Conductors, capacitors, and dielectrics

The source can respond

An electric field can move charge already present in a material. In a conductor, mobile carriers redistribute over macroscopic distances; in an insulating dielectric, local charge displacements can polarize the material without transporting charge freely from one electrode to the other. Both responses change the field, so the initial externally prescribed field need not be the final total field.

This lesson treats macroscopic stationary electrostatics with ideal conductors in equilibrium. Its concrete dielectric examples are linear, reversible, insulating materials with positive scalar permittivities. These are declared models, not claims that every metal, insulator or capacitor behaves ideally at arbitrary fields, frequencies or temperatures.

The previous potential lesson supplied a way to combine sources with boundary values. A conductor now supplies a physical boundary condition: charge moves until the ideal equilibrium condition is satisfied. A dielectric instead supplies a constitutive response that must be solved self-consistently with the field equations.

Keep each example’s constraints visible. A disconnected charged capacitor preserves its free electrode charge. A capacitor attached to an ideal voltage reservoir preserves its voltage while exchanging charge and energy with that reservoir. The same final capacitance can therefore accompany different fields, energies and measured work.

What equilibrium means in an ideal metal

In the ideal macroscopic electrostatic equilibrium model, the total electric field inside the conducting material is zero. A persistent bulk field would drive its mobile carriers and contradict that equilibrium. This statement concerns the sum of applied and induced fields, not the applied contribution by itself.

Since E = −∇V, each connected conducting component has one constant potential. Distinct conductors can be at different potentials. Surface charge need not be uniform; it adjusts to geometry, nearby sources and the imposed charge or voltage constraints. Sharp features often concentrate the field, but a universal quantitative enhancement factor cannot be inferred from shape alone.

A stationary Gaussian surface entirely within the ideal metal has zero electric flux. In a uniform homogeneous bulk this excludes net macroscopic volume charge there. Excess and induced charge are represented by surface distributions in this approximation. It does not mean that all conduction electrons have left the interior or that microscopic charge density vanishes atom by atom.

A real wire carrying a steady dissipative current is a different state: it generally needs a nonzero driving field. Finite-conductivity relaxation, contact and electrochemical effects require their own constitutive description. Zero electrostatic bulk field is neither a general statement about every metal in every state nor a definition of superconductivity.

A surface condition is a one-sided statement

At a smooth conductor-vacuum boundary in electrostatic equilibrium, the tangential field just outside is zero. Let n̂ point from metal into vacuum, and let σ be the signed total surface charge. The pillbox jump relation, with zero interior field, gives E_out = (σ/ε₀)n̂. Negative σ produces a field directed toward the metal.

The ideal surface charge is a sheet. Its two one-sided fields generally differ; the formula for E_out is not an assignment of a unique ordinary field at the sheet itself. Potential remains continuous for this ordinary surface-charge model, while its normal derivative can jump. Keep that distinction when plotting a conducting sphere.

The field that acts on a small element of surface charge excludes that element’s own singular self-field. In the vacuum sheet idealization, the appropriate average of the two one-sided fields is (E_out + E_in)/2. For an equilibrium conductor E_in = 0, this supplies the factor one-half in the normal electrical pressure.

Multiplying σ by the full exterior field would count a self-field contribution and double the vacuum pressure. Material stresses, finite microscopic layers and a dielectric in contact with a moving surface can require additional assumptions. The explicit pressure calculation later in this lesson uses vacuum to keep those assumptions separate.

Empty cavities, grounding and net charge

An empty closed cavity inside a connected conductor has a constant potential on its entire boundary. With no internal source, Laplace’s equation and the boundary-value uniqueness result admit the constant solution throughout the cavity. Hence its interior field is zero, even if charges outside the metal distort the outer-surface distribution.

This is an electrostatic result for an ideal closed conducting boundary. An aperture, an internal charge, finite-frequency fields or a material response outside the idealization changes the problem. A shielded cavity need not be at zero potential relative to infinity; a spatially constant cavity potential and a zero cavity field are compatible with a nonzero reference value.

Put a net charge q inside a cavity without touching the metal. A Gaussian surface in the metal enclosing that cavity still has zero flux, so the net induced charge on its inner wall is −q. If an isolated conductor’s total charge is Q_c, its outer surface must then carry Q_c + q. The distribution on either surface is not determined by this net-charge statement alone.

