Equations Not in the Data Booklet: HL edition

Fractional and percentage uncertainty

Symbolsx = measured value (any unit)Δx = absolute uncertainty in x (same unit as x)fractional, percentage = relative size of the uncertainty (no unit)

Uncertainty from repeated readings (half the range)

SymbolsΔx = absolute uncertainty (same unit as x)x_max, x_min = largest and smallest of the repeated readings (unit of x)

Multiplying by an exact constant

Symbolsy = calculated quantity (unit of y)k = exact constant, with no uncertainty of its own (unit of y/x)x = measured quantity (unit of x)Δx, Δy = absolute uncertainties in x and y (units of x, y)

Uncertainty in a gradient (from the steepest and shallowest lines through all error bars)

Symbolsm = gradient of the line of best fit (y-axis unit ÷ x-axis unit)m_max, m_min = gradients of the steepest and shallowest lines (same as m)Δm = uncertainty in the gradient (same as m)

Percentage difference from an accepted value

Symbolsx_exp = your experimental value (any unit)x_accepted = accepted (literature) value (same unit)% difference = size of the discrepancy (no unit)

SymbolsA_sphere = surface area of the sphere (m²)r = radius (m)

Small-angle approximation (θ in radians, roughly θ < 0.1 rad; behind s = λD/d and θ = λ/b)

Symbolsθ = angle (rad)

Arc length and angle

Symbolss = arc length (m)r = radius (m)θ = angle subtended at the centre (rad)ω = angular speed (rad s⁻¹)Δθ = angle turned through (rad)Δt = time taken (s)

Linearising for graphs

Symbolsy, x = the two variables plotted (any units)k = constantn = power of x (no unit)A = value of y when x = 0 (unit of y)λ = decay constant of the exponential (unit of 1/x)e = base of natural logarithms, ≈ 2.718 (no unit)ln = natural logarithm (log to base e)

Definitions of average speed, average velocity and acceleration (the booklet gives only the four suvat equations)

Symbolsdistance, time = total path length and time taken (m, s)v = average velocity (m s⁻¹)Δs = displacement (m)Δt = time interval (s)a = average acceleration (m s⁻²)Δv = change in velocity (m s⁻¹)

Projectile components (launch speed u at angle θ above the horizontal)

Symbolsu = launch speed (m s⁻¹)θ = launch angle above the horizontal (° or rad)u_x, u_y = horizontal and vertical components of the launch velocity (m s⁻¹)a_x, a_y = horizontal and vertical acceleration (upward positive) (m s⁻²)g = gravitational field strength (m s⁻² or N kg⁻¹)

Derived Level ground, no air resistance: time of flight, maximum height, range

SymbolsT = time of flight (s)H = maximum height above the launch point (m)R = horizontal range (m)u = launch speed (m s⁻¹)θ = launch angle above the horizontal (° or rad)g = gravitational field strength (m s⁻² or N kg⁻¹)

Equilibrium (Newton’s first law)

SymbolsΣF = resultant (net) force (N)a = acceleration (m s⁻²)

Conservation of momentum (isolated system)

Symbolsm₁, m₂ = masses of the two objects (kg)u₁, u₂ = velocities before the interaction (signed) (m s⁻¹)v₁, v₂ = velocities after the interaction (signed) (m s⁻¹)

Weight components on a slope (angle θ to the horizontal)

SymbolsF_∥ = component of the weight down the slope (N)F_⊥ = component of the weight perpendicular to the slope (N)m = mass (kg)g = gravitational field strength (m s⁻² or N kg⁻¹)θ = angle of the slope to the horizontal (° or rad)

Derived Centripetal force (from F = ma and a = v²/r)

SymbolsF = centripetal (resultant) force (N)m = mass (kg)v = speed (m s⁻¹)r = radius of the circle (m)ω = angular speed (rad s⁻¹)

Normal force in a lift (upward acceleration a positive)

SymbolsN = normal force from the floor (N)m = mass of the person or object (kg)g = gravitational field strength (m s⁻² or N kg⁻¹)a = acceleration of the lift, upward positive (m s⁻²)

