Build a disk field from ring elements
A disk is not a collection of equally charged rings when its surface density is uniform. A ring of radius r and width dr has area 2πr dr, so dq=σ2πr dr. Omitting this radial Jacobian gives the wrong amount of charge and the wrong axial field.
Learning outcome: At fixed σ, compare increasing R while keeping z fixed. Retain the changing Q as well as the field. Then hold R fixed and compare positive z values; test whether point or plane approximation meets a stated relative-error tolerance.
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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For σ=1 nC/m², R=1 m and z=0.5 m, Q=π nC and Ez=18π(1−1/√5)≈31.259335 N/C. The infinite-plane comparison is 18π≈56.548668 N/C, substantially larger. A finite disk has less field than that infinite source at the same σ.
Open the experimentPredict
Which charge element is correct for a uniform disk decomposed into rings?
Experiment
At fixed σ, compare increasing R while keeping z fixed. Retain the changing Q as well as the field. Then hold R fixed and compare positive z values; test whether point or plane approximation meets a stated relative-error tolerance.
Build a disk field from area-weighted rings
Worked teaching model · synthetic values · Uniform prescribed surface charge density 1 nC/m² on a disk in vacuum. Surface density, not total charge, stays fixed as radius changes. Use K=9×10^9 and dQ=σ 2πr dr. Observation stays strictly off the ideal zero-thickness charged surface.
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Finite disk axial field
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: ring-integrated disk field (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Finite disk axial field
ring-integrated disk field
| axial distance (m) | field (N/C) |
|---|---|
| 0.1 | 50.92187 |
| 0.14875 | 48.2286 |
| 0.1975 | 45.59195 |
| 0.24625 | 43.02748 |
| 0.295 | 40.5485 |
| 0.34375 | 38.16584 |
| 0.3925 | 35.8878 |
| 0.44125 | 33.72017 |
| 0.49 | 31.66639 |
| 0.53875 | 29.72782 |
| 0.5875 | 27.90399 |
| 0.63625 | 26.19293 |
| 0.685 | 24.59146 |
| 0.73375 | 23.09549 |
| 0.7825 | 21.70026 |
| 0.83125 | 20.40059 |
| 0.88 | 19.19104 |
| 0.92875 | 18.06609 |
| 0.9775 | 17.02024 |
| 1.02625 | 16.0481 |
| 1.075 | 15.14448 |
| 1.12375 | 14.30441 |
| 1.1725 | 13.52319 |
| 1.22125 | 12.79638 |
| 1.27 | 12.11986 |
| 1.31875 | 11.48975 |
| 1.3675 | 10.90248 |
| 1.41625 | 10.35474 |
| 1.465 | 9.843481 |
| 1.51375 | 9.365877 |
| 1.5625 | 8.919342 |
| 1.61125 | 8.501496 |
| 1.66 | 8.110155 |
| 1.70875 | 7.743312 |
| 1.7575 | 7.399129 |
| 1.80625 | 7.075918 |
| 1.855 | 6.77213 |
| 1.90375 | 6.486347 |
| 1.9525 | 6.217263 |
| 2.00125 | 5.963684 |
| 2.05 | 5.72451 |
| 2.09875 | 5.49873 |
| 2.1475 | 5.285416 |
| 2.19625 | 5.083714 |
| 2.245 | 4.892836 |
| 2.29375 | 4.712059 |
| 2.3425 | 4.540713 |
| 2.39125 | 4.378181 |
| 2.44 | 4.223895 |
| 2.48875 | 4.077327 |
| 2.5375 | 3.937991 |
| 2.58625 | 3.805437 |
| 2.635 | 3.679245 |
| 2.68375 | 3.559031 |
| 2.7325 | 3.444434 |
| 2.78125 | 3.33512 |
| 2.83 | 3.230781 |
| 2.87875 | 3.131127 |
| 2.9275 | 3.03589 |
| 2.97625 | 2.94482 |
| 3.025 | 2.857684 |
| 3.07375 | 2.774264 |
| 3.1225 | 2.694358 |
| 3.17125 | 2.617775 |
| 3.22 | 2.544338 |
| 3.26875 | 2.473882 |
| 3.3175 | 2.40625 |
| 3.36625 | 2.341297 |
| 3.415 | 2.278886 |
| 3.46375 | 2.218889 |
| 3.5125 | 2.161186 |
| 3.56125 | 2.105663 |
| 3.61 | 2.052214 |
| 3.65875 | 2.000738 |
| 3.7075 | 1.951143 |
| 3.75625 | 1.903337 |
| 3.805 | 1.857239 |
| 3.85375 | 1.812769 |
| 3.9025 | 1.769852 |
| 3.95125 | 1.728419 |
| 4 | 1.688402 |
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 disk charge at fixed density | 3.141593 | nC |
| Finite disk axial field | 31.25933 | N/C |
| Infinite-plane comparison | 56.54867 | N/C |
| Infinite-plane relative overestimate | 0.809017 | ratio |
| Point-charge comparison | 113.0973 | N/C |
The area element contributes a radial factor, and the finite disk approaches different limiting models near and far.
Integrate the ring kernel K z dq/(r²+z²)^(3/2) with dq=2πσr dr to obtain 2πKσ[1−z/√(R²+z²)] for z>0. The implementation rationalizes this difference as R²/[h(h+z)] to reduce cancellation. Near the positive side it tends to 2πKσ; far away it tends to KQ/z². The ideal charged surface itself is outside this model.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The area element contributes a radial factor, and the finite disk approaches different limiting models near and far.
Integrate the ring kernel K z dq/(r²+z²)^(3/2) with dq=2πσr dr to obtain 2πKσ[1−z/√(R²+z²)] for z>0. The implementation rationalizes this difference as R²/[h(h+z)] to reduce cancellation. Near the positive side it tends to 2πKσ; far away it tends to KQ/z². The ideal charged surface itself is outside this model.
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