IB Diploma Physics formulas

Space, time and motion

Kinematics

• average speed = distance ÷ time; average velocity = displacement ÷ time
• v = u + at; s = ut + ½at²; v² = u² + 2as; s = ½(u + v)t (constant acceleration only)
• Gradient of a displacement–time graph = velocity; gradient of a velocity–time graph = acceleration
• Area under a velocity–time graph = displacement; area under an acceleration–time graph = change in velocity
• Projectile launched at speed u and angle θ: horizontal component u cos θ stays constant; vertical component starts at u sin θ and changes by 9.8 m s⁻¹ every second
• At the highest point the vertical velocity is zero, but the horizontal velocity and the acceleration g are unchanged
• At terminal speed: drag = weight, resultant force = 0, acceleration = 0

Forces and momentum

• resultant force = mass × acceleration (F = ma); resultant force = change in momentum ÷ time (F = Δp/Δt)
• momentum = mass × velocity (p = mv), unit kg m s⁻¹; impulse = FΔt = Δp = area under a force–time graph
• weight = mg; Hooke's law: force = spring constant × extension (F = kx)
• Friction: static F_f ≤ μ_sF_N (it only matches the push until its maximum); dynamic F_f = μ_dF_N
• Stokes' law: drag on a sphere F = 6πηrv; buoyancy force = density of fluid × volume displaced × g (F = ρVg)
• Elastic collision: kinetic energy conserved. Inelastic: kinetic energy falls. Explosion: kinetic energy rises. Momentum is conserved in all three
• Circular motion: v = ωr; ω = 2π/T = 2πf (rad s⁻¹); a = v²/r = ω²r; F = mv²/r

Work, energy and power

• work done = force × distance × cos θ (W = Fs cos θ), unit J; θ is the angle between the force and the displacement
• Work done = area under a force–displacement graph
• kinetic energy = ½mv² = p²/2m; gravitational potential energy change = mgΔh; elastic potential energy = ½kx²
• power = work done ÷ time (P = W/t), unit W; for a constant velocity, P = Fv
• efficiency = useful energy (or power) out ÷ total energy (or power) in
• specific energy in J kg⁻¹; energy density in J m⁻³; energy density = specific energy × density
• Nuclear fuels such as uranium have a far higher specific energy than fossil fuels

The particulate nature of matter

Thermal energy transfers

• density = mass ÷ volume (ρ = m/V), unit kg m⁻³; T in K = temperature in °C + 273
• mean kinetic energy of a particle = (3/2)k_BT, with T in kelvin
• energy = mass × specific heat capacity × temperature change (Q = mcΔT); change of state: Q = mL
• Conduction: rate of energy transfer = kAΔT ÷ Δx, where k is the thermal conductivity, A the area and Δx the thickness
• Stefan–Boltzmann law: luminosity L = σAT⁴; for a sphere A = 4πr²
• Wien's law: λ_max × T = 2.9 × 10⁻³ m K
• apparent brightness b = L ÷ (4πd²), unit W m⁻²

Greenhouse effect

• Solar constant S is about 1360 W m⁻²; mean incoming intensity over the whole Earth = S/4
• albedo = total scattered power ÷ total incident power (no unit; Earth's mean is about 0.30; ice and cloud high, ocean and forest low)
• mean absorbed intensity = (1 − albedo) × S/4
• emissivity e = power radiated per unit area ÷ σT⁴, so power = eσAT⁴; a black body has e = 1
• Radiative equilibrium: power absorbed = power radiated, so the mean temperature stays constant
• Main greenhouse gases: water vapour, carbon dioxide, methane, nitrous oxide. Nitrogen and oxygen are not greenhouse gases
• Human sources: burning fossil fuels and deforestation (CO₂); cattle, rice fields and landfill (methane); fertilisers (nitrous oxide)

Gas laws

• pressure = force ÷ area (p = F/A), unit Pa; number of moles n = N ÷ N_A, with N_A = 6.02 × 10²³ mol⁻¹
• Fixed mass: pV = constant at constant T; V/T = constant at constant p; p/T = constant at constant V; in general p₁V₁/T₁ = p₂V₂/T₂
• Ideal gas equation: pV = nRT = Nk_BT (p in Pa, V in m³, T in K)
• Kinetic model: p = ⅓ρv², where ρ is the gas density and v² is the mean square speed
• Internal energy of an ideal monatomic gas: U = (3/2)nRT = (3/2)Nk_BT
• Real gases deviate most at high pressure and low temperature, where molecules are close together and intermolecular forces matter

