Conductors and electrostatic boundary conditions

A conductor contains charges that can move. At electrostatic equilibrium, with no sustained current or changing sources, a nonzero bulk E would accelerate mobile charges and contradict the assumed equilibrium. In the ideal conductor model the field within the conducting material vanishes:

A cavity cut into the conductor is not conducting material: an enclosed charge can produce a nonzero field inside the cavity even though E=0 in the surrounding metal. A real conductor carrying a steady current may sustain nonzero bulk E to drive J; do not apply this equilibrium statement to arbitrary DC circuits.

Place any closed Gaussian surface entirely within the conducting material. E=0 there, so its flux is zero and its enclosed NET charge must vanish. Any excess free charge of an ideal conductor at equilibrium therefore resides on its accessible surfaces rather than as a bulk excess density. When cavities contain charges, free charge can occupy inner as well as outer surfaces.

For a neutral isolated conductor enclosing a +q point charge in a cavity, a Gaussian surface embedded in the metal around that cavity requires total −q induced on the cavity wall; overall conductor neutrality then requires +q on the outer surface, assuming no other free charges. These are net charges on the respective surfaces, not claims of uniform local density.

At a locally smooth interface let n̂ point from medium 1 to medium 2. A closed pillbox straddles the interface, with top and bottom face area A and vanishing thickness. Bounded side fields give vanishing side flux. The top outward normal is +n̂, the bottom is −n̂, so Gauss's law becomes

Here σtotal is ALL microscopic charge localized in that sheet, including bound polarization charge if present; it is not automatically identical to a separately specified free sheet charge. The fields are one-sided limiting values, not a field assigned unambiguously on an ideal zero-thickness sheet.

Take a narrow rectangular loop crossing the interface, with long sides tangent to it and vanishing height. In electrostatics ∮E·dℓ=0. The short-side contributions vanish for bounded fields, leaving a difference of tangent components on the two sides:

For an ordinary finite sheet of charge without an ideal dipole layer, the potential itself is continuous across the interface even when its normal derivative jumps. These statements concern static fields; induction from a varying magnetic flux requires reexamining the circulation argument.

Inside a conductor at equilibrium E₁=0. For the vacuum just outside let medium 2 be vacuum and n̂ point OUT of metal. Tangential continuity forces Eout,t=0, while the pillbox jump with conductor surface charge σ yields

This is NOT the field σ/(2ε₀) on each side of an isolated nonconducting charged sheet: the conductor has zero bulk field on one side and its charge may rearrange to enforce equilibrium. A negative σ produces an inward external field relative to n̂.

In matter define polarization P (electric dipole moment per unit volume) and displacement D=ε₀E+P. Gauss's law for D counts free charge, not all polarization-bound charge. Across a static interface from material 1 to material 2:

Only with a local linear isotropic constitutive model may one additionally set Dᵢ=εᵢEᵢ. For zero free sheet charge this implies ε₂E₂n=ε₁E₁n; the normal E component need NOT be continuous if ε changes. Tangential E remains continuous even when the normal component jumps.

Consider an isolated perfectly conducting sphere of radius R with total charge Q, no cavity charge and no external sources. Rotational symmetry makes the equilibrium surface density σ=Q/(4πR²). E=0 in the metal; outside Gauss's law gives the point-charge field. With V(∞)=0:

The potential is continuous at R, while the normal field jumps across its charge-bearing surface. The exterior formula and uniform σ rely on spherical symmetry and an isolated sphere; a nearby external charge redistributes σ and destroys that simple field.

Let a planar conductor–vacuum interface carry σ=+2.00 nC/m² and be at electrostatic equilibrium. The outside normal field is Eout=σ/ε₀≈+2.26×10² N/C. An isolated uniform two-sided nonconducting sheet with the SAME specified σ has field magnitude σ/(2ε₀)≈1.13×10² N/C on each side. These are different source/boundary problems.

For two dielectrics with ε₂=4ε₁, no free sheet charge and E₁n=120 V/m, the normal D continuity gives E₂n=30 V/m (same chosen normal). Both media have the SAME tangential E at the boundary, but this tangential value need not be zero.

1. Declare static equilibrium, geometry, normal direction n̂ and which media lie on each side.
2. Ask whether the region is bulk metal, vacuum, a cavity or a dielectric, and whether conduction current is present.
3. For a conductor at equilibrium set its bulk E and internal potential gradient to zero, NOT the field in an empty cavity.
4. Apply the pillbox normal jump to total ε₀E charge or to free D charge, without mixing the definitions.
5. Apply the electrostatic tangential continuity and any declared constitutive laws.
6. Check one-sided values, field direction for negative charge, units, and limiting cases.

A metal carrying stationary transport current generally needs a finite internal electric field to sustain the current in a resistive description; “DC” alone is not equivalent to electrostatic equilibrium.

1. For an equilibrium conductor with σ=+2.00 nC/m² bordering vacuum, determine exterior normal and tangential E. Repeat the direction for negative σ.
2. For the same surface density, state the field on each side of an isolated infinite nonconducting sheet and explain the factor of two.
3. A neutral isolated conductor has one cavity enclosing +5.00 nC. Find the net inner-surface and outer-surface charges.
4. An isolated conducting sphere has Q=+1.00 nC, R=0.100 m. Find its σ, surface potential and immediately exterior E.
5. At a dielectric interface n̂ points from 1 to 2, σfree=0, ε₂=4ε₁, E₁n=120 V/m and E₁t=50 V/m. Find E₂n and E₂t.
6. Explain why E=0 within an equilibrium conductor cannot be imposed on a resistor carrying a nonzero steady current.

Chapter 05 turns these field and potential results into capacitance and stored electrostatic energy, while separating ideal static dielectric formulas from frequency-dispersive device physics.