Canary PAU Physics cheat sheet

Newton's law of gravitation: F = G m1 m2 / r^2, attractive, along the line joining the masses; G = 6.674×10^-11 N·m²/kg².

Gravitational field strength g = F/m = G M / r^2, directed toward the mass M; inside a uniform sphere g ∝ r, outside g ∝ 1/r².

Gravitational potential V = -G M / r, zero at infinity; potential energy U = -G M m / r.

Orbital speed for a circular orbit: v = √(G M / r); period T = 2π √(r³ / (G M)) - Kepler's third law T² ∝ r³.

Geostationary orbit: equatorial, prograde, period 24 h, radius ≈ 4.22×10^7 m (altitude ≈ 3.58×10^7 m).

Escape speed from a body of mass M and radius R: v_esc = √(2 G M / R); independent of direction and of the escaping mass.

Total energy of a bound orbit: E = -G M m / (2 r) for circular orbit; kinetic energy = -E, potential energy = 2E.

Kepler's second law: equal areas in equal times, equivalent to conservation of angular momentum r × v = constant.

Coulomb's law: F = k q1 q2 / r², k = 1/(4π ε0) ≈ 8.99×10^9 N·m²/C²; force is repulsive for like charges, attractive for unlike.

Electric field E = F/q0; for a point charge E = k Q / r², directed radially outward for positive Q.

Electric potential V = k Q / r; potential energy U = k q1 q2 / r; E = -dV/dr.

Gauss's law: ∮ E · dA = Q_enclosed / ε0; field inside a conductor in electrostatic equilibrium is zero.

Uniform field between parallel plates: E = V/d; force on charge q is F = qE, acceleration a = qE/m.

Capacitance C = Q/V; parallel-plate C = ε0 A / d; energy stored U = ½ C V² = ½ Q V = Q²/(2C).

Dielectric inserted into a capacitor at fixed charge reduces E and V by factor κ and increases C by κ.

Electric field lines start on positive charges and end on negative charges; they never cross and are perpendicular to conductor surfaces.

Magnetic force on a moving charge: F = q v × B, magnitude F = q v B sinθ; direction given by right-hand rule for positive q.

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

Charged particle in uniform B moves in a circle of radius r = m v / (|q| B); period T = 2π m / (|q| B) independent of speed.

Magnetic field of a long straight wire: B = μ0 I / (2π r); inside a solenoid: B = μ0 n I.

Force between parallel wires: F/L = μ0 I1 I2 / (2π d); currents in the same direction attract.

Ampere's law: ∮ B · dl = μ0 I_enclosed; magnetic field lines are closed loops (no magnetic monopoles).

Magnetic flux Φ = B · A = B A cosθ; unit tesla·meter² = weber (Wb).

A current loop in a magnetic field experiences torque τ = I A × B, magnitude τ = I A B sinθ.

Faraday's law: induced emf ε = -dΦ/dt = -N dΦ/dt; the minus sign is Lenz's law (induced current opposes the change in flux).

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

Magnetic flux through a loop: Φ = B A cosθ; changing B, A, or θ induces an emf.

Self-inductance: ε = -L dI/dt; energy stored in an inductor U = ½ L I².

An AC generator produces sinusoidal emf ε = N B A ω sin(ωt); peak emf = N B A ω.

Electromagnetic waves travel in vacuum at c = 1/√(μ0 ε0) ≈ 3.00×10^8 m/s; E and B are perpendicular to each other and to the propagation direction.

For an EM wave, E = c B; intensity I = ½ ε0 c E0² (average over a cycle).

EM spectrum from longest to shortest wavelength: radio, microwave, infrared, visible, ultraviolet, X-ray, gamma ray.

Wave speed v = f λ = λ/T; for a wave on a string v = √(T/μ), where T is tension and μ is linear mass density.

Sound speed in air at 20 °C ≈ 343 m/s; in a gas v = √(γ R T / M), so v increases with temperature.

Intensity level in decibels: β = 10 log10(I/I0), with I0 = 1×10^-12 W/m²; doubling intensity adds about 3 dB.

Doppler effect: f' = f (v ± v_o)/(v ∓ v_s); use + in numerator for observer moving toward source, + in denominator for source moving toward observer.

Standing waves on a string fixed at both ends: f_n = n v/(2L), n = 1,2,3,...; open pipe: f_n = n v/(2L); closed pipe: f_n = n v/(4L), n odd.

Beat frequency equals the absolute difference of the two source frequencies: f_beat = |f1 - f2|.

Intensity of a spherical wave: I = P/(4π r²), so I ∝ 1/r².

Transverse waves on a string are polarized; longitudinal sound waves in a fluid are not.

Snell's law: n1 sinθ1 = n2 sinθ2; refractive index n = c/v.

Total internal reflection occurs when light travels from higher to lower n and the incidence angle exceeds θ_c = arcsin(n2/n1).

Thin lens equation: 1/f = 1/d_o + 1/d_i; magnification m = -d_i/d_o; f > 0 for converging lens, f < 0 for diverging.

Mirror equation: 1/f = 1/d_o + 1/d_i with f = R/2; real images have d_i > 0 and are inverted.

Young's double-slit maxima: d sinθ = m λ; minima: d sinθ = (m + ½) λ; fringe spacing on a screen Δy = λ L/d.

Single-slit minima: a sinθ = m λ, m = ±1, ±2,...; central maximum is twice as wide as secondary maxima.

Diffraction grating: d sinθ = m λ gives sharp principal maxima; larger d gives smaller angular separation.

Brewster's angle: tanθ_B = n2/n1; reflected light is fully polarized perpendicular to the plane of incidence.

Lorentz factor γ = 1/√(1 - v²/c²); γ ≥ 1 and γ → ∞ as v → c.

Time dilation: Δt = γ Δt0, where Δt0 is the proper time between events at the same place.

Length contraction: L = L0/γ, where L0 is the proper length measured in the object's rest frame.

Relativistic momentum p = γ m v; total energy E = γ m c²; rest energy E0 = m c².

Kinetic energy K = (γ - 1) m c²; for v ≪ c, K ≈ ½ m v².

Energy-momentum relation: E² = (p c)² + (m c²)²; photons have m = 0 and E = p c.

Relativistic velocity addition: u' = (u - v)/(1 - u v/c²) for collinear motion.

Simultaneity is frame-dependent: events simultaneous in one inertial frame need not be simultaneous in another.

Photoelectric effect: K_max = h f - φ, where φ is the work function; threshold frequency f0 = φ/h; no emission below f0 regardless of intensity.

Photon energy E = h f = h c/λ; h = 6.63×10^-34 J·s; 1 eV = 1.60×10^-19 J.

de Broglie wavelength: λ = h/p = h/(m v); for electrons accelerated through V volts, λ ≈ 1.23/√V nm.

Bohr model: E_n = -13.6 eV / n² for hydrogen; transition photon energy equals the energy-level difference.

Heisenberg uncertainty principle: Δx Δp ≥ ħ/2, with ħ = h/(2π).

Radioactive decay: N = N0 e^(-λt); half-life T½ = ln2/λ; activity A = λ N.

Mass defect and binding energy: ΔE = Δm c²; 1 u = 931.5 MeV/c².

Nuclear reactions conserve charge (Z) and nucleon number (A); beta-minus decay converts a neutron to a proton and emits an electron and antineutrino.