CSS Physics formulas

Mathematical methods and mechanics

Vectors and vector analysis

• Resultant of two vectors at angle θ: R² = A² + B² + 2AB cos θ
• Dot product: A·B = AB cos θ = A_xB_x + A_yB_y + A_zB_z; work W = F·d
• Cross product: |A × B| = AB sin θ = area of the parallelogram on A and B; B × A = −(A × B)
• Unit vectors: i × j = k, j × k = i, k × i = j; i·i = 1, i·j = 0
• grad φ = (∂φ/∂x)i + (∂φ/∂y)j + (∂φ/∂z)k, a vector normal to the surfaces of constant φ
• div F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z (a scalar); curl F = ∇ × F (a vector); curl (grad φ) = 0 and div (curl F) = 0
• Gauss: ∮ F·dS = ∫ (∇·F) dV. Stokes: ∮ F·dl = ∫ (∇ × F)·dS

Kinematics

• Constant acceleration only: v = u + at; s = ut + ½at²; v² = u² + 2as; s = ½(u + v)t
• average speed = total distance ÷ total time (not the mean of the speeds)
• Projectile on level ground: time of flight T = 2u sin θ/g; greatest height H = u² sin²θ/(2g)
• Range R = u² sin 2θ/g; it is greatest at 45°, and angles θ and (90° − θ) give the same range
• At the highest point the vertical velocity is zero but the speed is still u cos θ
• Horizontal throw from height h: time to fall t = √(2h/g), whatever the horizontal speed
• Velocity of A relative to B: v_AB = v_A − v_B (a vector subtraction)

Newton's laws, work, energy and momentum

• net force = mass × acceleration (F = ma = dp/dt); weight = mg
• Friction: static f ≤ μ_sN; kinetic f = μ_kN; sliding down an incline at constant speed means μ_k = tan θ
• Centripetal force = mv²/r = mω²r; flat bend: v_max = √(μgr); top of a vertical circle on a string: v_min = √(gr)
• work = force × distance × cos θ (W = Fd cos θ); power = work ÷ time = Fv
• Kinetic energy = ½mv² = p²/(2m); gravitational PE = mgh; spring PE = ½kx²
• impulse = force × time = change in momentum (FΔt = Δp)
• Elastic head-on collision of equal masses: the velocities are exchanged. Perfectly inelastic: the bodies stick together and the loss of kinetic energy is greatest

Rotational motion and gravitation

• torque = force × perpendicular distance from the axis (τ = rF sin θ); τ = Iα
• Moment of inertia: hoop MR²; solid disc ½MR²; solid sphere (2/5)MR²; parallel axes: I = I_cm + Md²
• Angular momentum L = Iω; with no external torque I₁ω₁ = I₂ω₂
• Rotational kinetic energy = ½Iω²; rolling without slipping: v = ωR and total KE = ½Mv² + ½Iω²
• F = GMm/r²; surface gravity g = GM/R²; at height h, g′ = gR²/(R + h)²
• Orbital speed v = √(GM/r); period T² = 4π²r³/(GM); total energy of a satellite = −GMm/(2r)
• Escape speed = √(2GM/R) = √(2gR), which is √2 × the speed in a low orbit (about 11.2 km s⁻¹ for the Earth)

Oscillations and waves

• SHM: a = −ω²x; x = A sin(ωt + φ); period T = 2π/ω = 1/f
• Maximum speed = ωA; maximum acceleration = ω²A; v = ω√(A² − x²)
• Mass on a spring: T = 2π√(m/k), independent of g. Simple pendulum: T = 2π√(l/g), independent of mass
• Total energy = ½kA²; potential energy = ½kx²; kinetic energy = ½k(A² − x²)
• wave speed = frequency × wavelength (v = fλ); adjacent nodes are λ/2 apart
• String or open pipe: f_n = nv/(2L), n = 1, 2, 3 ... Pipe closed at one end: f_n = nv/(4L), n = 1, 3, 5 ... only
• Beat frequency = |f₁ − f₂|; Doppler: f′ = f(v ± v_o)/(v ∓ v_s), upper signs for approach; level in dB = 10 log₁₀(I/I₀)

