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AP Physics C: E&M Formula Sheet — Complete Equation Reference (2026)
Updated July 2026 · AP Physics C: Electricity & Magnetism · Calculus-based
AP Physics C: Electricity & Magnetism is built on four fundamental laws — Gauss's Law, Ampere's Law, Faraday's Law, and the Maxwell correction — collectively known as Maxwell's Equations. Here is every formula you need, with context on when and how to use each one.
Formula sheet provided: College Board provides an equation sheet for AP Physics C: E&M covering the major laws and constants. The challenge is not memorizing formulas — it's knowing which one applies and how to set up the integral or differential equation.
Constants You Must Know
| Constant | Symbol | Value |
|---|
| Coulomb's constant | k = 1/(4πε₀) | 8.99×10⁹ N·m²/C² |
| Permittivity of free space | ε₀ | 8.85×10⁻¹² C²/(N·m²) |
| Permeability of free space | μ₀ | 4π×10⁻⁷ T·m/A |
| Elementary charge | e | 1.60×10⁻¹⁹ C |
| Speed of light | c | 3.00×10⁸ m/s = 1/√(μ₀ε₀) |
| Electron mass | m_e | 9.11×10⁻³¹ kg |
Electrostatics
Coulomb's Law & Electric Field
| Equation | Variables | Notes |
|---|
| F = kq₁q₂/r² = q₁q₂/(4πε₀r²) | F = force, r = separation | Attractive if charges opposite |
| E = F/q = kQ/r² | E = electric field, Q = source charge | Field of a point charge |
| E⃗ = −∇V | V = electric potential | Field points from high V to low V |
| E = −dV/dx | 1-D form | Use when V is given as function of x |
| F = qE⃗ | Force on charge q in field E | |
Gauss's Law
| Equation | Notes |
|---|
| ∮E⃗·dA⃗ = Q_enc/ε₀ | Total flux through closed surface = enclosed charge / ε₀ |
| Φ_E = EA cosθ (uniform field) | θ between field and surface normal |
Gauss's Law results for symmetric charge distributions:
| Geometry | E field (outside, r > R) | E field (inside, r < R) |
|---|
| Point charge / conducting sphere | kQ/r² | 0 (inside conductor) |
| Uniformly charged solid sphere | kQ/r² | kQr/R³ (linear) |
| Infinite line charge (λ C/m) | λ/(2πε₀r) | — |
| Infinite plane (σ C/m²) | σ/(2ε₀) per side | — |
| Parallel plates (capacitor) | 0 outside | σ/ε₀ between plates |
Electric Potential
| Equation | Variables | Notes |
|---|
| V = kQ/r | Potential of point charge | Scalar — add without direction |
| V = −∫E⃗·dl⃗ | General definition | Choose path from reference (∞) to point |
| ΔV = V_B − V_A = −∫_A^B E⃗·dl⃗ | | |
| W = q(V_A − V_B) = −ΔU | Work by electric force | Moving from A to B |
| U = qV = kq₁q₂/r | U = potential energy | U = 0 at infinity |
Capacitors
| Equation | Variables | Notes |
|---|
| C = Q/V | C = capacitance (Farads) | Definition |
| C = ε₀A/d | A = plate area, d = separation | Parallel-plate capacitor (no dielectric) |
| C = κε₀A/d | κ = dielectric constant | With dielectric |
| U = ½QV = ½CV² = Q²/(2C) | Stored energy | All three forms equivalent |
| Series: 1/C_eq = Σ(1/Cᵢ) | | Smaller result than smallest C |
| Parallel: C_eq = ΣCᵢ | | Larger result |
| u = ½ε₀E² | u = energy density (J/m³) | Energy stored per unit volume |
DC Circuits
Ohm's Law & Resistance
| Equation | Variables | Notes |
|---|
| V = IR | V = voltage, I = current, R = resistance | Ohm's Law |
| R = ρL/A | ρ = resistivity, L = length, A = area | Resistance of a wire |
| P = IV = I²R = V²/R | P = power dissipated | |
| Series: R_eq = ΣRᵢ | | Current same through all |
| Parallel: 1/R_eq = Σ(1/Rᵢ) | | Voltage same across all |
Kirchhoff's Laws
| Law | Statement |
|---|
| KCL (Junction) | ΣI_in = ΣI_out — conservation of charge |
| KVL (Loop) | Σ(ΔV) = 0 around any closed loop — conservation of energy |
RC Circuits
| Equation | Notes |
|---|
| τ = RC | Time constant (seconds) |
| Charging: Q(t) = Cε(1 − e^(−t/τ)) | Q → Cε as t → ∞ |
| Charging: I(t) = (ε/R)e^(−t/τ) | Current starts at ε/R, decays to 0 |
| Discharging: Q(t) = Q₀e^(−t/τ) | Q → 0 as t → ∞ |
| Discharging: I(t) = I₀e^(−t/τ) | |
After 1τ: capacitor is 63% charged. After 5τ: 99.3% — considered fully charged. These time intervals appear directly in FRQ scoring.
