AP Physics C: E&M — Conductors and Capacitors (Unit 2) Deep Study Notes

Conductor

Material in which electric charges (typically electrons) can move freely through the material.

Electrostatic equilibrium

Steady state in a conductor where charges are no longer flowing; charge distribution has settled so no net motion occurs.

Zero electric field inside a conductor (equilibrium rule)

In electrostatic equilibrium, the electric field within the bulk conducting material is zero; otherwise charges would keep moving.

Excess charge resides on the surface (equilibrium rule)

Any net excess charge on a conductor must lie on its surface because the field inside the conductor is zero and Gauss’s law then implies zero enclosed net charge for interior Gaussian surfaces.

Equipotential conductor (equilibrium rule)

A conductor at electrostatic equilibrium has the same electric potential everywhere in the conducting material (and across its surface).

Perpendicular electric field at a conductor surface

At electrostatic equilibrium, the electric field just outside a conductor has no tangential component; it points normal (perpendicular) to the surface.

Tangential electric field component (why it must be zero on a conductor)

A field component parallel to the conductor surface would exert a force on surface charges and cause them to move, contradicting equilibrium.

Surface charge density (σ)

Charge per unit area on a surface: σ = dQ/dA (often treated as Q/A when uniform).

Permittivity of free space (ε0)

Constant relating electric fields to charge in vacuum; appears in Gauss’s law and capacitor formulas.

Conductor boundary condition: E⊥,outside = σ/ε0

For a conductor in electrostatic equilibrium, the perpendicular component of the electric field just outside the surface equals σ/ε0; inside the conductor E = 0.

Pillbox Gaussian surface argument

Using a thin cylindrical Gaussian surface straddling a surface to relate discontinuity in the normal electric field to surface charge density.

Charge concentration at sharp points

On conductors, regions with small radius of curvature (sharp edges/points) tend to have larger σ, producing larger electric fields locally.

Corona discharge

Ionization of air near a conductor caused by strong local electric fields, often near sharp points.

Lightning rod principle

A pointed conductor can produce large local electric fields (via high σ), promoting charge leakage/ionization and reducing lightning risk.

Field of a charged conducting sphere (outside)

For r > R, a charged conducting sphere produces E = (1/4π ε0)·(Q/r^2), as if all charge were at the center.

Field inside a charged conductor (sphere example)

For r < R inside the conducting material, E = 0 even if the conductor has net charge +Q.

Infinite nonconducting sheet field magnitude

For an infinite sheet of charge in space (nonconductor), the field magnitude on each side is σ/(2ε0).

Conductor vs sheet factor-of-2 distinction

At a conductor surface, Ejust outside = σ/ε0 because the field inside the conductor must be zero; this differs from σ/(2ε0) for an isolated infinite sheet.

Charging by contact (conduction)

Charging process where a charged object touches a conductor and charge transfers until potentials equalize (net charge of the isolated system is conserved).

Charging by induction

Charging without contact: an external charge causes charge redistribution; with grounding and then removing the ground, the conductor can end with net charge.

Polarization (of a conductor during induction)

Rearrangement of free charges in a conductor caused by a nearby external charge, producing separation of positive and negative regions.

Grounding

Connecting a conductor to Earth so charge can flow to/from a vast reservoir, effectively constraining the conductor’s potential (often taken as V = 0 reference).

Earth as charge reservoir

Model where Earth can supply/absorb charge with negligible change in its own potential due to its enormous size.

Induction with grounding sequence (typical result)

Bring external charge near → ground conductor (charge flows) → remove ground while external charge remains → remove external charge; conductor retains net charge opposite the external charge’s sign.

Electrostatic shielding

Effect where a conductor’s free charges rearrange to cancel electric fields within the conducting material, preventing static external fields from penetrating into protected regions.

Faraday cage

Conducting enclosure that provides electrostatic shielding, protecting its interior from external static electric fields (under electrostatic equilibrium assumptions).

Conductor with empty closed cavity (no internal charge)

At equilibrium, an empty closed cavity inside a conductor has no net charge induced on its inner surface and the electric field in the cavity is zero (electrostatic shielding).

Conductor cavity with internal point charge q

A point charge q inside a cavity induces total charge −q on the inner cavity surface to ensure E = 0 in the conductor’s bulk.

Outer surface charge when cavity contains charge

If the conductor’s net charge is Q and a cavity contains charge q, then the outer surface must carry total charge Q + q.

Gaussian surface in conductor material (cavity argument)

A Gaussian surface drawn within the conductor’s material has E = 0 everywhere, so it encloses net charge zero; used to determine induced charges on cavity walls.

Net induced charge vs distribution (cavity subtlety)

The total induced charge on a cavity surface is fixed (e.g., −q), but its distribution is generally nonuniform unless high symmetry applies (e.g., centered charge in spherical cavity).

Capacitor

Two conductors separated by an insulator (or vacuum) that store separated charge and electric field energy.

Capacitance (C)

Configuration property defined by C = Q/ΔV, relating stored charge magnitude to potential difference.

Farad (F)

Unit of capacitance: 1 F = 1 C/V.

Capacitance depends on geometry and dielectric

For ideal capacitors, C depends on conductor shapes/separation and the material permittivity between them, not on the particular values of Q or ΔV.

Parallel-plate capacitor (vacuum)

For large plates of area A separated by distance d with negligible fringing, C = ε0 A/d.

Parallel-plate field between plates (ideal)

With charges ±Q on plates of area A, E ≈ σ/ε0 = Q/(ε0 A) between the plates.

Isolated conducting sphere capacitance

Treating infinity as the other conductor, a sphere of radius R has C = 4π ε0 R.

Spherical capacitor (concentric spheres)

Two concentric conducting spheres radii a (inner) and b (outer) have C = 4π ε0 ab/(b − a) (vacuum).

Cylindrical (coaxial) capacitor

Two long coaxial cylinders (length L, radii a and b) have C = (2π ε0 L)/ln(b/a) when L ≫ b.

Capacitors in parallel (constraint)

Parallel capacitors share the same potential difference: ΔV1 = ΔV2 = ΔV.

Equivalent capacitance in parallel

For capacitors in parallel, Ceq = C1 + C2 + … because charges add at the same ΔV.

Capacitors in series (constraint)

Series capacitors carry the same charge magnitude on each: Q1 = Q2 = Q, due to charge conservation on the intermediate conductor.

Equivalent capacitance in series

For capacitors in series, 1/Ceq = 1/C1 + 1/C2 + … because voltages add at the same Q.

Energy stored in a capacitor (U)

Energy in the electric field of a capacitor: U = (1/2)C(ΔV)^2 = (1/2)QΔV = Q^2/(2C).

Energy density of an electric field (u)

Energy per volume in an electric field: u = (1/2) ε0 E^2 (useful for uniform-field regions like parallel plates).

Dielectric

Insulating material whose charges cannot flow freely but can shift slightly, allowing polarization when an electric field is applied.

Polarization (of a dielectric)

Small separation of positive and negative charge within atoms/molecules in response to an applied electric field, producing bound surface charges.

Dielectric constant (κ) / relative permittivity

Factor by which a dielectric increases capacitance when fully filling the region: C = κ C0.

Dielectric insertion: battery connected vs disconnected

With battery connected (fixed ΔV): C increases, Q increases (Q = CΔV), and for parallel plates E = ΔV/d stays the same. With battery disconnected (fixed Q): C increases, so ΔV = Q/C decreases, E decreases, and U = Q^2/(2C) decreases.