PAU Physics (Andalusia) cheat sheet

Newton's law of gravitation: F = G·m₁·m₂/r², always attractive and directed along the line joining the two masses; G = 6.67×10⁻¹¹ N·m²/kg².

Field strength at a point: g = F/m = G·M/r², pointing toward the source mass; for a uniform sphere it acts as if all mass were at the centre (outside the sphere).

Gravitational potential: V = −G·M/r, taken as zero at infinity; potential is a scalar, always negative near a mass, and adds algebraically.

Potential energy of two masses: U = −G·M·m/r; work done by the field equals the decrease in U, W = −ΔU.

Escape velocity from a body of mass M and radius R: v_e = √(2GM/R), independent of the escaping body's mass and direction (ignoring drag).

Circular orbit: gravity supplies the centripetal force, so G·M·m/r² = m·v²/r, giving v = √(GM/r) and T = 2π·√(r³/(GM)) (Kepler's third law).

Kepler's laws: elliptical orbits with the central mass at one focus; equal areas in equal times (conservation of angular momentum); T² ∝ a³.

For a satellite of mass m in orbit, kinetic energy is K = GMm/(2r) and total mechanical energy E = −GMm/(2r); E increases (becomes less negative) with larger r.

Variation of g with height h above the surface: g(h) = g₀·R²/(R+h)²; inside a uniform sphere g decreases linearly with distance from the centre.

Coulomb's law in vacuum: F = k·q₁·q₂/r² with k = 1/(4πε₀) ≈ 9×10⁹ N·m²/C²; force is repulsive for like signs, attractive for opposite signs.

Electric field of a point charge: E = k·Q/r², a vector pointing away from positive charge and toward negative charge; superposition applies to multiple charges.

Electric potential of a point charge: V = k·Q/r (scalar, zero at infinity); potential energy of a charge q in that field is U = q·V.

Uniform field between parallel plates: E = V/d, field lines run from the positive to the negative plate; a charge q experiences constant force qE.

Gauss's law: electric flux through a closed surface equals Q_enclosed/ε₀; use it for spheres, infinite lines (E = λ/(2πε₀r)) and infinite sheets (E = σ/(2ε₀)).

Relation between field and potential: E = −dV/dr; field points in the direction of decreasing potential, and equipotential surfaces are perpendicular to field lines.

Work done moving a charge between two points: W_ext = q·(V_B − V_A); the electrostatic force is conservative, so work around a closed path is zero.

Capacitance: C = Q/V; parallel-plate capacitor C = ε₀·ε_r·A/d, stored energy U = ½CV² = Q²/(2C).

Motion of a charge in a uniform field: acceleration a = qE/m, parabolic trajectory analogous to projectile motion when the initial velocity is perpendicular to E.

Magnetic force on a moving charge: F = q·v×B, magnitude qvB·sinθ; it is always perpendicular to v, so it does no work and cannot change speed.

Force on a straight current-carrying wire: F = I·L×B, magnitude BIL·sinθ; direction given by the right-hand rule (or left-hand rule for conventional current).

Field of a long straight wire: B = μ₀I/(2πr); inside a solenoid B = μ₀·n·I, with n the turns per unit length; μ₀ = 4π×10⁻⁷ T·m/A.

Charged particle moving perpendicular to a uniform B: circular path of radius r = mv/(qB) and period T = 2πm/(qB), independent of speed.

Magnetic flux: Φ = B·A·cosθ, with θ the angle between B and the surface normal; unit is the weber (Wb).

Faraday's law: induced emf ε = −dΦ/dt; Lenz's law fixes the sign - the induced current opposes the change in flux that produced it.

Motional emf for a rod of length L moving at speed v perpendicular to B: ε = B·L·v; if the circuit has resistance R, current I = BLv/R.

Self-induction: ε = −L·dI/dt; energy stored in an inductor U = ½LI²; a transformer obeys V_s/V_p = N_s/N_p for an ideal core.

Ampere's law for symmetric currents: ∮B·dl = μ₀·I_enclosed; use it for wires, solenoids and toroids rather than integrating Biot–Savart.

Wave equation: y(x,t) = A·sin(ωt − kx + φ), with ω = 2πf, k = 2π/λ, and propagation speed v = λf = ω/k.

Mechanical wave speed depends on the medium: on a string v = √(T/μ) (tension over linear density); sound in air v ≈ 340 m/s at room temperature.

Sound intensity level: β = 10·log(I/I₀) dB with I₀ = 10⁻¹² W/m²; intensity falls as 1/r² from a point source, so level drops about 6 dB per doubling of distance.

Doppler effect: f' = f·(v ± v_o)/(v ∓ v_r), signs chosen so approach raises the observed frequency; for light, redshift when source and observer separate.

Standing waves: nodes separated by λ/2; string fixed at both ends f_n = n·v/(2L); pipe open at both ends f_n = n·v/(2L); pipe closed at one end only odd harmonics f_n = n·v/(4L).

Snell's law: n₁·sinθ₁ = n₂·sinθ₂; refractive index n = c/v; total internal reflection occurs beyond the critical angle sinθ_c = n₂/n₁ when n₁ > n₂.

Thin lens and mirror equation: 1/f = 1/s' − 1/s (sign conventions vary); magnification m = −s'/s; power of a lens P = 1/f in dioptres.

Young's double slit: maxima at d·sinθ = mλ, fringe spacing Δy = λD/d on a screen at distance D; single-slit minima at a·sinθ = mλ.

Electromagnetic spectrum ordered by increasing frequency: radio, microwave, infrared, visible, ultraviolet, X-rays, gamma; all travel at c = 3×10⁸ m/s in vacuum.

Special relativity: postulates are the invariance of physical laws and of c in all inertial frames; γ = 1/√(1 − v²/c²).

Time dilation Δt = γ·Δt₀ and length contraction L = L₀/γ; relativistic mass-energy E = γmc², rest energy E₀ = mc².

Relativistic momentum p = γmv; total energy E² = (pc)² + (mc²)²; kinetic energy K = (γ − 1)mc².

Photoelectric effect: photon energy E = hf = hc/λ; maximum kinetic energy K_max = hf − W, with threshold frequency f₀ = W/h; h = 6.63×10⁻³⁴ J·s.

de Broglie wavelength λ = h/p = h/(mv); Compton scattering shows photons carry momentum p = E/c.

Bohr model: E_n = −13.6 eV/n² for hydrogen; photon emitted or absorbed has energy hf = E_f − E_i; spectral lines converge to the series limit.

Radioactive decay: N = N₀·e^(−λt); half-life T½ = ln2/λ; activity A = λN measured in becquerels.

Nuclear binding energy from mass defect: ΔE = Δm·c²; 1 u = 931.5 MeV/c²; fission and fusion release energy when binding energy per nucleon increases.

Mass–energy and momentum conservation in nuclear reactions: total Z and A balance; in beta-minus decay a neutron becomes a proton with emission of an electron and an antineutrino.