Separate a conducting material from its cavity

Establish the equilibrium mechanism

Mobile carriers in an ideal conductor rearrange when a field acts on them. In the electrostatic equilibrium approximation, this rapid relaxation is treated as negligible compared with the observation interval. A remaining tangential surface field or a field within the metal would drive further motion. Thus the metal has E = 0 and one constant potential; the exterior field is normal to its surface. This statement is about conducting material, not every empty region enclosed by it.

Use two separate charge accounts

A central source q sits in a spherical cavity of radius 1 m, surrounded by a shell with outer radius 2 m and net charge Q. A Gaussian sphere wholly in the metal has zero flux, so the inner surface carries −q. Conservation of the isolated conductor charge then gives outer charge Q + q. Excess negative charge corresponds to excess surface electrons; positive charge corresponds to an electron deficit, not mobile protons crossing the metal.

Match fields and potential

Inside the cavity away from the central point, E = kq/r². In the metal, E = 0. Outside the shell, E = k(Q + q)/r². The conductor potential relative to infinity is k(Q + q)/(2 m). The cavity potential is kq[1/r − 1/(1 m)] plus this constant. Potential is continuous at the surfaces while the normal field can jump. Separate plotted branches show one-sided field limits, not a smooth interpolation through a surface.

Test shielding without erasing an internal source

An empty closed cavity with no internal charge is shielded from static external fields in this ideal equilibrium model. If a source is inside the cavity, shielding does not delete its cavity field. Moving that source off-center changes local induced surface density, while the total inner induced charge remains −q. An external field can polarize a neutral conductor by redistributing its surface electrons without changing its net charge.

Worked original case

With q = +1 nC and Q = 0, the inner surface has −1 nC and the outer +1 nC. The metal field is zero, yet the cavity field at r = 0.5 m is 35.9502 N/C. The conductor potential is 4.49378 V, not zero merely because its field is zero.

Investigation

Compare q = −1, 0 and +1 nC at fixed shell charge. Retain inner/outer inventories and three spatial regions. Then set Q = −q and explain how exterior zero field can coexist with a nonzero cavity field. Sketch a noncentral-source alternative and identify which totals survive without claiming a spherical local field.

Changed-case practice

Sources and boundaries

Original cases checked against the official framework on 30 October 2026, with OpenStax verification on 31 October 2026 against the current official framework, Unit 10 and OpenStax capacitance. These are virtual supplied models, not instructions to charge, ground, wire or handle real equipment. Public answers are practice; retained plans and self-reports do not establish independent mastery, actual laboratory participation or AP authorization.