Current and circuits

• current = charge ÷ time (I = Δq/Δt); potential difference = work done ÷ charge (V = W/q); resistance R = V/I
• resistance = resistivity × length ÷ cross-sectional area (R = ρL/A), ρ in Ω m, A in m²
• power P = VI = I²R = V²/R; energy = power × time
• Series: R = R₁ + R₂, same current. Parallel: 1/R = 1/R₁ + 1/R₂, same potential difference
• Kirchhoff: ΣI = 0 at a junction (charge conserved); ΣV = 0 round a loop (energy conserved)
• Potential divider: V₁ = V × R₁ ÷ (R₁ + R₂). An NTC thermistor's resistance falls when it gets hotter; an LDR's resistance falls in brighter light
• ε = I(R + r); terminal potential difference V = ε − Ir

Wave behaviour

Simple harmonic motion

• Condition for SHM: acceleration ∝ −displacement (a = −ω²x)
• period T = time ÷ number of oscillations; frequency f = 1/T, unit Hz; ω = 2π/T = 2πf
• Mass–spring system: T = 2π√(m/k)
• Simple pendulum (small angles): T = 2π√(l/g); the mass of the bob does not matter
• At equilibrium: displacement and acceleration are zero, speed and kinetic energy are greatest
• At maximum displacement: speed and kinetic energy are zero, acceleration and potential energy are greatest
• total energy = kinetic energy + potential energy = constant (no damping)

Wave model

• wave speed = frequency × wavelength (v = fλ)
• frequency = 1 ÷ period (f = 1/T), unit Hz
• Wavelength λ: distance between adjacent crests (or adjacent compressions), measured on a displacement–distance graph
• Period T: time between adjacent peaks on a displacement–time graph
• Amplitude: maximum displacement from the equilibrium position, not the height from crest to trough
• Mechanical waves (sound, water, waves on a rope) need a medium; electromagnetic waves do not
• Electromagnetic waves are transverse and travel at c = 3.0 × 10⁸ m s⁻¹ in a vacuum

Wave phenomena

• refractive index n = c ÷ v (speed of light in vacuum ÷ speed in the medium)
• Snell's law: n₁ sin θ₁ = n₂ sin θ₂, with angles measured from the normal
• Critical angle: sin c = n₂ ÷ n₁ (= 1/n for a medium into air)
• Total internal reflection needs light going from the higher to the lower refractive index, at an angle of incidence greater than the critical angle
• Sources in phase: constructive interference when path difference = nλ; destructive when path difference = (n + ½)λ
• Double slit: fringe spacing s = λD ÷ d (D = slit-to-screen distance, d = slit separation)

Standing waves and resonance

• Distance between adjacent nodes (or adjacent antinodes) = half a wavelength (λ/2); node to the next antinode = λ/4
• Fixed end of a string and closed end of a pipe: node. Open end of a pipe: antinode
• String fixed at both ends, or pipe open at both ends: λ_n = 2L ÷ n and f_n = nv ÷ (2L), with n = 1, 2, 3, ...
• Pipe closed at one end: λ_n = 4L ÷ n and f_n = nv ÷ (4L), with odd n only (1, 3, 5, ...)
• Resonance: driving frequency = natural frequency, so the amplitude is a maximum
• More damping: the resonance peak is lower and broader
• Light damping: oscillates with slowly falling amplitude. Critical damping: returns to equilibrium in the shortest time with no oscillation. Heavy damping: returns slowly with no oscillation

Doppler effect

• Source and observer approaching: observed frequency higher, observed wavelength shorter
• Source and observer separating: observed frequency lower, observed wavelength longer
• Moving source: wavefronts are circles closer together in front of the source and further apart behind it
• For light: change in wavelength ÷ emitted wavelength ≈ relative speed ÷ speed of light (Δλ/λ = Δf/f ≈ v/c), valid when v is much smaller than c
• Δλ = observed wavelength − emitted (laboratory) wavelength. Positive: redshift, moving away. Negative: blueshift, moving towards
• The speed of the wave in the medium depends only on the medium, not on the motion of the source

Fields

Gravitational fields

• Gravitational force = G × mass 1 × mass 2 ÷ distance² (F = Gm₁m₂/r²), always attractive
• Field strength = force ÷ mass = G × mass of the body ÷ distance² (g = F/m = GM/r²), unit N kg⁻¹
• Kepler's first law: each planet moves in an ellipse with the Sun at one focus
• Kepler's second law: the line from the Sun to a planet sweeps out equal areas in equal times, so the planet is fastest when closest to the Sun
• Kepler's third law: period² is proportional to mean orbital radius³ (T² ∝ r³)
• Circular orbit: GMm/r² = mv²/r, which gives v = √(GM/r) and T² = 4π²r³ ÷ (GM)
• Field lines closer together mean a stronger field; radial lines spread out with distance

