AP Physics C: E&M Unit 1 — Electrostatics, Gauss’s Law, and Electric Potential (Teach-From-Scratch Notes)

Electric charge

A fundamental property of matter that causes objects to experience electric forces; comes in positive and negative types.

Electrostatics

The study of electric charges at rest and the forces/fields they produce.

Like-charge repulsion

Rule that charges with the same sign (both + or both −) exert forces that push them apart.

Opposite-charge attraction

Rule that charges with opposite signs (+ and −) exert forces that pull them together.

Conservation of charge

In an isolated system, the net electric charge cannot change; charge can be transferred but not created/destroyed overall.

Quantization of charge

Charge occurs in discrete units of the elementary charge e; macroscopic totals Q are often treated as continuous.

Elementary charge (e)

The magnitude of the charge of a proton (and of an electron with opposite sign); the basic unit of charge.

Coulomb’s law

Magnitude of electrostatic force between two point charges: F = k|q1 q2|/r^2.

Coulomb constant (k)

k = 1/(4πϵ0) ≈ 8.99×10^9 N·m^2/C^2.

Permittivity of free space (ϵ0)

A fundamental constant appearing in electrostatics; relates electric fields/flux to charge (e.g., in Gauss’s law).

Inverse-square law

Dependence where a quantity (like Coulomb force or point-charge field) scales as 1/r^2 with distance r.

Unit vector (r-hat)

A vector of magnitude 1 used to specify direction; in Coulomb/field formulas it points from source charge to the field point.

Vector form of Coulomb force

Force on charge 2 due to charge 1: F⃗{2←1} = k(q1 q2/r^2) r̂{1→2}, directed along the line joining the charges.

Superposition principle (force)

Net electrostatic force is the vector sum of forces from each individual charge: F⃗net = Σ F⃗i.

Newton’s third law in electrostatics

For two interacting charges, the force on 1 due to 2 equals the force on 2 due to 1 in magnitude and is opposite in direction.

Electric field (E⃗)

A vector field defined as force per unit positive test charge: E⃗ = F⃗/q_test.

Positive test charge

A small charge used to define E⃗ without significantly disturbing the source charges; E⃗ points the way it would accelerate.

Force from a known electric field

Relationship between force and field for a charge q: F⃗ = qE⃗.

Electric field of a point charge

Magnitude E = k|q|/r^2; vector form E⃗ = k(q/r^2) r̂ (radially outward for q>0, inward for q<0).

Superposition principle (field)

Net electric field is the vector sum of the fields from each source: E⃗net = Σ E⃗i.

Electric field lines

A visualization where the tangent gives E⃗ direction, line density indicates relative magnitude, lines start on + and end on − (or infinity), and lines never cross.

Electric flux (Φ_E)

A scalar measure of how much electric field passes through a surface: Φ_E = ∫ E⃗·dA⃗.

Area vector (dA⃗)

A vector normal to a surface patch with magnitude dA; for closed surfaces it points outward by convention.

Dot product in flux

E⃗·dA⃗ = E dA cosθ; only the component of E⃗ perpendicular to the surface contributes to flux.

Gaussian surface

An imaginary closed surface used to apply Gauss’s law; chosen to exploit symmetry so the flux integral simplifies.

Gauss’s law

The net electric flux through any closed surface equals enclosed charge divided by ϵ0: ∮ E⃗·dA⃗ = Q_enc/ϵ0.

Enclosed charge (Q_enc)

The total charge inside a chosen closed (Gaussian) surface; only this charge determines net flux in Gauss’s law.

Spherical symmetry (Gauss use)

Symmetry where E⃗ is radial and depends only on distance r from a center, allowing E to be constant on a spherical Gaussian surface.

Cylindrical symmetry (Gauss use)

Symmetry around a line/axis where E⃗ is radial from the axis and depends only on distance r, enabling a cylindrical Gaussian surface.

Planar symmetry (Gauss use)

Symmetry of an infinite sheet where E⃗ is perpendicular to the plane and constant in magnitude on either side.

Field of an infinite line charge

For linear charge density λ: E = λ/(2πϵ0 r), directed radially outward for λ>0.

Field of an infinite sheet of charge

For surface charge density σ: E = σ/(2ϵ0), perpendicular to the sheet (away if σ>0, toward if σ<0).

Pillbox Gaussian surface

A short cylinder used with planar symmetry; flux passes mainly through the two flat faces, not the curved side.

Uniformly charged solid sphere (outside field)

For r ≥ R, the field equals that of a point charge Q at the center: E = (1/(4πϵ0))·Q/r^2.

Uniformly charged solid sphere (inside field)

For r < R with uniform volume density ρ: E = ρr/(3ϵ0) (linear in r).

Continuous charge distribution

A model where charge is spread along a line/surface/volume and calculations use integrals over charge elements dq.

Linear charge density (λ)

Charge per unit length: λ = dq/dl, so dq = λ dl.

Surface charge density (σ)

Charge per unit area: σ = dq/dA, so dq = σ dA.

Volume charge density (ρ)

Charge per unit volume: ρ = dq/dV, so dq = ρ dV.

Differential electric field element (dE⃗)

Contribution from a small charge element dq at distance r: dE⃗ = k(dq/r^2) r̂ (direction set by geometry and sign).

Component-cancellation by symmetry

Strategy where certain vector components sum to zero due to symmetric placement of charge, reducing the integral/calculation to remaining components.

Electrostatic equilibrium (conductor)

State in which charges in a conductor have finished moving; no net force drives further motion of free charges.

Zero field inside a conductor (equilibrium)

In electrostatic equilibrium, E⃗ = 0 everywhere within the conducting material; otherwise charges would move.

Excess charge on a conductor’s surface

In electrostatic equilibrium, any excess net charge resides on the surface of a conductor, not in its bulk.

Equipotential conductor

A conductor in electrostatic equilibrium has the same electric potential everywhere within it and on its surface.

Perpendicular field at conductor surface

Just outside a conductor in electrostatic equilibrium, E⃗ has no tangential component and is perpendicular to the surface.

Induced charge in a cavity

A charge placed inside a conductor’s cavity induces equal-magnitude opposite-sign charge on the inner surface to keep E⃗ = 0 in the conductor material (with corresponding outer-surface charge if needed).

Electric potential energy (U) for point charges

For two point charges separated by r (zero at infinity): U = k(q1 q2)/r; sign indicates repulsive (U>0) or bound/attractive (U<0) configuration.

Electric potential (V)

Potential energy per unit charge: V = U/q; relates to work via ΔV = −W_field/q and to energy via ΔU = qΔV.

Field–potential relationship

Potential difference is the negative line integral of field: V(B)−V(A) = −∫A^B E⃗·dl⃗; in 1D, Ex = −dV/dx (conceptually E⃗ = −∇V).