Derived Terminal velocity of a sphere (weight = drag + buoyancy, using Stokes’ law)

Symbolsm = mass of the sphere (kg)g = gravitational field strength (m s⁻² or N kg⁻¹)η = viscosity of the fluid (Pa s)r = radius of the sphere (m)v_t = terminal speed (m s⁻¹)ρ_f = density of the fluid (kg m⁻³)V = volume of the sphere (m³)ρ_s = density of the sphere (kg m⁻³)

Conservation of energy (no dissipative forces)

SymbolsE_k = kinetic energy (J)E_p = gravitational potential energy (J)m = mass (kg)v = speed (m s⁻¹)g = gravitational field strength (m s⁻² or N kg⁻¹)Δh = change in height (m)

Work–energy theorem

SymbolsW_net = work done by the resultant force (J)ΔE_k = change in kinetic energy (J)

Power as rate of energy transfer (the booklet writes power in terms of work done; this is the energy form)

SymbolsP = power (W)E = energy transferred (J)t = time taken (s)

Rolling without slipping (v = ωr is in the booklet under circular motion and also holds for the centre of a rolling object; the acceleration form is not printed)

Symbolsa = linear acceleration of the centre (m s⁻²)α = angular acceleration (rad s⁻²)r = radius of the rolling object (m)

Total kinetic energy of a rolling object

SymbolsE_k = total kinetic energy (J)m = mass (kg)v = speed of the centre of mass (m s⁻¹)I = moment of inertia about the axis through the centre (kg m²)ω = angular speed (rad s⁻¹)

Derived Rolling down a height h from rest

Symbolsm = mass (kg)g = gravitational field strength (m s⁻² or N kg⁻¹)h = vertical height descended (m)v = speed of the centre at the bottom (m s⁻¹)I = moment of inertia (kg m²)r = radius of the rolling object (m)

Rotational work and power

SymbolsW = work done by the torque (J)τ = torque (N m)Δθ = angle turned through (rad)P = power (W)ω = angular speed (rad s⁻¹)

Rotational equilibrium (both conditions together)

SymbolsΣF = resultant force (N)Στ = resultant torque about any axis (N m)

Conservation of angular momentum (no external torque) and angular momentum of a point mass

SymbolsI₁, I₂ = moments of inertia before and after (kg m²)ω₁, ω₂ = angular speeds before and after (rad s⁻¹)L = angular momentum (kg m² s⁻¹)m = mass of the point object (kg)v = its speed (m s⁻¹)r = perpendicular distance from the axis to its line of motion (m)

Inverse Lorentz transformations (from S′ back to S)

Symbolsx, t = position and time of an event in frame S (m, s)x′, t′ = position and time of the same event in frame S′ (m, s)v = velocity of S′ relative to S (m s⁻¹)γ = Lorentz factor (no unit)c = speed of light in a vacuum (constant, m s⁻¹)

Interval forms (use these for two events)

SymbolsΔx, Δt = distance and time between two events in S (m, s)Δx′, Δt′ = distance and time between the same events in S′ (m, s)v = velocity of S′ relative to S (m s⁻¹)γ = Lorentz factor (no unit)c = speed of light in a vacuum (constant, m s⁻¹)

Derived Relativity of simultaneity (events simultaneous in S, Δt = 0)

SymbolsΔt′ = time between the events in S′ (s)γ = Lorentz factor (no unit)v = velocity of S′ relative to S (m s⁻¹)Δx = distance between the events in S (m)c = speed of light in a vacuum (constant, m s⁻¹)

Heating with a known power (e.g. an electric heater, P = IV)

SymbolsP = power of the heater (W)t = heating time (s)m = mass heated (kg)c = specific heat capacity (J kg⁻¹ K⁻¹)ΔT = temperature change (K or °C)L = specific latent heat (J kg⁻¹)

Mixing: energy lost by the hot body = energy gained by the cold body (no losses)