Electric and magnetic fields

• Coulomb's law: force = k × charge 1 × charge 2 ÷ distance² (F = kq₁q₂/r²), k = 8.99 × 10⁹ N m² C⁻²
• Electric field strength = force ÷ charge (E = F/q), unit N C⁻¹ or V m⁻¹
• Uniform field between parallel plates: field strength = potential difference ÷ plate separation (E = V/d); lines are parallel and equally spaced, from the positive plate to the negative plate
• Charge is quantized: q = ne, with n a whole number and e = 1.6 × 10⁻¹⁹ C
• Stationary oil drop in Millikan's experiment: electric force = weight (qE = mg)
• Identical conducting spheres that touch share the total charge equally
• Straight wire: field lines are concentric circles around the wire. Solenoid: strong, uniform field inside, parallel to its axis, like a bar magnet outside

Motion in electromagnetic fields

• Electric force = charge × field strength (F = qE); kinetic energy gained = charge × potential difference (qV)
• Magnetic force on a moving charge: F = qvB sin θ, where θ is the angle between the velocity and the field
• Circular path: qvB = mv²/r, so radius r = mv ÷ (qB)
• Crossed fields, no deflection: qE = qvB, so speed v = E ÷ B, for any charge
• Force on a wire: F = BIL sin θ, where θ is the angle between the current and the field
• Parallel wires: force per unit length F/L = μ₀I₁I₂ ÷ (2πr); currents in the same direction attract, opposite directions repel
• The magnetic force is at right angles to both the field and the velocity (or the current); on a moving charge it does no work, so the speed stays constant

Nuclear and quantum physics

Structure of the atom

• Nuclide notation ^A_ZX: A = protons + neutrons, Z = protons, neutrons = A − Z; a neutral atom has Z electrons
• Photon energy = Planck constant × frequency (E = hf) = hc ÷ λ
• Photon energy in a transition = difference between the two energy levels (ΔE = E_upper − E_lower)
• 1 eV = 1.60 × 10⁻¹⁹ J
• Emission spectrum: bright lines on a dark background, from a hot low-pressure gas
• Absorption spectrum: dark lines on a continuous spectrum, at the same wavelengths as the emission lines of that gas
• Most alpha particles pass straight through: mostly empty space. A few deflected by more than 90°: small, dense, positive nucleus

Radioactive decay

• Mass defect = total mass of the separate nucleons − mass of the nucleus; binding energy = mass defect × c² (E = Δmc²); 1 u = 931.5 MeV c⁻²
• Alpha decay: emits ⁴₂He, so A falls by 4 and Z falls by 2
• Beta-minus: neutron → proton + electron + antineutrino; A unchanged, Z rises by 1
• Beta-plus: proton → neutron + positron + neutrino; A unchanged, Z falls by 1
• Gamma: a photon is emitted; A and Z do not change
• Alpha: most ionizing, stopped by paper. Beta: stopped by a few mm of aluminium. Gamma: least ionizing, reduced by thick lead
• After n half-lives the fraction remaining is (1/2)ⁿ; subtract the background count before using half-life

Fission

• In a fission equation the total nucleon number and the total proton number are the same on both sides
• Energy released = mass lost × c²; about 200 MeV per fission of uranium-235
• Fuel rods: contain the fissile material, such as uranium-235
• Moderator (water or graphite): slows fast neutrons so they are more likely to cause fission
• Control rods (boron or cadmium): absorb neutrons; lowering them reduces the rate of fission
• Heat exchanger: passes thermal energy from the reactor coolant to water, making steam for the turbines
• Shielding (thick concrete and steel): absorbs neutrons and gamma radiation. Waste stays radioactive for thousands of years and needs secure long-term storage

Fusion and stars

• Distance in parsecs = 1 ÷ parallax angle in arcseconds (d = 1/p); works only for nearby stars, because small angles are hard to measure
• Luminosity = σ × surface area × temperature⁴ (L = σAT⁴ = 4πR²σT⁴), with T in kelvin
• Comparing two stars: R is proportional to √L ÷ T²
• Sun-like star: main sequence → red giant → planetary nebula → white dwarf
• Massive star: main sequence → red supergiant → supernova → neutron star or black hole
• HR diagram: upper left hot and luminous; upper right cool but luminous, so very large; lower left hot but dim, so very small