Choose the correct branch of enclosed cylindrical charge
For an infinite cylindrical source, symmetry makes the field radial and constant on a coaxial cylindrical side. The end caps have normals along the axis, perpendicular to the field, so they contribute no flux. This reasoning depends on the infinite-source idealization and absence of an external field; a finite tube near an end need not behave this way.
Learning outcome: Compare observations below the core radius, inside the material and outside it. Derive both boundary values independently and check continuity. Explain which new premise would be needed before applying the zero-core conclusion to a finite charged tube.
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 core radius a=0.5 m, the field is zero for r≤0.5 m. At r=1 m, λenc=0.75π nC/m and E=13.5π N/C. At the outer boundary r=2 m, E=33.75π N/C. At r=3 m it is 22.5π N/C, consistent with an exterior inverse-radius law.
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Why is the exterior field proportional to 1/r in this model?
Experiment
Compare observations below the core radius, inside the material and outside it. Derive both boundary values independently and check continuity. Explain which new premise would be needed before applying the zero-core conclusion to a finite charged tube.
Use the correct enclosed charge in a hollow volume source
Worked teaching model · synthetic values · Infinite prescribed cylindrical volume source with rho=1 nC/m³ between core radius a and outer radius2 m, zero elsewhere. No external fields, axial/end effects or conducting redistribution. Cylinder has infinite total charge but finite charge per unit length; this is not a finite apparatus.
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Hollow-core, charged-annulus and exterior branches
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: infinite-source symmetry field (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Hollow-core, charged-annulus and exterior branches
infinite-source symmetry field
| radius (m) | radial field (N/C) |
|---|---|
| 0 | 0 |
| 0.0375 | 0 |
| 0.075 | 0 |
| 0.1125 | 0 |
| 0.15 | 0 |
| 0.1875 | 0 |
| 0.225 | 0 |
| 0.2625 | 0 |
| 0.3 | 0 |
| 0.3375 | 0 |
| 0.375 | 0 |
| 0.4125 | 0 |
| 0.45 | 0 |
| 0.4875 | 0 |
| 0.525 | 2.760114 |
| 0.5625 | 6.675884 |
| 0.6 | 10.36726 |
| 0.6375 | 13.87383 |
| 0.675 | 17.2264 |
| 0.7125 | 20.44929 |
| 0.75 | 23.56194 |
| 0.7875 | 26.58012 |
| 0.825 | 29.51669 |
| 0.8625 | 32.38231 |
| 0.9 | 35.18584 |
| 0.9375 | 37.93473 |
| 0.975 | 40.63529 |
| 1.0125 | 43.29289 |
| 1.05 | 45.91213 |
| 1.0875 | 48.49698 |
| 1.125 | 51.05088 |
| 1.1625 | 53.57682 |
| 1.2 | 56.07743 |
| 1.2375 | 58.555 |
| 1.275 | 61.01158 |
| 1.3125 | 63.44895 |
| 1.35 | 65.86873 |
| 1.3875 | 68.27233 |
| 1.425 | 70.66103 |
| 1.4625 | 73.03599 |
| 1.5 | 75.39822 |
| 1.5375 | 77.74867 |
| 1.575 | 80.08817 |
| 1.6125 | 82.41749 |
| 1.65 | 84.73732 |
| 1.6875 | 87.0483 |
| 1.725 | 89.35099 |
| 1.7625 | 91.64594 |
| 1.8 | 93.93362 |
| 1.8375 | 96.21448 |
| 1.875 | 98.48893 |
| 1.9125 | 100.7573 |
| 1.95 | 103.0201 |
| 1.9875 | 105.2774 |
| 2.025 | 104.7198 |
| 2.0625 | 102.8158 |
| 2.1 | 100.9798 |
| 2.1375 | 99.20819 |
| 2.175 | 97.4977 |
| 2.2125 | 95.8452 |
| 2.25 | 94.24778 |
| 2.2875 | 92.70273 |
| 2.325 | 91.20753 |
| 2.3625 | 89.75979 |
| 2.4 | 88.35729 |
| 2.4375 | 86.99795 |
| 2.475 | 85.6798 |
| 2.5125 | 84.401 |
| 2.55 | 83.15981 |
| 2.5875 | 81.95459 |
| 2.625 | 80.78381 |
| 2.6625 | 79.64601 |
| 2.7 | 78.53982 |
| 2.7375 | 77.46393 |
| 2.775 | 76.41712 |
| 2.8125 | 75.39822 |
| 2.85 | 74.40614 |
| 2.8875 | 73.43983 |
| 2.925 | 72.49829 |
| 2.9625 | 71.58059 |
| 3 | 70.68583 |
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 |
|---|---|---|
| Enclosed line charge | 2.356194 | nC/m |
| Radial electric field | 42.4115 | N/C |
| Total source line charge | 11.78097 | nC/m |
| Inner boundary field | 0 | N/C |
| Outer boundary field | 106.0288 | N/C |
A Gaussian cylinder counts the annular volume enclosed, not all charge in the source at every radius.
For r≤a, no charge is enclosed and cylindrical symmetry gives E=0. For a<r<2, line charge is rhoπ(r²−a²); outside it is rhoπ(4−a²). Flux through a coaxial length ell is E2πr ell, with zero end-cap flux, so E=2Klambdaenc/r. The exterior decays as 1/r, not the 1/r² of a finite localized source. The zero-core result depends on the full infinite-source symmetry and no external field.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
A Gaussian cylinder counts the annular volume enclosed, not all charge in the source at every radius.
For r≤a, no charge is enclosed and cylindrical symmetry gives E=0. For a<r<2, line charge is rhoπ(r²−a²); outside it is rhoπ(4−a²). Flux through a coaxial length ell is E2πr ell, with zero end-cap flux, so E=2Klambdaenc/r. The exterior decays as 1/r, not the 1/r² of a finite localized source. The zero-core result depends on the full infinite-source symmetry and no external field.
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