Electric Charges, Fields, and Gauss’s Law

What this unit covers

The topics below follow the published Physics C: E&M course framework for Unit 1. This unit is worth 17–23% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Electric Charge & Coulomb’s Law12 min · 3 objectivesDescribe charge, its quantization, and conservation · Apply Coulomb’s law to find the force between point charges · Add electric forces as vectors using superposition
• The Electric Field12 min · 3 objectivesDefine the electric field and relate it to the force on a test charge · Compute the field of one or more point charges by superposition · Interpret electric field lines and the field of a dipole
• Fields from Continuous Charge Distributions15 min · 3 objectivesSet up field integrals using charge density (λ, σ, ρ) and dq · Exploit symmetry to cancel components before integrating · Derive the field of a charged ring and an infinite line of charge
• Gauss’s Law14 min · 3 objectivesDefine electric flux and compute it through a surface · State Gauss’s law and explain why flux depends only on enclosed charge · Use Gaussian surfaces to find fields of symmetric distributions

Formulas in Unit 1

Every term in Unit 1

All 22 terms we publish for Electric Charges, Fields, and Gauss’s Law, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Gauss's law

∮E·dA = q_enc/ε₀. Always true, but only useful when symmetry makes E constant and parallel or perpendicular over the surface.

Field of a continuous charge distribution

E = ∫k dq/r², integrating over the distribution. Express dq using linear, surface or volume charge density before integrating.

Coulomb's law in vector form

F = kq₁q₂r̂/r², directed along the line joining the charges. Superposition means total force is the vector sum of the individual pair forces.

Electric field of a point charge

E = kq/r², away from positive and toward negative charge. Field is force per unit positive test charge.

Charge densities

λ = charge per length, σ = charge per area, ρ = charge per volume. Choosing the right one sets up the integral correctly.

Field on the axis of a charged ring

E = kQx/(x² + R²)^(3/2). Zero at the center by symmetry, and it approaches a point-charge field far away.

Electric flux

Φ = ∫E·dA. Only the field component perpendicular to the surface contributes, since the dot product removes the parallel part.

Choosing a Gaussian surface

Sphere for point or spherical symmetry, cylinder for line or cylindrical symmetry, pillbox for a plane. The wrong choice makes the integral intractable.

Field of an infinite line of charge

E = λ/2πε₀r, falling as 1/r rather than 1/r² because the source extends infinitely in one dimension.

Field of an infinite plane of charge

E = σ/2ε₀, independent of distance — the field does not weaken as you move away from an infinite sheet.

Field inside a conductor in electrostatic equilibrium

Zero. Any internal field would drive charge until it canceled, so all excess charge resides on the surface.

Field inside a uniformly charged insulating sphere

E = kQr/R³, rising linearly with r inside because only the enclosed charge contributes, then falling as 1/r² outside.

Gauss's law vs direct integration

Use Gauss when the charge distribution has spherical, cylindrical or planar symmetry; integrate dE directly when it does not.

Why symmetry is required for Gauss

The law is always true, but E only comes out of the integral when it is constant in magnitude and fixed in orientation over the surface.

Flux through a closed surface with no enclosed charge

Zero, however strong the external field, because every field line entering also leaves.

Superposition of fields

Total field is the vector sum of contributions. Resolve into components before adding; magnitudes never add directly unless collinear.

Symmetry arguments to eliminate components

For a charged ring on its axis, the perpendicular components cancel in pairs, leaving only the axial component to integrate.

Field of a finite line of charge

Requires integrating dE = k dq/r² with dq = λ dx and resolving components. Reduces to λ/2πε₀r in the infinite limit.

Conductor with a cavity

Charge q inside a cavity induces −q on the cavity wall and +q on the outer surface, keeping the conductor's interior field zero.

Shielding

A conducting shell blocks external fields from its interior because charge redistributes to cancel them — the principle of the Faraday cage.

Charge on a conductor surface

Concentrates where curvature is greatest, so the field is strongest at sharp points. This is why lightning rods are pointed.

Checking a field expression

Far from any bounded distribution, the field must approach kQ_total/r². If yours does not, the integration is wrong.

What examiners penalize here

• On free-response problems, always resolve each Coulomb force into components before summing. Only add magnitudes directly when every force lies along the same line, as it did above. Otherwise sum Fₓ and Fᵧ separately, then recombine.
• A continuous-distribution FRP earns its points in the setup: state dq in terms of a density, draw dE and identify which component survives by symmetry, write correct integration limits, and only then integrate. Show the symmetry argument explicitly — graders reward it.
• For a Gaussian-surface FRP: (1) name the symmetry and pick a matching surface, (2) argue E is constant over the part carrying flux, (3) write ∮ E·dA = E·A, (4) find Q_enc — using ρ, σ, or λ times the enclosed volume/area/length — and (5) solve E·A = Q_enc/ε₀. Missing the Q_enc step is the usual lost point.

Practice Physics C: E&M

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 6 units of AP Physics C: E & M

1. Unit 1 · Electric Charges, Fields, and Gauss’s Law
2. Unit 2 · Electric Potential
3. Unit 3 · Conductors, Capacitors, and Dielectrics
4. Unit 4 · Electric Circuits
5. Unit 5 · Magnetic Fields
6. Unit 6 · Electromagnetic Induction