Symbolsm₁, m₂ = masses of the hot and cold bodies (kg)c₁, c₂ = their specific heat capacities (J kg⁻¹ K⁻¹)T₁ = initial temperature of the hot body (K or °C)T₂ = initial temperature of the cold body (K or °C)T_f = final common temperature (K or °C)

Derived Power radiated per unit area by a black body

SymbolsP = power radiated (W)A = surface area of the emitter (m²)σ = Stefan–Boltzmann constant (constant, W m⁻² K⁻⁴)T = absolute temperature of the surface (K)

Intensity

SymbolsI = intensity (W m⁻²)P = power (W)A = area perpendicular to the direction of energy flow (m²)

Mean incoming solar intensity on the whole Earth (disc area πR² intercepts, sphere area 4πR² shares)

SymbolsI_mean = incoming intensity averaged over the whole surface of the Earth (W m⁻²)S = solar constant: intensity arriving at the top of the atmosphere, on a surface facing the Sun (W m⁻²)

Derived Energy balance for an equilibrium planet (albedo α, emissivity e)

Symbolsα = albedo (fraction reflected) (no unit)S = solar constant (W m⁻²)e = emissivity (no unit)σ = Stefan–Boltzmann constant (constant, W m⁻² K⁻⁴)T = mean surface temperature of the planet (K)

Amount of substance from mass (M = molar mass in kg mol⁻¹)

Symbolsn = amount of substance (mol)m = mass of gas (kg)M = molar mass (kg mol⁻¹)

Derived Single-change gas laws (fixed amount of ideal gas)

Symbolsp₁, p₂ = pressure before and after (Pa)V₁, V₂ = volume before and after (m³)T₁, T₂ = absolute temperature before and after (K)

Link between the constants

SymbolsR = molar gas constant (constant, J mol⁻¹ K⁻¹)N_A = Avogadro constant (constant, mol⁻¹)k_B = Boltzmann constant (constant, J K⁻¹)

Derived rms speed of gas molecules (from p = ⅓ρv² and pV = nRT)

Symbolsv_rms = root mean square speed of the molecules (m s⁻¹)R = molar gas constant (constant, J mol⁻¹ K⁻¹)T = absolute temperature (K)M = molar mass (kg mol⁻¹)k_B = Boltzmann constant (constant, J K⁻¹)m_molecule = mass of one molecule (kg)

Derived Adiabatic change for a monatomic ideal gas, in terms of T

SymbolsT = absolute temperature (K)V = volume (m³)2/3 = exponent for a monatomic gas (no unit)

Heat engines: energy in, work out and energy rejected (the booklet gives efficiency as useful work over energy input; combine it with the energy balance)

SymbolsQ_H = energy taken in from the hot reservoir (J)W = useful work done by the engine (J)Q_C = energy rejected to the cold reservoir (J)η = efficiency (no unit)

Second law (isolated system)

SymbolsΔS = change in entropy of the isolated system (J K⁻¹)

Kirchhoff’s laws

SymbolsΣI_in, ΣI_out = total current into and out of a junction (A)Σε = sum of the emfs around a closed loop (V)ΣIR = sum of the potential differences across the resistances in that loop (V)I = current (A)R = resistance (Ω)ε = emf (V)

Potential divider

SymbolsV_out = output potential difference, across R₂ (V)V_in = potential difference across the whole divider (V)R₁, R₂ = the two resistances (Ω)

Derived Terminal potential difference (lost volts = Ir)

SymbolsV = terminal potential difference (V)ε = emf of the cell (V)I = current (A)r = internal resistance (Ω)

Charge and energy (the booklet gives I = Δq/Δt; these are not printed)

SymbolsE = energy transferred (J)V = potential difference (V)I = current (A)t = time (s)P = power (W)q = charge (C)N = number of electrons (no unit)e = elementary charge (constant, C)

Derived ω for the two standard oscillators

Symbolsω = angular frequency (rad s⁻¹)k = spring constant (N m⁻¹)m = oscillating mass (kg)g = gravitational field strength (m s⁻² or N kg⁻¹)l = length of the pendulum (m)

