Separate conductor equilibrium from grounding
A connected conductor in electrostatic equilibrium has zero field inside its material and one potential throughout that material. Neither statement says that a cavity contains zero field when charge lies inside it. This original concentric shell has inner radius 1 m and outer radius 2 m, with a point charge q at its center and isolated shell net charge Q.
Learning outcome: Compare isolated Q=−1,0,+1 nC with fixed central q=1 nC. Retain inner charge, outer charge, metal potential and cavity field separately. Explain which observations change and why replacing isolation by grounding is a new physical constraint.
Starting recap
Review charge conservation, conductor equipotential conditions, definite field integrals and energy derivatives under a stated electrical constraint. Convert capacitance and charge units before interpreting small energies.
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For q=1 nC and Q=0, the inner and outer charges are −1 and +1 nC. The metal potential is 4.5 V, its field is zero, and the cavity field at r=0.5 m is 36 N/C outward. Cavity potential there is 13.5 V; exterior potential at r=3 m is 3 V. The grounded alternative would have net shell charge −1 nC and zero exterior field.
Open the experimentPredict
For q=1 nC and an isolated neutral shell, which surface-charge pair is required?
Experiment
Compare isolated Q=−1,0,+1 nC with fixed central q=1 nC. Retain inner charge, outer charge, metal potential and cavity field separately. Explain which observations change and why replacing isolation by grounding is a new physical constraint.
Reconcile cavity charge and a conductor potential
Worked teaching model · synthetic values · Concentric conducting spherical shell inner radius1 m, outer radius2 m, central point charge q, isolated shell net charge Q. Electrostatic equilibrium in vacuum, no external sources. Potential zero at infinity. The charge location is central, so induced surface densities are uniform; no off-center inference. Rounded K=9×10⁹.
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Continuous potential through cavity and conductor
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: isolated concentric configuration (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Continuous potential through cavity and conductor
isolated concentric configuration
| radius (m) | potential with infinity reference (V) |
|---|---|
| 0.25 | 31.5 |
| 0.284375 | 27.14835 |
| 0.31875 | 23.73529 |
| 0.353125 | 20.98673 |
| 0.3875 | 18.72581 |
| 0.421875 | 16.83333 |
| 0.45625 | 15.22603 |
| 0.490625 | 13.84395 |
| 0.525 | 12.64286 |
| 0.559375 | 11.58939 |
| 0.59375 | 10.65789 |
| 0.628125 | 9.828358 |
| 0.6625 | 9.084906 |
| 0.696875 | 8.414798 |
| 0.73125 | 7.807692 |
| 0.765625 | 7.255102 |
| 0.8 | 6.75 |
| 0.834375 | 6.286517 |
| 0.86875 | 5.859712 |
| 0.903125 | 5.465398 |
| 0.9375 | 5.1 |
| 0.971875 | 4.76045 |
| 1.00625 | 4.5 |
| 1.040625 | 4.5 |
| 1.075 | 4.5 |
| 1.109375 | 4.5 |
| 1.14375 | 4.5 |
| 1.178125 | 4.5 |
| 1.2125 | 4.5 |
| 1.246875 | 4.5 |
| 1.28125 | 4.5 |
| 1.315625 | 4.5 |
| 1.35 | 4.5 |
| 1.384375 | 4.5 |
| 1.41875 | 4.5 |
| 1.453125 | 4.5 |
| 1.4875 | 4.5 |
| 1.521875 | 4.5 |
| 1.55625 | 4.5 |
| 1.590625 | 4.5 |
| 1.625 | 4.5 |
| 1.659375 | 4.5 |
| 1.69375 | 4.5 |
| 1.728125 | 4.5 |
| 1.7625 | 4.5 |
| 1.796875 | 4.5 |
| 1.83125 | 4.5 |
| 1.865625 | 4.5 |
| 1.9 | 4.5 |
| 1.934375 | 4.5 |
| 1.96875 | 4.5 |
| 2.003125 | 4.49298 |
| 2.0375 | 4.417178 |
| 2.071875 | 4.343891 |
| 2.10625 | 4.272997 |
| 2.140625 | 4.20438 |
| 2.175 | 4.137931 |
| 2.209375 | 4.07355 |
| 2.24375 | 4.011142 |
| 2.278125 | 3.950617 |
| 2.3125 | 3.891892 |
| 2.346875 | 3.834887 |
| 2.38125 | 3.779528 |
| 2.415625 | 3.725744 |
| 2.45 | 3.673469 |
| 2.484375 | 3.622642 |
| 2.51875 | 3.573201 |
| 2.553125 | 3.525092 |
| 2.5875 | 3.478261 |
| 2.621875 | 3.432658 |
| 2.65625 | 3.388235 |
| 2.690625 | 3.344948 |
| 2.725 | 3.302752 |
| 2.759375 | 3.261608 |
| 2.79375 | 3.221477 |
| 2.828125 | 3.18232 |
| 2.8625 | 3.144105 |
| 2.896875 | 3.106796 |
| 2.93125 | 3.070362 |
| 2.965625 | 3.034773 |
| 3 | 3 |
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 |
|---|---|---|
| Inner surface charge | -1 | nC |
| Outer surface charge | 1 | nC |
| Conductor potential | 4.5 | V |
| Cavity field at radius .5 | 36 | N/C |
| Field in conductor material | 0 | N/C |
| Exterior field at radius 3 | 1 | N/C |
| Potential at cavity radius .5 | 13.5 | V |
| Potential at exterior radius 3 | 3 | V |
| Net charge required if instead grounded | -1 | nC |
Zero field in the conductor fixes its inner surface charge, while isolation fixes the remaining outer charge.
A Gaussian surface inside the metal encloses q+Qinner=0, so Qinner=−q. Conservation of isolated shell charge gives Qouter=Q+q. The entire conductor has potential K(Q+q)/2; inside the cavity add Kq(1/r−1), and outside use K(Q+q)/r. If instead grounded with no external sources, outer charge is zero and shell net charge becomes−q; that is a changed boundary condition, not the same isolated state.
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Zero field in the conductor fixes its inner surface charge, while isolation fixes the remaining outer charge.
A Gaussian surface inside the metal encloses q+Qinner=0, so Qinner=−q. Conservation of isolated shell charge gives Qouter=Q+q. The entire conductor has potential K(Q+q)/2; inside the cavity add Kq(1/r−1), and outside use K(Q+q)/r. If instead grounded with no external sources, outer charge is zero and shell net charge becomes−q; that is a changed boundary condition, not the same isolated state.
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