Basque 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; for a uniform sphere, outside the sphere it acts as if all mass were at the center.

Gravitational potential V = -G M / r (zero at infinity); potential energy U = -G M m / r. Potential is a scalar; field is the negative gradient of potential: g = -dV/dr.

Kepler's laws: 1) orbits are ellipses with the central mass at one focus; 2) equal areas in equal times (conservation of angular momentum); 3) T^2 = (4π²/GM) a^3, where a is the semi-major axis.

For circular orbits: v = sqrt(GM/r), T = 2π sqrt(r^3/GM), and orbital energy E = -GMm/(2r). Escape velocity from a body of mass M and radius R: v_esc = sqrt(2GM/R).

Geostationary orbit: circular, equatorial, period 24 h, radius ≈ 4.22×10^7 m from Earth's center; appears fixed above a point on the equator.

Gravitational field inside a uniform spherical shell is zero; inside a uniform solid sphere, g = (G M / R^3) r, linear with r.

Weight measured in a lift: apparent weight N = m(g ± a), with + when accelerating upward and − when accelerating downward; in free fall N = 0 (apparent weightlessness).

Coulomb's law: F = k q1 q2 / r^2, k = 1/(4π ε0) ≈ 8.99×10^9 N·m²/C²; force is along the line joining charges, repulsive for like signs, attractive for opposite signs.

Electric field E = F/q0 = k Q / r^2 for a point charge, directed radially outward for positive Q and inward for negative Q. Superposition: E_total = Σ E_i vectorially.

Electric potential V = k Q / r for a point charge (zero at infinity); potential energy U = k q1 q2 / r. Potential is scalar; E = -dV/dr.

Electric flux through a closed surface: Φ = ∮ E·dA = Q_enclosed / ε0 (Gauss's law). Useful for spheres, infinite lines (E = λ/(2π ε0 r)), and infinite sheets (E = σ/(2 ε0)).

Uniform field between parallel plates: E = V/d, directed from the positive to the negative plate; a charge q experiences force qE and acceleration qE/m.

Work done by the electric field on a charge moving from A to B: W = q(V_A - V_B); if only electric forces act, kinetic energy increases by q(V_A - V_B).

Equipotential surfaces are always perpendicular to electric field lines; no work is done moving a charge along an equipotential.

Electric dipole: moment p = q d, torque τ = p × E, potential energy U = -p·E. Field on the axis of a dipole falls as 1/r^3.

Magnetic force on a moving charge: F = q v × B, magnitude q v B sinθ; direction given by the right-hand rule. Magnetic force does no work because it is always perpendicular to velocity.

Force on a straight current-carrying wire: F = I L × B, magnitude B I L sinθ. Two parallel wires carrying currents in the same direction attract; opposite directions repel.

Biot–Savart law: dB = (μ0 I / 4π) (dl × r̂)/r^2. For a long straight wire: B = μ0 I / (2π r); at the center of a circular loop: B = μ0 I / (2R); inside a solenoid: B = μ0 n I.

Ampère's law: ∮ B·dl = μ0 I_enclosed. It gives the field of symmetric current distributions (long wire, solenoid, toroid).

Magnetic flux through a surface: Φ = B·A cosθ, unit weber (Wb). Faraday's law: induced emf ε = -dΦ/dt; for a coil of N turns, ε = -N dΦ/dt.

Lenz's law: the induced current opposes the change in flux that produces it (conservation of energy).

Motional emf for a rod of length L moving with speed v perpendicular to B: ε = B L v; the induced current direction follows from v × B.

A charged particle moving perpendicular to a uniform B field follows a circular path of radius r = m v / (|q| B) and period T = 2π m / (|q| B), independent of speed.

Wave equation for a harmonic wave: y(x,t) = A sin(ωt - kx + φ), with ω = 2πf, k = 2π/λ, and speed v = λf = ω/k. The sign before kx gives the direction of propagation.

Wave speed on a stretched string: v = sqrt(T/μ), where T is tension and μ is linear mass density. Sound speed in air increases with temperature: v ≈ 331 + 0.6 T(°C) m/s.

Superposition principle: the displacement at any point is the vector sum of the displacements due to each wave. Constructive interference when path difference Δr = nλ; destructive when Δr = (n + ½)λ.

Standing waves on a string fixed at both ends: λ_n = 2L/n, f_n = n v/(2L), n = 1,2,3,...; nodes at the ends. For a pipe open at both ends the same formula applies; for a pipe closed at one end, only odd harmonics: f_n = n v/(4L), n = 1,3,5,...