Magnetostatics
Magnetic Force
| Equation | Variables | Notes |
|---|
| F⃗ = qv⃗ × B⃗ | q = charge, v = velocity, B = field | |F| = qvB sinθ; zero if v‖B |
| F⃗ = IL⃗ × B⃗ | I = current, L = length vector | Force on current-carrying wire |
| r = mv/(qB) | Radius of circular orbit | Magnetic force provides centripetal force |
| ω_c = qB/m | Cyclotron frequency | Independent of speed |
Biot-Savart Law & Ampere's Law
| Equation | Notes |
|---|
| dB⃗ = (μ₀/4π)(I dl⃗ × r̂)/r² | Biot-Savart Law: field from current element |
| B = μ₀I/(2πr) | Infinite straight wire at perpendicular distance r |
| B = μ₀nI | Solenoid (n = turns/length); uniform inside, ~0 outside |
| B = μ₀NI/(2πr) | Toroid at radius r inside |
| ∮B⃗·dl⃗ = μ₀I_enc | Ampere's Law (magnetostatics form) |
Magnetic Flux & Dipoles
| Equation | Notes |
|---|
| Φ_B = ∫B⃗·dA⃗ = BA cosθ (uniform) | Magnetic flux through a surface |
| μ = NIA | Magnetic dipole moment (N turns, area A) |
| τ = μ × B = μB sinθ | Torque on magnetic dipole |
Electromagnetic Induction
Faraday's & Lenz's Laws
| Equation | Notes |
|---|
| EMF = −dΦ_B/dt | Faraday's Law — negative sign = Lenz's Law |
| EMF = −N dΦ_B/dt | N-turn coil |
| EMF = BLv | Motional EMF: rod of length L moving at v ⊥ B |
Lenz's Law: The induced current flows in the direction that opposes the change in flux that caused it. Use this to determine current direction before calculating magnitude.
Self-Inductance & RL Circuits
| Equation | Variables | Notes |
|---|
| EMF = −L dI/dt | L = self-inductance (Henries) | Inductor opposes current changes |
| L = μ₀n²V = μ₀N²A/ℓ | n = turns/length, V = volume, ℓ = length | Inductance of solenoid |
| U_L = ½LI² | Energy stored in inductor | |
| u_B = B²/(2μ₀) | Magnetic energy density (J/m³) | |
| τ = L/R | RL circuit time constant | |
| I(t) = (ε/R)(1 − e^(−t/τ)) [building] | | Switch closes at t=0 |
| I(t) = I₀e^(−t/τ) [decaying] | | Source removed at t=0 |
Mutual Inductance & Transformers
| Equation | Notes |
|---|
| EMF₂ = −M dI₁/dt | M = mutual inductance |
| V_s/V_p = N_s/N_p | Transformer voltage ratio |
| I_s/I_p = N_p/N_s | Transformer current ratio (ideal) |
| P_p = P_s | Power conserved (ideal transformer) |
Maxwell's Equations (Complete Form)
| Law | Integral Form | Meaning |
|---|
| Gauss's Law (E) | ∮E⃗·dA⃗ = Q_enc/ε₀ | Electric field lines originate on charges |
| Gauss's Law (B) | ∮B⃗·dA⃗ = 0 | No magnetic monopoles exist |
| Faraday's Law | ∮E⃗·dl⃗ = −dΦ_B/dt | Changing B creates E |
| Ampere-Maxwell Law | ∮B⃗·dl⃗ = μ₀(I + ε₀ dΦ_E/dt) | Changing E creates B; displacement current |
Displacement current: I_D = ε₀ dΦ_E/dt. Maxwell added this term to fix Ampere's Law in regions where no real current flows (e.g., between capacitor plates). It also predicts electromagnetic waves: c = 1/√(μ₀ε₀).
