AP Physics C: E&M Unit 3: Conductors and Capacitors Notes

Introduction

Electrostatic equilibrium fixes the field and potential of conductors, while capacitance describes how geometry and materials relate separated charge to voltage and energy.

Mobile charge in a conductor rearranges until the electric field inside the conducting material is zero. That one condition leads to an equipotential conductor, surface charge, and a field immediately outside that is perpendicular to the surface.

A capacitor uses two conductors to maintain separated charge. Geometry sets its capacitance; the electrical connection determines whether charge or voltage stays fixed when the geometry or dielectric changes.

Mobile charge at rest

zero field in conductor material

Zero internal field

constant conductor potential

Geometry and material

capacitance

Charge and voltage

stored field energy

Battery connection

fixed-voltage or fixed-charge constraint

Current exam guidance: Aligned to the current 2026–27 AP Physics C: Electricity and Magnetism framework, with calculus-based field, circuit, and induction models, explicit assumptions, and experimental reasoning.

10.1

Electrostatics with Conductors

What must be true inside and immediately outside a conductor in electrostatic equilibrium?

Short answer

The field is zero within the conducting material, the connected conductor is equipotential, excess charge lies on surfaces, and the exterior field is normal to the surface.

Know firstElectric field · Electric potential · Gauss’s law

Equilibrium removes the tangential field

If a field remained inside a conductor, mobile charges would accelerate. They therefore redistribute until $\vec E=0$ throughout the material. Because $\Delta V=-\int\vec E\cdot d\vec\ell$ , every point in one connected conductor has the same potential. That potential need not be zero.

A tiny Gaussian pillbox across a surface gives $E_n=\sigma/\epsilon_0$ just outside in vacuum, while the tangential component must vanish. An empty enclosed cavity has zero field when no charge is inside and the conductor is in electrostatic equilibrium. A charge placed in the cavity induces charge on its inner wall, so that simpler conclusion no longer applies.

Physics model

Free charge rearranges until a conductor’s interior field is zero

Increase the applied field across an ideal, very wide conducting slab. In this planar approximation, uniform induced surface charge creates a field that cancels the applied field inside.

Applied field4 kN/C

Applied field: 4 kN/C
Planar surface density: ±35.416 nC/m²
Interior field: 0 N/C

applied fieldnegative surface chargepositive surface chargeconducting slab

\vec E_{inside}=0,\qquad \sigma=\epsilon_0E_{outside,n}

Read this figure without the graphic

An applied field points toward an ideal wide conducting slab. Uniform opposite surface charges appear on its planar faces while the slab interior is marked as zero field.

Current quantity	Value
Applied field	4 kN/C
Planar surface density	±35.416 nC/m²
Interior field	0 N/C
Model relation	$\vec E_{inside}=0,\qquad \sigma=\epsilon_0E_{outside,n}$

Large historic electrostatic generator with glass disks and conducting terminals in the Teylers Museum. — A conductor reaches equilibrium by redistributing mobile charge
The machine transfers charge to conducting terminals whose mobile charges redistribute until the conductor reaches electrostatic equilibrium. Shape then affects the surface distribution even when the conductor is at one potential.
Use this visual to see
Connect electrostatic equilibrium and surface-charge redistribution to a large conducting apparatus.
What it cannot show
The photograph shows glass disks, supports, and rounded conducting terminals but cannot establish zero interior field, equal potential, surface-charge density, field strength, or charge sign.

Read a charged spherical shell by region

System: A concentric conducting spherical shell with empty cavity, outer radius $R$ , and net charge $Q$ .
Frame: Origin at the common center.
Assumptions: Electrostatic equilibrium, no external field, vacuum outside, and spherical symmetry.
Givens: The cavity contains no charge.

Use the conductor equilibrium condition in the metal.
Apply Gauss’s law to a spherical surface outside the shell.
Integrate the exterior field from infinity to obtain potential.
Use continuity of potential at $R$ and zero field inward to extend the constant value through the shell and empty cavity.

Result

\begin{aligned}E&=0,\quad V=\frac{k_eQ}{R}&&\text{in the empty cavity and metal},\\ E&=\frac{k_eQ}{r^2},\quad V=\frac{k_eQ}{r}&&\text{for }r>R\end{aligned}

Units:

E

in N/C and

V

in V.

Boundary cases

The electric field jumps by $\sigma/\epsilon_0$ at the charged outer surface.
An off-center charge in the cavity destroys the simple zero-field cavity result.

Field is zero through the empty cavity and metal, then falls as one over radius squared outside; potential stays flat inside and falls as one over radius outside.

Keep field and potential distinct

Region	Electric field	Potential
empty cavity	$0$	constant
conducting material	$0$	same constant
just outside surface	normal, $E_n=\sigma/\epsilon_0$	continuous
far exterior	set by all source charge	approaches chosen reference

Analyze a conductor configuration

Mark conductor material, cavities, and exterior as separate regions.
Apply $\vec E=0$ only inside conductor material at equilibrium.
Use Gauss’s law where symmetry supports it.
Enforce constant conductor potential and continuous potential at boundaries.
Add induced inner-surface charge if a cavity contains charge.

Keep these three ideas

Conducting material has zero internal field at equilibrium.
A connected conductor is equipotential, not necessarily at zero volts.
Surface charge creates a normal field discontinuity.

Conductors and Capacitors