Compare direct shell integration with Gauss symmetry
A spherical shell provides a useful independent benchmark because its field can be found both by a direct source integral and by symmetry with Gauss law. The original shell has radius R=1 m and total charge Q=2 nC. Observers lie on the positive z axis, strictly away from the charged surface. The model does not assign an on-surface self-field.
Learning outcome: Predict the exact field at each allowed observer before evaluating the midpoint integral. Compare 20,40,100 cells at z=0.5,0.75,1.25,2 m. Retain signed errors and explain why simply increasing cells does not validate an omitted surface-charge or external-field assumption.
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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At observer z=0.5 m the exact field is zero despite nonzero individual shell contributions. At z=2 m it is 18/4=4.5 N/C. At the center, opposite source contributions cancel exactly analytically. Comparing 20 and 100 latitude cells near z=0.75 m tests numerical resolution without changing the physical shell.
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A coarse direct quadrature reports a small nonzero field inside the stipulated uniform shell. What should be checked first?
Experiment
Predict the exact field at each allowed observer before evaluating the midpoint integral. Compare 20,40,100 cells at z=0.5,0.75,1.25,2 m. Retain signed errors and explain why simply increasing cells does not validate an omitted surface-charge or external-field assumption.
Compare direct shell quadrature with a symmetry result
Worked teaching model · synthetic values · Uniform thin spherical shell radius1 m and charge2 nC, observed on positive z axis away from the charged surface. Integrate over u=cos(theta) with dq=Q du/2. Midpoint quadrature is compared with an independent spherical-symmetry result. On-surface self-field is excluded, not assigned an averaged value.
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Latitude-ring contribution after the area substitution
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: signed source contribution (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Latitude-ring contribution after the area substitution
signed source contribution
| u = cos(theta) | field integrand (N/C) |
|---|---|
| -1 | 2.938776 |
| -0.975 | 2.950826 |
| -0.95 | 2.962986 |
| -0.925 | 2.975256 |
| -0.9 | 2.987635 |
| -0.875 | 3.000123 |
| -0.85 | 3.012719 |
| -0.825 | 3.025422 |
| -0.8 | 3.038231 |
| -0.775 | 3.051143 |
| -0.75 | 3.064157 |
| -0.725 | 3.077271 |
| -0.7 | 3.090481 |
| -0.675 | 3.103784 |
| -0.65 | 3.117177 |
| -0.625 | 3.130655 |
| -0.6 | 3.144213 |
| -0.575 | 3.157845 |
| -0.55 | 3.171544 |
| -0.525 | 3.185304 |
| -0.5 | 3.199115 |
| -0.475 | 3.212969 |
| -0.45 | 3.226853 |
| -0.425 | 3.240756 |
| -0.4 | 3.254664 |
| -0.375 | 3.268561 |
| -0.35 | 3.28243 |
| -0.325 | 3.29625 |
| -0.3 | 3.31 |
| -0.275 | 3.323653 |
| -0.25 | 3.337182 |
| -0.225 | 3.350555 |
| -0.2 | 3.363734 |
| -0.175 | 3.37668 |
| -0.15 | 3.389346 |
| -0.125 | 3.40168 |
| -0.1 | 3.413623 |
| -0.075 | 3.425108 |
| -0.05 | 3.436058 |
| -0.025 | 3.446389 |
| 0 | 3.456 |
| 0.025 | 3.464781 |
| 0.05 | 3.472603 |
| 0.075 | 3.479319 |
| 0.1 | 3.484762 |
| 0.125 | 3.488739 |
| 0.15 | 3.491026 |
| 0.175 | 3.491366 |
| 0.2 | 3.489459 |
| 0.225 | 3.484959 |
| 0.25 | 3.47746 |
| 0.275 | 3.466485 |
| 0.3 | 3.451477 |
| 0.325 | 3.431775 |
| 0.35 | 3.406594 |
| 0.375 | 3.375 |
| 0.4 | 3.335873 |
| 0.425 | 3.28786 |
| 0.45 | 3.229321 |
| 0.475 | 3.158252 |
| 0.5 | 3.072186 |
| 0.525 | 2.968059 |
| 0.55 | 2.842035 |
| 0.575 | 2.689264 |
| 0.6 | 2.503542 |
| 0.625 | 2.27684 |
| 0.65 | 1.998622 |
| 0.675 | 1.654854 |
| 0.7 | 1.226512 |
| 0.725 | 0.6872933 |
| 0.75 | 0 |
| 0.775 | -0.8893906 |
| 0.8 | -2.061819 |
| 0.825 | -3.643164 |
| 0.85 | -5.838291 |
| 0.875 | -9 |
| 0.9 | -13.78146 |
| 0.925 | -21.51411 |
| 0.95 | -35.30356 |
| 0.975 | -64.03612 |
| 1 | -144 |
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 |
|---|---|---|
| Actual axial observer coordinate | 0.75 | m |
| Midpoint direct-integral field | 0.4661142 | N/C |
| Independent symmetry field | 0 | N/C |
| Signed quadrature error | 0.4661142 | N/C |
| Cosine-coordinate cell width | 0.05 | ratio |
Interior cancellation is exact for the full shell but need not be exact on a coarse quadrature grid.
A latitude strip has dq=Q du/2 after the spherical area Jacobian is included. Its axial contribution is KQ(z−Ru)du/[2(R²+z²−2Rzu)^(3/2)]. The independent spherical-symmetry solution is zero inside and KQ/z² outside. Near-surface integrands vary rapidly, so refining a source grid matters. The charged surface itself is excluded; quadrature residue must not be promoted to a physical violation of Gauss law.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Interior cancellation is exact for the full shell but need not be exact on a coarse quadrature grid.
A latitude strip has dq=Q du/2 after the spherical area Jacobian is included. Its axial contribution is KQ(z−Ru)du/[2(R²+z²−2Rzu)^(3/2)]. The independent spherical-symmetry solution is zero inside and KQ/z² outside. Near-surface integrands vary rapidly, so refining a source grid matters. The charged surface itself is excluded; quadrature residue must not be promoted to a physical violation of Gauss law.
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