Electromagnetic Waves
| Equation | Notes |
|---|
| c = 1/√(μ₀ε₀) = 3×10⁸ m/s | Speed of light in vacuum |
| E₀ = cB₀ | Amplitudes of E and B in an EM wave |
| E⃗ ⊥ B⃗ ⊥ direction of propagation | Transverse wave |
| S = (1/μ₀)E⃗ × B⃗ | Poynting vector — energy flux (W/m²) |
| I = S_avg = E₀B₀/(2μ₀) = E₀²/(2μ₀c) | Intensity of EM wave |
What's NOT on the Formula Sheet (Memorize These)
- Direction rules: right-hand rule for B around wire, magnetic force F = qv × B
- Lenz's Law direction — must determine induced current direction yourself
- Boundary condition at conductors: E inside = 0; surface charge on conductors only
- Superposition principle: fields and potentials add as vectors/scalars
- Capacitor with dielectric disconnected vs. connected: different behavior for Q and V
- RC/RL circuit behavior at t = 0⁺ and t → ∞: capacitor = open (charging) or wire (charged); inductor = wire (steady) or open (switching)
- Hall effect: V_H = IB/(nqt) — drift velocity and carrier type
Common AP Physics C: E&M FRQ Tasks
- Gauss's Law problems: draw a symmetric Gaussian surface, apply ∮E·dA = Q_enc/ε₀, solve for E as a function of r
- Ampere's Law problems: choose a circular Amperian loop, apply ∮B·dl = μ₀I_enc, find B(r)
- Circuit analysis: write KVL/KCL equations, solve for I and V; for RC/RL find time constant and write Q(t) or I(t)
- Induction setup: find Φ_B as a function of time, differentiate to get EMF, use Lenz's law for direction, then I = EMF/R
- Solenoid/inductor energy: U = ½LI², L = μ₀n²(volume); compare to capacitor energy U = ½CV²
AP Physics C: E&M vs. AP Physics 2 Comparison
| AP Physics 2 | AP Physics C: E&M |
|---|
| Math level | Algebra | Calculus (integrals, derivatives, differential equations) |
| Gauss's Law | Conceptual | Full integral form, solve for E(r) |
| Circuits | Ohm's Law, basic RC | KVL/KCL, RC and RL differential equations |
| Induction | Qualitative Lenz's Law | Faraday's Law, motional EMF, mutual inductance |
| Maxwell | Not covered | Full Maxwell's Equations, displacement current |
Score Calculator
Use our AP Physics C: E&M Score Calculator to convert your raw score to an AP score estimate.
Score cutoffs (2026):
| AP Score | Notes |
|---|
| 5 | ~27–30 correct on 40-question MCQ + strong FRQ |
| 4 | ~21–26 correct |
| 3 | ~15–20 correct |
| 2 | ~9–14 correct |
AP Physics C: E&M has one of the highest 5-rates among AP exams (~30%) because the student pool is self-selected — these are students who took AP Calculus and chose a second physics course.
Frequently Asked Questions
Does AP Physics C: E&M provide a formula sheet on the exam?
Yes. College Board provides an equation sheet for the entire AP Physics C: E&M exam, available for both the multiple choice and free response sections. However, knowing when and how to apply each formula still requires deep understanding — the sheet doesn't tell you which law to use in a given problem.
Is AP Physics C: E&M harder than Mechanics?
Most students find E&M harder. The field laws (Gauss's, Ampere's, Faraday's) require more abstract thinking about flux and surfaces. The circuit analysis also adds complexity with RC and RL transients. However, both exams have similarly high 5-rates because of self-selection in the student population.
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