Sound intensity level: β = 10 log10(I/I0) dB, with I0 = 10^-12 W/m². Intensity I = P/(4πr²) for a point source in free space, so level drops by 6 dB when distance doubles.

Doppler effect for sound: f' = f (v ± v_o)/(v ∓ v_s), where v is sound speed, v_o observer speed, v_s source speed; use upper signs for approach. For light, redshift/blueshift: Δf/f ≈ v/c for v << c.

Diffraction is significant when the wavelength is comparable to the obstacle or slit width; for a single slit of width a, first minimum at sinθ = λ/a.

Energy transported by a mechanical wave is proportional to the square of the amplitude and to the square of the frequency.

Law of reflection: angle of incidence equals angle of reflection; both angles measured from the normal. For a plane mirror, the image is virtual, upright, same size, and as far behind the mirror as the object is in front.

Snell's law: n1 sinθ1 = n2 sinθ2, with n = c/v. Light bends toward the normal when entering a denser medium (n2 > n1).

Total internal reflection occurs when light travels from a denser to a less dense medium and the incidence angle exceeds the critical angle θc = arcsin(n2/n1).

Thin lens/mirror equation: 1/f = 1/s + 1/s', with sign conventions: for lenses, f > 0 for converging (convex), f < 0 for diverging (concave); for mirrors, f = R/2, f > 0 for concave. Magnification m = -s'/s = y'/y.

Real images are formed on the opposite side of a converging lens (or in front of a concave mirror) and can be projected; virtual images are upright and on the same side as the object (for lenses).

Ray tracing rules for a converging lens: a ray parallel to the axis passes through the far focal point; a ray through the center continues undeviated; a ray through the near focal point emerges parallel to the axis.

The eye: normal near point ≈ 25 cm; a magnifying glass gives angular magnification M = 25/f (f in cm) when the image is at infinity. A compound microscope uses an objective and an eyepiece; a telescope's angular magnification is M = f_objective / f_eyepiece.

Dispersion: refractive index depends on wavelength, so white light separates in a prism; violet bends more than red because n_violet > n_red.

Young's double-slit interference: bright fringes at d sinθ = nλ, dark at d sinθ = (n + ½)λ; fringe spacing on a screen at distance L: Δy = λL/d (small angles).

Diffraction grating: maxima at d sinθ = nλ, where d is the slit spacing; sharper and more separated maxima than a double slit.

Photoelectric effect: maximum kinetic energy K_max = hf - Φ, where Φ is the work function; threshold frequency f0 = Φ/h; no emission below f0 regardless of intensity. Stopping potential V_s satisfies e V_s = K_max.

Photon energy E = hf = hc/λ; h = 6.63×10^-34 J·s, and hc = 1240 eV·nm. Photon momentum p = h/λ.

de Broglie wavelength: λ = h/p = h/(mv). For an electron accelerated through potential difference V, λ ≈ 1.23 nm / sqrt(V in volts).

Bohr model for hydrogen: E_n = -13.6 eV / n²; photon energy for a transition is |E_f - E_i| = hc/λ. Ionization energy from ground state is 13.6 eV.

Relativity basics: γ = 1/sqrt(1 - v²/c²); time dilation Δt = γ Δt0, length contraction L = L0/γ, relativistic mass-energy E = γ m c², rest energy E0 = m c².

The photoelectric effect and Compton scattering demonstrate the particle nature of light; interference and diffraction demonstrate its wave nature.

For repeated measurements of the same quantity, the best estimate is the mean: x̄ = (Σ x_i)/n; the standard deviation of the mean is s/√n, where s is the sample standard deviation.

For a single measurement with an analogue instrument, the absolute uncertainty is typically half the smallest scale division; for a digital instrument, it is the last digit's place value (or as stated by the manufacturer).

When adding or subtracting quantities, absolute uncertainties add: if z = x ± y, then Δz = Δx + Δy (worst case). When multiplying or dividing, relative uncertainties add: Δz/z = Δx/x + Δy/y.

For a power law z = x^n, the relative uncertainty is |n| times the relative uncertainty of x: Δz/z = |n| Δx/x.

Graphical analysis: plot the dependent variable on the y-axis and the independent variable on the x-axis; the slope of a straight-line fit gives the rate constant, and the intercept gives the initial value. Error bars represent uncertainties.

To linearize data, choose variables so the expected relation is y = m x + b; for example, T² vs. L for a pendulum gives a straight line with slope 4π²/g.

Significant figures: the result of a calculation should have the same number of significant figures as the least precise input; uncertainties are usually quoted to one significant figure.

Systematic errors shift all measurements in the same direction and affect accuracy; random errors scatter measurements and affect precision, reducing with averaging.