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AP Physics C: E&M — Equations & Constants

6 sections · 53 entries · print it and keep it beside your practice sets

The calculus-based E&M sheet: Gauss, Ampère, Faraday, and the RC/LR/LC transients. The integral forms are supplied, but choosing the symmetry (and therefore the Gaussian or Amperian surface) is the graded skill.

Practise with the sheet, not from memory. The College Board hands out its own version of this page on exam day, so nothing here is worth memorising for its own sake. What earns points is speed: knowing which section a quantity lives in, and reading off the right line without breaking your train of thought. Keep this open (or printed) for every practice set you do.

Electrostatics

Quantity / relationshipWhat it means & how to use it

F = (1/4πε₀) (q₁ q₂ / r²) r̂

Coulomb force

E = F / q

Electric field

E_point

E = (1/4πε₀) (q/r²) r̂

Field of a point charge

Gauss

∮ E · dA = Q_enclosed / ε₀

Flux through a closed surface equals enclosed charge over ε₀

E_sheet

E = σ / (2ε₀)

Infinite sheet of surface charge density σ

Between the plates of a parallel-plate capacitor the two sheets add: E = σ/ε₀.

E_line

E = λ / (2π ε₀ r)

Infinite line of linear charge density λ

ΔV

ΔV = − ∫ E · dl

Potential difference from the field

E from V

E_x = − ∂V/∂x

Field as the negative gradient of potential

V = (1/4πε₀) Σ qᵢ / rᵢ

Potential of a collection of point charges

U_E

U_E = q V = (1/4πε₀) q₁ q₂ / r

Potential energy of a charge pair

Capacitance & dielectrics

Quantity / relationshipWhat it means & how to use it

C = Q / V

Definition of capacitance

C_plates

C = κ ε₀ A / d

Parallel plates, dielectric constant κ

U_C

U_C = ½ Q V = ½ C V² = Q²/(2C)

Stored energy

u_E

u_E = ½ ε₀ E²

Energy density of an electric field

Combinations

1/C_s = Σ 1/Cᵢ, C_p = Σ Cᵢ

Series and parallel capacitors

Current, resistance, DC circuits

Quantity / relationshipWhat it means & how to use it

I = dQ/dt = ∫ J · dA

Current as charge flow

J = I/A = n q v_d

Current density; n is carrier number density

R = ρ ℓ / A, E = ρ J

Resistance from resistivity ρ

V = IR

V = I R

Ohm’s law

P = I V = I²R = V²/R

Power

Combinations

R_s = Σ Rᵢ, 1/R_p = Σ 1/Rᵢ

Series and parallel resistors

RC and LR transients

Quantity / relationshipWhat it means & how to use it

RC charging

Q(t) = Cε (1 − e^(−t/RC)), I(t) = (ε/R) e^(−t/RC)

Capacitor charging through R from emf ε; τ = RC

RC discharging

Q(t) = Q₀ e^(−t/RC)

Capacitor discharging through R

LR growth

I(t) = (ε/R)(1 − e^(−Rt/L))

Current in an inductor; τ = L/R

Limits

t → 0: C acts as a wire, L as a break. t → ∞: C acts as a break, L as a wire.

Steady state shortcuts

Magnetic fields

Quantity / relationshipWhat it means & how to use it

F_B

F_B = q v × B

Force on a moving charge

F_wire

F = ∫ I dl × B

Force on a current-carrying wire

Biot–Savart

dB = (μ₀/4π) (I dl × r̂)/r²

Field of a current element

Ampère

∮ B · dl = μ₀ I_enclosed

Line integral of B around a closed loop

B_wire

B = μ₀ I / (2π r)

Long straight wire

B_solenoid

B = μ₀ n I

Inside a long solenoid, n turns per metre

r_cyclotron

r = m v / (|q| B)

Radius of a charged particle’s circular path

Induction & inductance

Quantity / relationshipWhat it means & how to use it

Φ_B

Φ_B = ∫ B · dA

Magnetic flux

ε = − dΦ_B / dt

Faraday’s law; the minus sign is Lenz’s law

ε_motional

ε = B ℓ v

Rod of length ℓ moving at v perpendicular to B

L = N Φ_B / I, ε = − L dI/dt

Self-inductance

L_solenoid

L = μ₀ N² A / ℓ

Inductance of a long solenoid

U_L

U_L = ½ L I²

Energy stored in an inductor

u_B

u_B = B² / (2μ₀)

Energy density of a magnetic field

ω = 1/√(L C)

Angular frequency of an LC oscillator

# Constants & conversions

e = 1.60 × 10⁻¹⁹ C

Elementary charge (magnitude of electron charge)

m_e

m_e = 9.11 × 10⁻³¹ kg

Mass of an electron

m_p

m_p = 1.67 × 10⁻²⁷ kg

Mass of a proton

ε₀

ε₀ = 8.85 × 10⁻¹² C²/(N·m²)

Vacuum permittivity

k = 1/(4πε₀)

k = 8.99 × 10⁹ N·m²/C²

Coulomb’s law constant

Use 9.0 × 10⁹ for hand arithmetic.

μ₀

μ₀ = 4π × 10⁻⁷ T·m/A = 1.26 × 10⁻⁶ T·m/A

Vacuum permeability

μ₀/(4π)

μ₀/(4π) = 1 × 10⁻⁷ T·m/A

Magnetostatic constant

c = 3.00 × 10⁸ m/s

Speed of light in vacuum

h = 6.63 × 10⁻³⁴ J·s = 4.14 × 10⁻¹⁵ eV·s

Planck’s constant

k_B

k_B = 1.38 × 10⁻²³ J/K

Boltzmann’s constant

R = 8.31 J/(mol·K)

Universal gas constant

N_A

N_A = 6.02 × 10²³ mol⁻¹

Avogadro’s number

1 eV

1 eV = 1.60 × 10⁻¹⁹ J

Electron-volt in joules

Practise with the sheet open Cheatsheet