Maximum acceleration (at the extremes, from a = −ω²x)

Symbolsa_max = maximum magnitude of acceleration (m s⁻²)a = acceleration (m s⁻²)ω = angular frequency (rad s⁻¹)x = displacement from equilibrium (m)x₀ = amplitude (m)

HL Maximum speed (at the equilibrium position)

Symbolsv_max = maximum speed (m s⁻¹)ω = angular frequency (rad s⁻¹)x₀ = amplitude (m)

HL Phase difference from a time difference

SymbolsΔϕ = phase difference (rad)Δt = time between corresponding points of the two oscillations (s)T = period (s)

HL Derived Kinetic energy at displacement x (the booklet gives E_T and E_p only)

SymbolsE_k = kinetic energy (J)m = mass (kg)ω = angular frequency (rad s⁻¹)x₀ = amplitude (m)x = displacement from equilibrium (m)

Phase difference between two points on a wave

SymbolsΔϕ = phase difference (rad)Δx = separation of the two points along the direction of travel (m)λ = wavelength (m)

Intensity and amplitude

SymbolsI = intensity (W m⁻²)A = amplitude (m, or unit of the oscillating quantity)

Inverse-square law for a point source

SymbolsI = intensity at distance r (W m⁻²)P = power emitted by the source (W)r = distance from the source (m)

Absolute refractive index

Symbolsn = absolute refractive index of the medium (no unit)c = speed of light in a vacuum (constant, m s⁻¹)v = speed of light in the medium (m s⁻¹)

Critical angle (light going from medium n₁ into a less dense medium n₂)

Symbolsθ_c = critical angle (°)n₁ = refractive index of the medium the light is in (the denser one) (no unit)n₂ = refractive index of the less dense medium (no unit)n = refractive index of the medium, with air taken as 1 (no unit)

HL Grating spacing and highest order visible (N lines per metre)

Symbolsd = grating spacing (distance between adjacent lines) (m)N = number of lines per metre (m⁻¹)n_max = highest order seen, a whole number (no unit)λ = wavelength (m)

HL Single slit: width of the central maximum (screen at distance D)

Symbolsλ = wavelength (m)b = slit width (m)D = distance from slit to screen (m)angular width (rad)linear width (m)

String fixed at both ends, or pipe open at both ends (n = 1, 2, 3, …)

Symbolsλ_n = wavelength of the nth harmonic (m)L = length of the string or pipe (m)n = harmonic number (no unit)f_n = frequency of the nth harmonic (Hz)v = wave speed (m s⁻¹)

Pipe closed at one end (odd harmonics only, n = 1, 2, 3, …)

Symbolsλ = wavelength (m)L = length of the pipe (m)n = counting number; 2n − 1 is the harmonic number (no unit)f = frequency (Hz)v = speed of sound in the air column (m s⁻¹)

Spacings

Symbolsλ = wavelength (m)

SL+HL Derived Reflection from a moving object (radar, ultrasound; v ≪ wave speed c)

SymbolsΔf = difference between emitted and received frequency (Hz)v = speed of the reflector along the line of sight (m s⁻¹)c = wave speed (speed of light for radar, speed of sound for ultrasound) (m s⁻¹)f = emitted frequency (Hz)

HL Source and observer both moving (upper signs when they approach each other)

Symbolsf′ = frequency received by the observer (Hz)f = frequency emitted by the source (Hz)v = speed of the wave in the medium (m s⁻¹)u_o = speed of the observer (m s⁻¹)u_s = speed of the source (m s⁻¹)

SL+HL Orbital radius from the planet’s centre

Symbolsr = orbital radius, measured from the planet’s centre (m)R_planet = radius of the planet (m)h = height above the surface (m)

SL+HL Derived Circular orbit (gravity provides the centripetal force; the orbital speed itself is printed in the higher level part of the booklet, but the derivation is not)

SymbolsG = gravitational constant (constant, N m² kg⁻²)M = mass of the planet (central body) (kg)m = mass of the satellite (kg)r = orbital radius (m)v = orbital speed (m s⁻¹)

