Explain a conductor’s zero field without erasing its potential

Establish the model and evidence

A conductor in electrostatic equilibrium cannot sustain an internal electric field: mobile charges would continue moving if a field remained. For a solid conductor with no cavity, excess charge resides on its surface. At the surface the external field is perpendicular; a tangential component would drive further surface charge motion. These statements concern equilibrium, not a conductor carrying a steady current.

Zero internal field implies no potential difference between two interior points, not necessarily zero potential relative to infinity. For an isolated spherical conductor of radius R and charge Q, V=kQ/R throughout the interior. Outside, spherical symmetry gives the same field and potential as a point charge at the center: radial E=kQ/r² and V=kQ/r. The algebraic sign of E specifies outward or inward direction.

The ideal surface is a boundary. Potential is continuous, but the normal electric field changes from zero just inside to a nonzero value just outside. The ledger explicitly uses the outside limit when the selected radius equals R. Do not apply the outside inverse-square law at small radii inside the metal, where it would invent a singular field that is not present.

An insulator does not obey the same free-charge equilibrium argument. Charge can remain in its interior, and its internal electric field may be nonzero; only qualitative interior reasoning is required here. Connecting two conductors allows redistribution until their potentials agree. Equal potential does not generally mean equal charge: differently sized isolated spheres can require different charges to reach the same potential.

Worked evidence

For Q=+2 nC and R=0.15 m, interior potential is 119.87 V relative to infinity, while internal field is zero. At r=0.30 m, V=59.93 V and the outward field is 199.78 N/C. At the surface, the outside limiting field is 799.11 N/C.

Predict and investigate

Can two locations have the same nonzero potential and zero potential difference between them?

Capture r=0.05, 0.10, 0.15 and 0.30 m for Q=2 nC. Identify the interior plateau, continuous potential boundary and selected outside field limit. Reverse Q and track signs without changing magnitude.

Transfer to changed conditions

Two widely separated conducting spheres of radii 0.10 and 0.20 m are connected by a thin wire, with total charge +6 nC. Neglect their mutual influence. Find final charges.

Compare the worked transfer

Equal potentials require Q₁/R₁=Q₂/R₂, so Q₂=2Q₁. Conservation gives Q₁=2 nC and Q₂=4 nC. The far-separation approximation and negligible wire charge are essential.

Explain the boundary

Which parts of this argument fail if the object is an insulator or if current is flowing?

Sources and authorship

Physical relationship reference. Current AP Physics 2 framework, checked 30 October 2026. These cases, diagrams and numerical records are original. All experiments are bounded browser calculations or synthetic data, not measured laboratory evidence. Do not construct or handle charged capacitors, high-voltage sources or particle-beam equipment for these activities.