Integrate a nonuniform field through an oriented patch
Flux is an oriented surface integral, not a count of actual particles and not the magnitude of the electric field. For each small patch, use E dot n times its area. A tangent field can be large while contributing no flux. For an open patch, reversing the chosen normal reverses flux; enclosed-charge language applies only after specifying a closed surface.
Learning outcome: Compare β=−1,0,1,2 at L=2 m. Retain signed flux, center approximation and its error before inspecting the graph. Explain why a zero net flux through part of the patch would not prove that the local field vanishes.
Starting recap
Review oriented normals, definite integrals, source volume measures and partial derivatives as one-coordinate rates. Potential differences use an explicit reference and field work uses a signed line integral.
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For L=2 m and β=1 N/C/m², exact flux is 20/3 N m²/C, and the area-average normal field is 10/3 N/C. Center-field times area gives 6 N m²/C, an error −2/3. The tangent 5 N/C contributes zero. Reversing the surface normal gives −20/3.
Open the experimentPredict
For L=2 m and β=1, what is the exact flux through the +z patch?
Experiment
Compare β=−1,0,1,2 at L=2 m. Retain signed flux, center approximation and its error before inspecting the graph. Explain why a zero net flux through part of the patch would not prove that the local field vanishes.
Integrate a nonuniform normal field over a patch
Worked teaching model · synthetic values · Patch z=0,0≤x≤L,0≤y≤1 m with chosen +z normal. Prescribed field on the patch is E=(5,0,2+βx²) N/C with dimensioned coefficients. The tangent component has no flux through this patch. A local patch is not a closed surface and does not alone determine enclosed charge.
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Nonuniform normal component on the patch
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: prescribed patch field (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Nonuniform normal component on the patch
prescribed patch field
| x coordinate (m) | normal field (N/C) |
|---|---|
| 0 | 2 |
| 0.025 | 2.000625 |
| 0.05 | 2.0025 |
| 0.075 | 2.005625 |
| 0.1 | 2.01 |
| 0.125 | 2.015625 |
| 0.15 | 2.0225 |
| 0.175 | 2.030625 |
| 0.2 | 2.04 |
| 0.225 | 2.050625 |
| 0.25 | 2.0625 |
| 0.275 | 2.075625 |
| 0.3 | 2.09 |
| 0.325 | 2.105625 |
| 0.35 | 2.1225 |
| 0.375 | 2.140625 |
| 0.4 | 2.16 |
| 0.425 | 2.180625 |
| 0.45 | 2.2025 |
| 0.475 | 2.225625 |
| 0.5 | 2.25 |
| 0.525 | 2.275625 |
| 0.55 | 2.3025 |
| 0.575 | 2.330625 |
| 0.6 | 2.36 |
| 0.625 | 2.390625 |
| 0.65 | 2.4225 |
| 0.675 | 2.455625 |
| 0.7 | 2.49 |
| 0.725 | 2.525625 |
| 0.75 | 2.5625 |
| 0.775 | 2.600625 |
| 0.8 | 2.64 |
| 0.825 | 2.680625 |
| 0.85 | 2.7225 |
| 0.875 | 2.765625 |
| 0.9 | 2.81 |
| 0.925 | 2.855625 |
| 0.95 | 2.9025 |
| 0.975 | 2.950625 |
| 1 | 3 |
| 1.025 | 3.050625 |
| 1.05 | 3.1025 |
| 1.075 | 3.155625 |
| 1.1 | 3.21 |
| 1.125 | 3.265625 |
| 1.15 | 3.3225 |
| 1.175 | 3.380625 |
| 1.2 | 3.44 |
| 1.225 | 3.500625 |
| 1.25 | 3.5625 |
| 1.275 | 3.625625 |
| 1.3 | 3.69 |
| 1.325 | 3.755625 |
| 1.35 | 3.8225 |
| 1.375 | 3.890625 |
| 1.4 | 3.96 |
| 1.425 | 4.030625 |
| 1.45 | 4.1025 |
| 1.475 | 4.175625 |
| 1.5 | 4.25 |
| 1.525 | 4.325625 |
| 1.55 | 4.4025 |
| 1.575 | 4.480625 |
| 1.6 | 4.56 |
| 1.625 | 4.640625 |
| 1.65 | 4.7225 |
| 1.675 | 4.805625 |
| 1.7 | 4.89 |
| 1.725 | 4.975625 |
| 1.75 | 5.0625 |
| 1.775 | 5.150625 |
| 1.8 | 5.24 |
| 1.825 | 5.330625 |
| 1.85 | 5.4225 |
| 1.875 | 5.515625 |
| 1.9 | 5.61 |
| 1.925 | 5.705625 |
| 1.95 | 5.8025 |
| 1.975 | 5.900625 |
| 2 | 6 |
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 |
|---|---|---|
| Patch area | 2 | m² |
| Flux with positive-z normal | 6.666667 | N m²/C |
| Flux with reversed normal | -6.666667 | N m²/C |
| Area-averaged normal field | 3.333333 | N/C |
| Center-field times area approximation | 6 | N m²/C |
| Center approximation minus exact flux | -0.6666667 | N m²/C |
| Tangent-field contribution to flux | 0 | N m²/C |
Flux integrates the normal component over oriented area, not field magnitude times area.
The y integration contributes its 1 m width. Integrating 2+βx² from 0 to L gives 2L+βL³/3. Using the center value gives 2L+βL³/4, differing by −βL³/12; the midpoint shortcut happens to be exact for a linear profile, not this quadratic. Reversing the normal changes the sign of every contribution. The tangent component contributes zero regardless of its magnitude.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Flux integrates the normal component over oriented area, not field magnitude times area.
The y integration contributes its 1 m width. Integrating 2+βx² from 0 to L gives 2L+βL³/3. Using the center value gives 2L+βL³/4, differing by −βL³/12; the midpoint shortcut happens to be exact for a linear profile, not this quadratic. Reversing the normal changes the sign of every contribution. The tangent component contributes zero regardless of its magnitude.
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