Grounding specifies potential relative to a large charge reservoir and permits charge exchange; it does not specify that the conductor’s net charge remains zero. For a grounded closed shell with no external sources and a compatible zero-at-infinity reference, the exterior can have zero field even with a charge in the cavity. An isolated shell with fixed total charge has a different exterior constraint.

The empty-cavity theorem is not a claim that an internal charge’s influence is undetectable in every exterior experiment. Outer charge, grounding, external sources and conductor geometry must be stated. Use uniqueness with the actual boundary conditions rather than treating the word shielding as a substitute for them.

Define capacitance with a reference and a geometry

For a linear two-conductor capacitor with fixed geometry and specified linear dielectric arrangement, put +Q on one electrode and −Q on the other. With no independent background source producing an additional charge offset, Q = CΔV defines the capacitance. Here ΔV is the potential of the positive electrode relative to the other; C has units farads, or C/V.

Capacitance describes how a complete electrostatic boundary problem stores charge for a voltage difference. It is not merely a property of one plate’s material or area. Nearby conductors, grounding, dielectric geometry and the reference electrode affect it. In this linear setting, doubling the voltage doubles Q while C remains unchanged.

An isolated conductor can instead have a capacitance relative to infinity when its reference problem permits that definition. This is different from the capacitance of a pair. If other fixed sources are present, the total electrode charge can contain a bias term as well as the part proportional to applied voltage; an incremental response must not be confused with the full Q/ΔV ratio.

For a nonlinear device, distinguish a ratio Q/ΔV from a differential capacitance dQ/d(ΔV) about a chosen operating state. Hysteresis can make the response path dependent as well. The simple constant-C formulas below belong to their stated linear reversible regime.

A conducting sphere is not a uniformly charged ball

Take an isolated conducting sphere of radius 2L₀ with total charge 3q₀, where q₀ and L₀ are positive scales. Define V₀ = kₑq₀/L₀, E₀ = V₀/L₀ and C_* = 4πε₀L₀ = q₀/V₀. Assume vacuum and V → 0 at infinity, with no nearby bodies that would break spherical symmetry.

All excess charge lies on the sphere’s surface in this model. For R = r/L₀ > 2, Gauss’s law gives E_r/E₀ = 3/R², and integration gives V/V₀ = 3/R. For R < 2, the potential is the constant 3/2 and the field is zero. Potential matches continuously at the surface, while the field has the one-sided limits 0 and 3E₀/4.

The signed surface charge density is σ = 3q₀/(16πL₀²), consistent with E_out = σ/ε₀. The capacitance relative to infinity is C = Q/V_surface = 2q₀/V₀ = 2C_*, or 4πε₀a for radius a = 2L₀.

A uniformly charged insulating solid ball has charge throughout its volume and a nonzero interior radial field except at the center. It can share an exterior inverse-square field with a conducting sphere having the same Q and radius, while having a different interior potential and energy. The identical exterior formula does not make the source models interchangeable.

Derive a concentric two-conductor capacitor

For a separate configuration, retain an inner conducting sphere of radius a = 2L₀ and place it concentrically inside a closed conducting shell whose inner radius is b = 5L₀. Give the inner conductor +3q₀ and the outer conductor −3q₀. With no other sources, the total charge is zero, the exterior field vanishes and the outer conductor may use potential zero relative to infinity.

In the vacuum gap a < r < b, the radial field is kₑQ/r² with Q = 3q₀. Therefore ΔV = ∫ from a to b E_r dr = kₑQ(1/a − 1/b). The capacitance is C = 4πε₀/(1/a − 1/b) = 4πε₀ab/(b − a). No field from the outer shell is added as a local radial contribution inside the gap, but its constant potential contribution is essential to the voltage reference.

For the selected radii, ΔV = (9/10)V₀ and C = (10/3)C_*. In the gap V/V₀ = 3/R − 3/5; inside the inner conductor it is 9/10, and in the outer conductor and exterior it is zero. The field’s one-sided values at R = 2 are 0 and 3/4 in units E₀; at R = 5 they are 3/25 and 0.

As b tends to infinity with a fixed, the pair capacitance tends to the isolated-sphere result 4πε₀a. As the vacuum gap becomes small compared with a, the expression approaches a parallel-plate form using the sphere’s local area. These are limits of a specified geometry, not permission to use one geometry’s formula for any two electrodes.