Conductors in Electrostatic Equilibrium (AP Physics C: E&M Unit 2 Notes)

Conductor

Material with mobile charge carriers (typically electrons) that can move freely through the material.

Electrostatic Equilibrium (for a conductor)

State where mobile charges have finished moving; charge distribution is time-independent, there is no net motion of charge (no current), and fields are steady.

Zero Electric Field in a Conductor (bulk)

In electrostatic equilibrium, the electric field everywhere within the conducting material is (\mathbf{E}=\mathbf{0}); otherwise charges would accelerate and produce current.

Equipotential Conductor

In electrostatic equilibrium, the entire conductor (bulk and surface) is at a single electric potential value (no potential difference between any two points on it).

Potential–Field Relation

(\Delta V=-\int \mathbf{E}\cdot d\mathbf{l}); if (\mathbf{E}=0) along a path, then (\Delta V=0).

Excess (Net) Charge on a Conductor

Any added free charge on a conductor in electrostatic equilibrium resides on the surface(s), not in the bulk.

Surface Charge Density ((\sigma))

Charge per unit area on a surface: (\sigma=\frac{dQ}{dA}) (units: C/m(^2)).

Tangential Electric Field at a Conductor Surface

In electrostatic equilibrium, the tangential component must be zero: (E_{\text{tangent}}=0); otherwise surface charges would move along the surface.

Normal (Perpendicular) Electric Field at a Conductor Surface

Just outside a conductor in electrostatic equilibrium, (\mathbf{E}) points perpendicular (normal) to the surface.

Gauss’s Law (Electric Flux Law)

(\PhiE=\oint \mathbf{E}\cdot d\mathbf{A}=\frac{Q{\text{enc}}}{\epsilon_0}).

Gaussian Pillbox (surface boundary method)

A tiny Gaussian surface that straddles a conductor surface, used to relate the field just outside to the enclosed surface charge.

Field Just Outside a Conductor (Boundary Condition)

For a conductor surface in electrostatic equilibrium: (E{\perp}=\frac{\sigma}{\epsilon0}), where (E_{\perp}) is the normal field immediately outside.

Permittivity of Free Space ((\epsilon_0))

Constant in electrostatics appearing in Gauss’s law and boundary conditions; units (\text{C}^2/(\text{N}\cdot\text{m}^2)).

Electrostatic Shielding

Property of conductors in electrostatic equilibrium where external electrostatic influences do not produce an electric field within the conductor material; can protect interior regions.

Empty Cavity in a Conductor (no charge inside)

A closed cavity containing no charge can have zero electric field in electrostatics; the conductor enforces an equipotential boundary leading to constant potential in the cavity.

Charge in a Cavity (isolated from conductor)

If a charge (q) is placed inside a cavity without touching the conductor, induced charge appears on the cavity wall to keep (\mathbf{E}=0) in the conductor material.

Induced Charge on Inner Cavity Surface

Total induced charge on the inner surface of a cavity equals minus the charge inside: (Q_{\text{inner surface}}=-q) (from Gauss’s law with (\mathbf{E}=0) in the metal).

Outer Surface Charge with Cavity Charge

If the conductor’s net charge is (Q{\text{net}}) and a cavity contains charge (q), then (Q{\text{outer surface}}=Q_{\text{net}}+q).

Grounded Conductor

A conductor connected to Earth (a large charge reservoir) so charges can flow on/off until its potential is fixed, typically modeled as (V=0).

Isolated Conductor

A conductor not connected to ground; its total charge is fixed (cannot change by charge flow to Earth).

Charged Conducting Sphere: Electric Field Outside

For a conducting sphere of radius (R) with net charge (Q): for (r>R), (E(r)=\frac{1}{4\pi\epsilon_0}\frac{Q}{r^2}) (radial).

Charged Conducting Sphere: Electric Field Inside (metal)

For (r<R) within the conducting material, (E=0) in electrostatic equilibrium.

Charged Conducting Sphere: Electric Potential

With (V(\infty)=0): for (r\ge R), (V(r)=\frac{1}{4\pi\epsilon0}\frac{Q}{r}); for (r\le R), (V=\frac{1}{4\pi\epsilon0}\frac{Q}{R}) (constant inside).

Uniform Surface Charge on an Isolated Conducting Sphere

By spherical symmetry, (\sigma) is constant: (\sigma=\frac{Q}{4\pi R^2}).

Charge Crowding (Curvature Effect)

Surface charge density is larger on sharper regions (smaller radius of curvature); larger (\sigma) implies larger local (E{\perp}) via (E{\perp}=\sigma/\epsilon_0).