Audit source quadrature against an analytic benchmark
Approximating a continuous source by cells is a numerical method, not a new physical charge measurement. For the uniform rod −1≤s≤1 m, divide the length into N equal cells of width Δs=2/N and place each cell charge λΔs at its midpoint. Sum the signed vector kernel rather than adding magnitudes.
Learning outcome: At y=1 m, compare N=4,8,16,32 and record signed errors against the exact integral. Repeat at y=0.1 m before asserting a convergence rate. State which discrepancy comes from discretization and which hypothetical independent observation could challenge the source distribution itself.
Starting recap
Review signed vector components, definite integrals, radians, density units and elementary derivatives. Distinguish observer coordinates from source integration coordinates.
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At y=1 m the exact field is 18/√2≈12.727922 N/C. For N=4, midpoint samples are at −0.75,−0.25,0.25,0.75 m with Δs=0.5 m. Their weighted north contributions sum to approximately 12.826 N/C; evaluate the displayed grid rather than replacing it by the exact integral. Symmetric east terms cancel analytically.
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Agreement between N and 2N midpoint calculations alone establishes:
Experiment
At y=1 m, compare N=4,8,16,32 and record signed errors against the exact integral. Repeat at y=0.1 m before asserting a convergence rate. State which discrepancy comes from discretization and which hypothetical independent observation could challenge the source distribution itself.
Test spatial midpoint quadrature against an analytic field
Worked teaching model · synthetic values · Uniform rod −1≤s≤1 m with λ=1 nC/m; observation at (0,y). Composite midpoint rule approximates the signed field components. The exact integral is available independently. N and 2N discrepancy diagnoses discretization, not calibration or statistical confidence.
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Source-kernel resolution near the observer projection
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: analytic field integrand (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Source-kernel resolution near the observer projection
analytic field integrand
| source coordinate (m) | north kernel (N/C/m) |
|---|---|
| -1 | 1.697159 |
| -0.975 | 1.825612 |
| -0.95 | 1.967209 |
| -0.925 | 2.123651 |
| -0.9 | 2.296911 |
| -0.875 | 2.489276 |
| -0.85 | 2.703415 |
| -0.825 | 2.942446 |
| -0.8 | 3.21003 |
| -0.775 | 3.510481 |
| -0.75 | 3.848903 |
| -0.725 | 4.231356 |
| -0.7 | 4.665076 |
| -0.675 | 5.158729 |
| -0.65 | 5.722752 |
| -0.625 | 6.369768 |
| -0.6 | 7.115125 |
| -0.575 | 7.977567 |
| -0.55 | 8.98011 |
| -0.525 | 10.15115 |
| -0.5 | 11.52592 |
| -0.475 | 13.14828 |
| -0.45 | 15.07318 |
| -0.425 | 17.36962 |
| -0.4 | 20.12461 |
| -0.375 | 23.448 |
| -0.35 | 27.47846 |
| -0.325 | 32.39057 |
| -0.3 | 38.40232 |
| -0.275 | 45.78201 |
| -0.25 | 54.85127 |
| -0.225 | 65.9782 |
| -0.2 | 79.54951 |
| -0.175 | 95.90358 |
| -0.15 | 115.2 |
| -0.125 | 137.2042 |
| -0.1 | 160.9969 |
| -0.075 | 184.7006 |
| -0.05 | 205.4419 |
| -0.025 | 219.8277 |
| 0 | 225 |
| 0.025 | 219.8277 |
| 0.05 | 205.4419 |
| 0.075 | 184.7006 |
| 0.1 | 160.9969 |
| 0.125 | 137.2042 |
| 0.15 | 115.2 |
| 0.175 | 95.90358 |
| 0.2 | 79.54951 |
| 0.225 | 65.9782 |
| 0.25 | 54.85127 |
| 0.275 | 45.78201 |
| 0.3 | 38.40232 |
| 0.325 | 32.39057 |
| 0.35 | 27.47846 |
| 0.375 | 23.448 |
| 0.4 | 20.12461 |
| 0.425 | 17.36962 |
| 0.45 | 15.07318 |
| 0.475 | 13.14828 |
| 0.5 | 11.52592 |
| 0.525 | 10.15115 |
| 0.55 | 8.98011 |
| 0.575 | 7.977567 |
| 0.6 | 7.115125 |
| 0.625 | 6.369768 |
| 0.65 | 5.722752 |
| 0.675 | 5.158729 |
| 0.7 | 4.665076 |
| 0.725 | 4.231356 |
| 0.75 | 3.848903 |
| 0.775 | 3.510481 |
| 0.8 | 3.21003 |
| 0.825 | 2.942446 |
| 0.85 | 2.703415 |
| 0.875 | 2.489276 |
| 0.9 | 2.296911 |
| 0.925 | 2.123651 |
| 0.95 | 1.967209 |
| 0.975 | 1.825612 |
| 1 | 1.697159 |
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 |
|---|---|---|
| Coarse midpoint east field | -2.66454e-15 | N/C |
| Coarse midpoint north field | 84.75563 | N/C |
| Doubled-grid midpoint north field | 88.22668 | N/C |
| Analytic north field | 88.25226 | N/C |
| Coarse signed integration error | -3.496631 | N/C |
| Fine signed integration error | -0.02558548 | N/C |
| Fine minus coarse discrepancy | 3.471045 | N/C |
| Cell width divided by observer distance | 1.25 | ratio |
A sharply varying near-source kernel can defeat a coarse spatial discretization even when the total charge is correct.
Each cell contributes K λ Δs (−s,y)/(s²+y²)^(3/2), evaluated at its midpoint. Paired horizontal terms cancel analytically, though floating summation may leave tiny residue. Compare N and 2N with the independent exact integral. A small mesh discrepancy alone is not a guaranteed error bound, and it says nothing about sensor calibration or a wrong source model.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
A sharply varying near-source kernel can defeat a coarse spatial discretization even when the total charge is correct.
Each cell contributes K λ Δs (−s,y)/(s²+y²)^(3/2), evaluated at its midpoint. Paired horizontal terms cancel analytically, though floating summation may leave tiny residue. Compare N and 2N with the independent exact integral. A small mesh discrepancy alone is not a guaranteed error bound, and it says nothing about sensor calibration or a wrong source model.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
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Which geometry, calibration or source assumption could fail while an exact arithmetic fit remains possible?
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