Heat, electricity and light

Thermodynamics and kinetic theory

• Zeroth law: two bodies each in thermal equilibrium with a third are in thermal equilibrium with each other
• First law: ΔU = Q − W (Q = heat supplied to the gas, W = work done by the gas); isothermal ideal gas: ΔU = 0, so Q = W; adiabatic: Q = 0
• Entropy change at constant temperature: ΔS = Q/T (J K⁻¹); total entropy rises in any irreversible process
• Third law: the entropy of a perfect crystal tends to zero as T tends to 0 K
• Ideal gas: pV = nRT with T in kelvin; C_p − C_v = R
• Kinetic theory: p = ⅓ρ⟨c²⟩; mean kinetic energy per molecule = (3/2)kT; v_rms = √(3RT/M)
• Carnot efficiency = 1 − T_cold/T_hot; efficiency = work out ÷ heat in

Electrostatics and current electricity

• Coulomb's law: F = kq₁q₂/r², with k = 1/(4πε₀) = 9 × 10⁹ N m² C⁻²
• Point charge: field E = kQ/r² (N C⁻¹ or V m⁻¹); potential V = kQ/r; work done W = qV
• Gauss's law: total flux through a closed surface = Q_enclosed/ε₀; E = 0 inside a charged conducting shell
• capacitance = charge ÷ p.d. (C = Q/V); parallel plates C = ε₀A/d; energy = ½CV² = Q²/(2C)
• Capacitors: in parallel C = C₁ + C₂; in series 1/C = 1/C₁ + 1/C₂. Resistors are the other way round
• V = IR; R = ρL/A; power P = VI = I²R = V²/R; terminal p.d. V = E − Ir
• Kirchhoff: sum of currents into a junction = sum out; sum of e.m.f.s round a loop = sum of IR drops. Balanced Wheatstone bridge: P/Q = R/S

Magnetism, induction and Maxwell's equations

• Force on a charge: F = qvB sin θ; circular path of radius r = mv/(qB)
• Force on a wire: F = BIL sin θ; parallel currents in the same direction attract, in opposite directions repel
• Biot–Savart: dB = (μ₀/4π) I dl sin θ/r². Long straight wire: B = μ₀I/(2πr). Solenoid (from Ampère's law ∮B·dl = μ₀I): B = μ₀nI
• Faraday: e.m.f. = −N dΦ/dt, with flux Φ = BA cos θ (Wb); rod moving across a field: e.m.f. = BLv
• Maxwell: ∇·E = ρ/ε₀ (Gauss); ∇·B = 0 (no isolated magnetic poles)
• Maxwell: ∇ × E = −∂B/∂t (Faraday); ∇ × B = μ₀J + μ₀ε₀∂E/∂t (Ampère–Maxwell)
• Electromagnetic waves: transverse, E ⊥ B, travel along E × B; c = 1/√(μ₀ε₀) = fλ = E₀/B₀ = 3 × 10⁸ m s⁻¹

Optics

• Reflection: angle of incidence = angle of reflection; turning a mirror through θ turns the reflected ray through 2θ
• Snell's law: n₁ sin θ₁ = n₂ sin θ₂; n = c/v; critical angle: sin C = 1/n (dense medium to air)
• Mirrors and thin lenses: 1/f = 1/u + 1/v (real distances positive, virtual negative); mirror f = R/2; magnification m = v/u
• power of a lens = 1 ÷ focal length in metres (dioptres); thin lenses in contact: 1/f = 1/f₁ + 1/f₂
• Young's double slit: fringe spacing = λD/d; bright where path difference = nλ, dark where it is (n + ½)λ
• Single slit minima: a sin θ = nλ. Diffraction grating maxima: d sin θ = nλ, with d = 1 ÷ (lines per metre)
• Malus's law: I = I₀ cos²θ for polarised light; an ideal polariser passes half of unpolarised light; Brewster's angle: tan θ_B = n