SL+HL Derived Kepler’s third law

SymbolsT = orbital period (s)G = gravitational constant (constant, N m² kg⁻²)M = mass of the central body (kg)r = orbital radius (m)

HL Derived Energies of a satellite in circular orbit

SymbolsE_k = kinetic energy of the satellite (J)E_total = total (kinetic + potential) energy (J)G = gravitational constant (constant, N m² kg⁻²)M = mass of the planet (kg)m = mass of the satellite (kg)r = orbital radius (m)

HL Derived Energy needed to move to a higher circular orbit

SymbolsΔE = energy that must be supplied (J)G = gravitational constant (constant, N m² kg⁻²)M = mass of the planet (kg)m = mass of the satellite (kg)r₁ = initial orbital radius (m)r₂ = final orbital radius (m)

SL+HL Derived Field of a point charge (from F = kq₁q₂/r² and E = F/q)

SymbolsE = electric field strength (N C⁻¹ or V m⁻¹)k = Coulomb constant (constant, N m² C⁻²)Q = the point charge creating the field (C)r = distance from the charge (m)

SL+HL Quantisation of charge and the electronvolt

Symbolsq = charge (C)n = number of elementary charges, a whole number (no unit)e = elementary charge (constant, C)eV = electronvolt, a unit of energy

SL+HL Derived Magnetic field of a long straight wire (from the force per unit length between two wires)

SymbolsB = magnetic flux density (T)μ₀ = permeability of free space (constant, T m A⁻¹)I = current in the wire (A)r = perpendicular distance from the wire (m)

Acceleration of a charge in an electric field (from E = F/q, which is in the booklet)

Symbolsa = acceleration (m s⁻²)q = charge of the particle (C)E = electric field strength (N C⁻¹ or V m⁻¹)m = mass of the particle (kg)

Acceleration through a potential difference V (from rest)

Symbolsq = charge of the particle (C)V = accelerating potential difference (V)m = mass of the particle (kg)v = final speed (m s⁻¹)

Derived Circular motion in a uniform magnetic field (qvB = mv²/r)

Symbolsr = radius of the circular path (m)m = mass of the particle (kg)v = speed (m s⁻¹)q = charge (C)B = magnetic flux density (T)T = period of the circular motion (s)

Derived Velocity selector (crossed fields, undeflected particle: qE = qvB)

Symbolsv = speed of the particles that pass undeflected (m s⁻¹)E = electric field strength (V m⁻¹)B = magnetic flux density (T)

Flux linkage

SymbolsN = number of turns (no unit)Φ = magnetic flux through one turn (Wb)flux linkage (Wb)

Coil of N turns, area A, rotating at ω in a uniform field B

Symbolsε = induced emf at time t (V)ε_max = peak emf (V)N = number of turns (no unit)B = magnetic flux density (T)A = area of the coil (m²)ω = angular speed of the coil (rad s⁻¹)t = time, from when the coil’s plane is perpendicular to the field (s)

SL+HL Photon energy from wavelength (the booklet gives E = hf and a value for hc, but not this form)

SymbolsE = photon energy (J)h = Planck constant (constant, J s)c = speed of light in a vacuum (constant, m s⁻¹)λ = wavelength (m)

SL+HL Photon emitted or absorbed in a transition

SymbolsΔE = energy difference between the levels, equal to the photon energy (J or eV)E_upper, E_lower = energies of the two levels (J or eV)h = Planck constant (constant, J s)f = photon frequency (Hz)c = speed of light in a vacuum (constant, m s⁻¹)λ = photon wavelength (m)

HL Derived Distance of closest approach (head-on alpha scattering, all E_k becomes E_p)

SymbolsE_k = initial kinetic energy of the alpha particle (J)k = Coulomb constant (constant, N m² C⁻²)Q = charge of the nucleus (C)q = charge of the alpha particle (C)d = distance of closest approach (m)

HL Derived Hydrogen transition wavelength (from E = −13.6/n² eV)

