AP Physics C: E&M cheat sheet

Coulomb's law: F = kq₁q₂/r² with k = 1/(4πε₀) ≈ 8.99×10⁹ N·m²/C²; force is along the line joining the charges, repulsive for like signs.

Electric field of a point charge: E = kq/r², directed radially outward from positive charge and inward toward negative charge.

Electric flux through a surface: Φ_E = ∫E·dA; only the field component normal to the surface contributes.

Gauss's law: ∮E·dA = Q_enc/ε₀; holds for any closed surface, with ε₀ ≈ 8.85×10⁻¹² C²/(N·m²).

Infinite line of charge (linear density λ): E = λ/(2πε₀r), radial, falls off as 1/r.

Infinite sheet of charge (surface density σ): E = σ/(2ε₀), uniform and normal to the sheet, independent of distance.

Uniform spherical shell of charge: E = 0 inside the shell; outside it E = kQ/r², identical to a point charge at the center.

Solid uniform sphere of charge Q, radius R: inside, E = kQr/R³ (linear in r); outside, E = kQ/r².

Electric dipole moment p = qd points from negative to positive charge; in a uniform field the torque is τ = p×E.

Potential difference: ΔV = V_b − V_a = −∫E·ds from a to b; the field points toward decreasing potential.

Potential of a point charge with V = 0 at infinity: V = kq/r; potential is a scalar so contributions add algebraically.

Potential energy of a charge q in a potential: U = qV; for two charges U = kq₁q₂/r.

Relation between field and potential: E = −dV/dx in one dimension, or E = −∇V in general.

Work done by an external agent moving charge q slowly from a to b is W_ext = qΔV = q(V_b − V_a).

Equipotential surfaces are everywhere perpendicular to field lines; no work is done moving a charge along one.

Uniform field between parallel plates separated by d: magnitude E = V/d, field directed from the higher- to lower-potential plate.

Potential at the center of a uniformly charged ring of radius R and charge Q: V = kQ/R; the field there is zero by symmetry.

Electric potential energy of a system equals the work required to assemble the charges from infinity.

In electrostatic equilibrium the electric field inside a conductor is zero and any excess charge resides on the surface.

The electric field just outside a conductor is E = σ/ε₀, normal to the surface; the surface is an equipotential.

Capacitance is C = Q/V, defined by geometry and dielectric, independent of the charge stored.

Parallel-plate capacitor: C = ε₀A/d; inserting a dielectric of constant κ multiplies C by κ.

Capacitors in parallel add: C_eq = ΣCᵢ; capacitors in series add reciprocally: 1/C_eq = Σ(1/Cᵢ).

Energy stored in a capacitor: U = ½QV = ½CV² = Q²/(2C).

Energy density in an electric field: u = ½ε₀E² (or ½κε₀E² in a dielectric).

For an isolated charged capacitor, inserting a dielectric at constant Q lowers V and E by factor κ while C rises by κ.

A spherical capacitor or isolated sphere has C = 4πε₀R for a solitary sphere of radius R.

Ohm's law for a resistor: V = IR; resistance of a uniform wire is R = ρL/A.

Resistors in series add: R_eq = ΣRᵢ; resistors in parallel add reciprocally: 1/R_eq = Σ(1/Rᵢ).

Junction rule (charge conservation): the sum of currents into a node equals the sum leaving.

Loop rule (energy conservation): the sum of potential changes around any closed loop is zero.

Electrical power dissipated in a resistor: P = IV = I²R = V²/R.

RC charging from emf ε: q(t) = Cε(1 − e^(−t/RC)) and i(t) = (ε/R)e^(−t/RC); time constant τ = RC.

RC discharging: q(t) = Q₀e^(−t/RC) and i(t) = −(Q₀/RC)e^(−t/RC).

An ideal ammeter has zero resistance and is placed in series; an ideal voltmeter has infinite resistance and is placed in parallel.

Terminal voltage of a real source: V = ε − Ir, where r is the internal resistance.

Magnetic force on a moving charge: F = qv×B, magnitude qvB sinθ; the force is perpendicular to both v and B and does no work.

Magnetic force on a current-carrying wire: F = I L×B, magnitude BIL sinθ.

Radius of circular motion of a charge in a uniform B field: r = mv/(|q|B); period T = 2πm/(|q|B) is independent of speed.

Biot–Savart law: dB = (μ₀/4π)(I ds×r̂)/r², with μ₀ = 4π×10⁻⁷ T·m/A.

Long straight wire: B = μ₀I/(2πr); the field lines are circles around the wire given by the right-hand rule.

Center of a circular loop of radius R: B = μ₀I/(2R); inside a solenoid: B = μ₀nI with n turns per unit length.

Ampère's law: ∮B·ds = μ₀I_enc; useful when symmetry gives a constant B along the loop.

Magnetic flux through a surface: Φ_B = ∫B·dA, measured in webers (T·m²).

Magnetic dipole moment of a current loop: μ = IA, with torque τ = μ×B and potential energy U = −μ·B.

Faraday's law: emf = −dΦ_B/dt; the emf equals the negative rate of change of magnetic flux through a circuit.

Lenz's law: the induced current opposes the change in flux that produced it, a statement of energy conservation.

Motional emf for a rod of length L moving at speed v perpendicular to B: emf = BLv.

Magnetic flux through a loop of area A in uniform field: Φ_B = BA cosθ, where θ is between B and the area normal.

Induced emf in a coil of N turns: emf = −N dΦ_B/dt.

Self-induced emf in an inductor: emf = −L dI/dt; the henry is the unit of inductance.

Energy stored in an inductor: U = ½LI²; energy density in a magnetic field: u = B²/(2μ₀).

LR circuit time constant: τ = L/R; current rises as I(t) = (ε/R)(1 − e^(−Rt/L)).

A changing electric flux acts as a displacement current, allowing Ampère's law to hold inside a charging capacitor.