4 | -1.386 |
| 1.425 | -1.43591 |
| 1.45 | -1.487156 |
| 1.475 | -1.539762 |
| 1.5 | -1.59375 |
| 1.525 | -1.649145 |
| 1.55 | -1.705969 |
| 1.575 | -1.764246 |
| 1.6 | -1.824 |
| 1.625 | -1.885254 |
| 1.65 | -1.948031 |
| 1.675 | -2.012355 |
| 1.7 | -2.07825 |
| 1.725 | -2.145738 |
| 1.75 | -2.214844 |
| 1.775 | -2.28559 |
| 1.8 | -2.358 |
| 1.825 | -2.432098 |
| 1.85 | -2.507906 |
| 1.875 | -2.585449 |
| 1.9 | -2.66475 |
| 1.925 | -2.745832 |
| 1.95 | -2.828719 |
| 1.975 | -2.913434 |
| 2 | -3 |
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 |
|---|---|---|
| Initial magnetic flux | 9.424778 | Wb |
| Flux time derivative | 4.712389 | Wb/s |
| Counterclockwise emf | -4.712389 | V |
| Azimuthal electric field on contour | -0.75 | V/m |
| Circulation reconstruction residual | 0 | V |
Changing flux is an area integral; neither center nor edge field times area is generally correct.
Integrate Bz over 2pi r dr to get Phi=(2+beta t)pi(R²+R⁴/2) in the stated metre convention. Faraday gives emf=−dPhi/dt, and circular symmetry gives Etheta=emf/(2pi R). The induced electric field can circulate even without a conducting wire. A nonzero closed electric circulation cannot be represented by a single electrostatic scalar potential throughout this region.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Changing flux is an area integral; neither center nor edge field times area is generally correct.
Integrate Bz over 2pi r dr to get Phi=(2+beta t)pi(R²+R⁴/2) in the stated metre convention. Faraday gives emf=−dPhi/dt, and circular symmetry gives Etheta=emf/(2pi R). The induced electric field can circulate even without a conducting wire. A nonzero closed electric circulation cannot be represented by a single electrostatic scalar potential throughout this region.
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