Modern physics

Special relativity

• Lorentz factor: γ = 1/√(1 − v²/c²), always ≥ 1
• Useful values: v = 0.6c gives γ = 1.25; v = 0.8c gives γ = 5/3; γ = 2 at v = (√3/2)c = 0.866c
• Time dilation: Δt = γΔt₀, where the proper time Δt₀ is measured by a clock present at both events
• Length contraction: L = L₀/γ, along the direction of motion only (L₀ = proper length, measured at rest)
• Momentum p = γmv; total energy E = γmc²; rest energy E₀ = mc²
• Kinetic energy = (γ − 1)mc²
• E² = (pc)² + (mc²)²; for a photon E = pc

Quantum physics and atomic structure

• Photon energy E = hf = hc/λ, with h = 6.6 × 10⁻³⁴ J s and hc = 1240 eV nm; photon momentum p = h/λ
• Photoelectric equation: KE_max = hf − φ = eV_s; threshold frequency f₀ = φ/h. Intensity changes the number of electrons, not their energy
• de Broglie wavelength = Planck constant ÷ momentum (λ = h/p = h/(mv))
• Compton shift: Δλ = (h/m_ec)(1 − cos θ); the scattered photon has the longer wavelength
• Uncertainty principle: ΔxΔp ≥ ħ/2, where ħ = h/(2π)
• Bohr: angular momentum L = nh/(2π); E_n = −13.6 Z²/n² eV; orbit radius ∝ n²; photon energy = difference of two levels
• 1/λ = R(1/n₁² − 1/n₂²): Lyman ends on n = 1 (ultraviolet), Balmer on n = 2 (visible), Paschen on n = 3 (infrared). Infinite well: E_n ∝ n²

Nuclear and particle physics

• Mass number A = Z + N; nuclear radius R = R₀A^1/3, with R₀ ≈ 1.2 × 10⁻¹⁵ m
• binding energy = mass defect × c²; 1 u is equivalent to 931.5 MeV; binding energy per nucleon peaks at about 8.8 MeV for A near 56 to 62
• Decay law: N = N₀e^−λt = N₀(½)^t/T; half-life T = 0.693/λ; mean life = 1/λ; activity = λN (Bq)
• Alpha: A falls by 4, Z falls by 2. Beta-minus: A unchanged, Z rises by 1. Gamma: A and Z unchanged
• Ionising power: alpha > beta > gamma; penetrating power is the reverse; gamma is not deflected by electric or magnetic fields
• Reactor: slow neutrons cause fission of U-235; the moderator slows the neutrons and the control rods absorb them. Fusion needs very high temperature to overcome the repulsion of the positive nuclei
• Proton = uud, neutron = udd; leptons include the electron, muon, tau and neutrinos; carriers: gluon (strong), photon (electromagnetic), W and Z (weak); beta-minus: d → u + e⁻ + antineutrino

Solid-state physics and electronics

• Energy gap: about 1.1 eV for silicon and 0.7 eV for germanium; several eV for an insulator. Light creates an electron–hole pair only if hc/λ ≥ E_g
• n-type: donor (Group V) impurity, majority carriers are electrons. p-type: acceptor (Group III) impurity, majority carriers are holes
• Forward bias narrows the depletion layer and the diode conducts (about 0.7 V across silicon); reverse bias widens it and almost no current flows
• Zener diode: used in reverse breakdown to hold a voltage steady. Ripple frequency: half-wave = supply frequency; full-wave = 2 × supply frequency
• Transistor: I_E = I_B + I_C; β = I_C/I_B; α = I_C/I_E = β/(β + 1). Active region: emitter–base junction forward biased, collector–base junction reverse biased
• Gates: AND gives 1 only if both inputs are 1; OR gives 0 only if both are 0; NAND and NOR are their inverses. De Morgan: NOT (A AND B) = (NOT A) OR (NOT B)
• Voltage gain in dB = 20 log₁₀(V_out/V_in); a common-emitter amplifier inverts (180° phase shift); inverting op-amp gain = −R_f/R_in