Symbolsh = Planck constant (constant, J s)c = speed of light in a vacuum (constant, m s⁻¹)λ = wavelength of the photon (m)n_i, n_f = initial and final energy levels, whole numbers (no unit)13.6 eV = ionisation energy of hydrogen from the ground state

HL Derived Nuclear density is about the same for all nuclei (from R = R₀A^1/3)

Symbolsρ = nuclear density (kg m⁻³)m_u = unified atomic mass unit (constant, kg)R₀ = nuclear radius constant, about 1.2 × 10⁻¹⁵ m (m)

Threshold frequency and stopping potential

Symbolsf₀ = threshold frequency (Hz)Φ = work function (J)h = Planck constant (constant, J s)e = elementary charge (constant, C)V_s = stopping potential (V)E_max = maximum kinetic energy of the photoelectrons (J)

Momentum of a photon (the booklet gives the de Broglie relation λ = h/p, not this energy form)

Symbolsp = momentum of the photon (kg m s⁻¹)E = photon energy (J)c = speed of light in a vacuum (constant, m s⁻¹)

Derived de Broglie wavelength from kinetic energy (non-relativistic; second form for a charge accelerated through V)

Symbolsλ = de Broglie wavelength (m)h = Planck constant (constant, J s)m = mass of the particle (kg)E_k = kinetic energy (J)e = charge of the particle (e for an electron or proton) (C)V = accelerating potential difference (V)

SL+HL Mass defect and binding energy

SymbolsΔm = mass defect (kg or u)Z = proton number (no unit)m_p = mass of a proton (kg or u)A = nucleon number (no unit)m_n = mass of a neutron (kg or u)M_nucleus = mass of the nucleus (kg or u)E_B = binding energy (J)c = speed of light in a vacuum (constant, m s⁻¹)

SL+HL Binding energy per nucleon

SymbolsE_B = binding energy of the nucleus (J or MeV)A = nucleon number (no unit)

SL+HL Unit shortcut (mass in u, energy in MeV; the booklet lists the energy equivalent of 1 u, this is how to use it)

SymbolsE = energy equivalent (MeV)Δm = mass difference (u)931.5 = energy equivalent of 1 u, in MeV

SL+HL Remaining fraction after n half-lives (n = t / T_½)

SymbolsN = number of undecayed nuclei remaining (no unit)N₀ = initial number of undecayed nuclei (no unit)n = number of half-lives elapsed (no unit)t = time elapsed (s)T_½ = half-life (s)

HL Mass and count rate decay the same way

Symbolsm = mass of the radioactive isotope remaining (kg)m₀ = initial mass (kg)C = count rate (s⁻¹)C₀ = initial count rate (s⁻¹)λ = decay constant (s⁻¹)t = time (s)e = base of natural logarithms, ≈ 2.718 (no unit)

HL Linear form for graphs (gradient = −λ)

SymbolsN = number of undecayed nuclei (no unit)N₀ = initial number (no unit)λ = decay constant (minus the gradient) (s⁻¹)t = time (s)ln = natural logarithm

Energy released in a reaction

SymbolsQ = energy released (J)Σm_before, Σm_after = total rest mass before and after the reaction (kg)c = speed of light in a vacuum (constant, m s⁻¹)

Power output of a reactor or power station

SymbolsP_out = useful (electrical) output power (W)η = overall efficiency (no unit)fissions per second (s⁻¹)energy per fission (J)

Derived Comparing two stars (from L = σAT⁴ with A = 4πR²)

SymbolsL₁, L₂ = luminosities of the two stars (W)R₁, R₂ = their radii (m)T₁, T₂ = their surface temperatures (K)

Derived Radius of a star

SymbolsR = radius of the star (m)L = luminosity (W)σ = Stefan–Boltzmann constant (constant, W m⁻² K⁻⁴)T = surface temperature (K)

Derived Distance from apparent brightness

Symbolsd = distance to the star (m)L = luminosity (W)b = apparent brightness (W m⁻²)