Check inductance against a field-volume integral
A flux through one turn is not the same as total flux linkage. In the long-solenoid approximation, turns per length n, length ell and area A give total turns N=n ell and internal field B=μ0nI. This packet uses ell=2 m, A=0.01 m² and a vacuum interior; external field and end corrections are neglected.
Learning outcome: Compare n=100,200,400 at the same current, then set current to zero. Record flux per turn, linkage, inductance and both energy accounts. Explain why doubling N at fixed length differs from doubling length at fixed n.
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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At n=200 per metre and I=2 A, N=400 turns, B≈0.000502655 T and L=0.00032π H≈0.001005310 H. Stored energy is 0.00064π J≈0.002010619 J. Integrating the internal field density over 0.02 m³ gives the same value.
Open the experimentPredict
At fixed length, area and current, doubling turn density changes stored energy by:
Experiment
Compare n=100,200,400 at the same current, then set current to zero. Record flux per turn, linkage, inductance and both energy accounts. Explain why doubling N at fixed length differs from doubling length at fixed n.
Reconcile flux linkage with field energy
Worked teaching model · synthetic values · Long-solenoid approximation length2 m,area.01 m²,vacuum. Uniform internal field and negligible external field are stipulated approximations; total turns n times length. Rounded mu0. No finite-end, core saturation, real coil heating or handling model.
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Quadratic magnetic storage at fixed geometry
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: half L I squared (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Quadratic magnetic storage at fixed geometry
half L I squared
| current (A) | stored energy (J) |
|---|---|
| 0 | 0 |
| 0.05 | 1.25664e-6 |
| 0.1 | 5.02655e-6 |
| 0.15 | 1.13097e-5 |
| 0.2 | 2.01062e-5 |
| 0.25 | 3.14159e-5 |
| 0.3 | 4.52389e-5 |
| 0.35 | 6.15752e-5 |
| 0.4 | 8.04248e-5 |
| 0.45 | 0.0001017876 |
| 0.5 | 0.0001256637 |
| 0.55 | 0.0001520531 |
| 0.6 | 0.0001809557 |
| 0.65 | 0.0002123717 |
| 0.7 | 0.0002463009 |
| 0.75 | 0.0002827433 |
| 0.8 | 0.0003216991 |
| 0.85 | 0.0003631681 |
| 0.9 | 0.0004071504 |
| 0.95 | 0.000453646 |
| 1 | 0.0005026548 |
| 1.05 | 0.0005541769 |
| 1.1 | 0.0006082123 |
| 1.15 | 0.000664761 |
| 1.2 | 0.0007238229 |
| 1.25 | 0.0007853982 |
| 1.3 | 0.0008494867 |
| 1.35 | 0.0009160884 |
| 1.4 | 0.0009852035 |
| 1.45 | 0.001056832 |
| 1.5 | 0.001130973 |
| 1.55 | 0.001207628 |
| 1.6 | 0.001286796 |
| 1.65 | 0.001368478 |
| 1.7 | 0.001452672 |
| 1.75 | 0.00153938 |
| 1.8 | 0.001628602 |
| 1.85 | 0.001720336 |
| 1.9 | 0.001814584 |
| 1.95 | 0.001911345 |
| 2 | 0.002010619 |
| 2.05 | 0.002112407 |
| 2.1 | 0.002216708 |
| 2.15 | 0.002323522 |
| 2.2 | 0.002432849 |
| 2.25 | 0.00254469 |
| 2.3 | 0.002659044 |
| 2.35 | 0.002775911 |
| 2.4 | 0.002895292 |
| 2.45 | 0.003017186 |
| 2.5 | 0.003141593 |
| 2.55 | 0.003268513 |
| 2.6 | 0.003397947 |
| 2.65 | 0.003529894 |
| 2.7 | 0.003664354 |
| 2.75 | 0.003801327 |
| 2.8 | 0.003940814 |
| 2.85 | 0.004082814 |
| 2.9 | 0.004227327 |
| 2.95 | 0.004374354 |
| 3 | 0.004523893 |
| 3.05 | 0.004675947 |
| 3.1 | 0.004830513 |
| 3.15 | 0.004987592 |
| 3.2 | 0.005147185 |
| 3.25 | 0.005309292 |
| 3.3 | 0.005473911 |
| 3.35 | 0.005641044 |
| 3.4 | 0.00581069 |
| 3.45 | 0.005982849 |
| 3.5 | 0.006157522 |
| 3.55 | 0.006334707 |
| 3.6 | 0.006514407 |
| 3.65 | 0.006696619 |
| 3.7 | 0.006881345 |
| 3.75 | 0.007068583 |
| 3.8 | 0.007258336 |
| 3.85 | 0.007450601 |
| 3.9 | 0.00764538 |
| 3.95 | 0.007842672 |
| 4 | 0.008042477 |
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 |
|---|---|---|
| Total turns | 400 | count |
| Approximate internal magnetic field | 0.0005026548 | T |
| Flux per turn | 5.02655e-6 | Wb |
| Flux linkage | 0.002010619 | Wb turn |
| Self inductance | 0.00100531 | H |
| Circuit magnetic energy | 0.002010619 | J |
| Internal field energy density | 0.100531 | J/m³ |
| Field-volume energy | 0.002010619 | J |
| Energy-account residual | 0 | J |
Flux linkage determines inductance, and the same approximation must be used in the field-volume energy check.
The long-solenoid field is B=mu0 n I and N=n ell, so flux linkage N B A=mu0 n² A ell I. Hence L=mu0 n² A ell. Integrating B²/(2mu0) over the approximate internal volume A ell gives LI²/2. Agreement checks consistency of one ideal geometry; it does not validate neglected fringe fields or a nonlinear magnetic core.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Flux linkage determines inductance, and the same approximation must be used in the field-volume energy check.
The long-solenoid field is B=mu0 n I and N=n ell, so flux linkage N B A=mu0 n² A ell I. Hence L=mu0 n² A ell. Integrating B²/(2mu0) over the approximate internal volume A ell gives LI²/2. Agreement checks consistency of one ideal geometry; it does not validate neglected fringe fields or a nonlinear magnetic core.
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