Integrate enclosed charge in a nonuniform sphere
A spherical Gaussian surface simplifies the field integral only when the complete charge distribution and boundary conditions have spherical symmetry. In this original insulating sphere, radial density is ρ0(1+αr²/R²), radius R=1 m and total charge Q=3 nC. No external field or conducting rearrangement is included.
Learning outcome: Compare α=0,1,2,3 at radii 0,0.5,1,2 m. Retain density normalization, enclosed charge and field separately. Identify which measurements distinguish density shape and which are identical because they depend only on total charge.
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 uniform density α=0, the enclosed charge at r=0.5 m is 3(0.5³)=0.375 nC and E=13.5 N/C. For α=2 with the same total Q, the enclosed charge there becomes 39/176≈0.221591 nC and E=351/44≈7.977273 N/C. Both models give 27 N/C at r=1 m and 6.75 N/C at r=2 m.
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Changing the radial density shape at fixed total charge leaves which field unchanged here?
Experiment
Compare α=0,1,2,3 at radii 0,0.5,1,2 m. Retain density normalization, enclosed charge and field separately. Identify which measurements distinguish density shape and which are identical because they depend only on total charge.
Normalize and integrate enclosed spherical charge
Worked teaching model · synthetic values · Insulating sphere radius1 m, fixed total charge3 nC and radial volume density rho(r)=rho0(1+αr²/R²) inside, zero outside. No external field. Exact spherical symmetry permits a radial Gaussian field. At center the continuous field is exactly zero; no division by zero is evaluated.
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Normalized volume-source field through its boundary
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: piecewise Gaussian field (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Normalized volume-source field through its boundary
piecewise Gaussian field
| radius (m) | radial field (N/C) |
|---|---|
| 0 | 0 |
| 0.025 | 0.3070483 |
| 0.05 | 0.6154773 |
| 0.075 | 0.9266676 |
| 0.1 | 1.242 |
| 0.125 | 1.562855 |
| 0.15 | 1.890614 |
| 0.175 | 2.226656 |
| 0.2 | 2.572364 |
| 0.225 | 2.929116 |
| 0.25 | 3.298295 |
| 0.275 | 3.681281 |
| 0.3 | 4.079455 |
| 0.325 | 4.494196 |
| 0.35 | 4.926886 |
| 0.375 | 5.378906 |
| 0.4 | 5.851636 |
| 0.425 | 6.346457 |
| 0.45 | 6.86475 |
| 0.475 | 7.407895 |
| 0.5 | 7.977273 |
| 0.525 | 8.574264 |
| 0.55 | 9.20025 |
| 0.575 | 9.856611 |
| 0.6 | 10.54473 |
| 0.625 | 11.26598 |
| 0.65 | 12.02175 |
| 0.675 | 12.81342 |
| 0.7 | 13.64236 |
| 0.725 | 14.50997 |
| 0.75 | 15.41761 |
| 0.775 | 16.36668 |
| 0.8 | 17.35855 |
| 0.825 | 18.39459 |
| 0.85 | 19.4762 |
| 0.875 | 20.60476 |
| 0.9 | 21.78164 |
| 0.925 | 23.00822 |
| 0.95 | 24.28589 |
| 0.975 | 25.61602 |
| 1 | 27 |
| 1.025 | 25.69899 |
| 1.05 | 24.4898 |
| 1.075 | 23.36398 |
| 1.1 | 22.31405 |
| 1.125 | 21.33333 |
| 1.15 | 20.41588 |
| 1.175 | 19.55636 |
| 1.2 | 18.75 |
| 1.225 | 17.9925 |
| 1.25 | 17.28 |
| 1.275 | 16.609 |
| 1.3 | 15.97633 |
| 1.325 | 15.37914 |
| 1.35 | 14.81481 |
| 1.375 | 14.28099 |
| 1.4 | 13.77551 |
| 1.425 | 13.2964 |
| 1.45 | 12.84185 |
| 1.475 | 12.41023 |
| 1.5 | 12 |
| 1.525 | 11.60978 |
| 1.55 | 11.23829 |
| 1.575 | 10.88435 |
| 1.6 | 10.54687 |
| 1.625 | 10.22485 |
| 1.65 | 9.917355 |
| 1.675 | 9.623524 |
| 1.7 | 9.342561 |
| 1.725 | 9.073724 |
| 1.75 | 8.816327 |
| 1.775 | 8.569728 |
| 1.8 | 8.333333 |
| 1.825 | 8.106587 |
| 1.85 | 7.88897 |
| 1.875 | 7.68 |
| 1.9 | 7.479224 |
| 1.925 | 7.28622 |
| 1.95 | 7.100592 |
| 1.975 | 6.921968 |
| 2 | 6.75 |
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 |
|---|---|---|
| Density normalization | 0.3255442 | nC/m³ |
| Enclosed charge | 0.2215909 | nC |
| Radial field | 7.977273 | N/C |
| Boundary field at radius 1 | 27 | N/C |
| Center field | 0 | N/C |
| Total sphere charge | 3 | nC |
Normalize the full volume source, then integrate only the charge inside the selected Gaussian sphere.
Q=4πrho0 R³(1/3+α/5). For r≤R, Qenc=4πrho0[r³/3+αr⁵/(5R²)], and E=KQenc/r² by full spherical symmetry. At r=0 the limiting field is zero. No surface sheet is present, so the field is continuous at R even though volume density jumps; outside, only total Q enters. A thin shell with the same Q has a different interior field.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Normalize the full volume source, then integrate only the charge inside the selected Gaussian sphere.
Q=4πrho0 R³(1/3+α/5). For r≤R, Qenc=4πrho0[r³/3+αr⁵/(5R²)], and E=KQenc/r² by full spherical symmetry. At r=0 the limiting field is zero. No surface sheet is present, so the field is continuous at R even though volume density jumps; outside, only total Q enters. A thin shell with the same Q has a different interior field.
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