Physics Data Booklet

Physics data booklet For use during the course and in the examinations First assessment 2025 Version 1.1 Annotated by YPhysics (Version 1.0) Contents Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Mathematical equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Fundamental constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Metric (SI) multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Unit conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Electrical circuit symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Electromagnetic spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 A. Space, time and motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 B. The particulate nature of matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 C. Wave behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 D. Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 E. Nuclear and quantum physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Physics data booklet Mathematical equations 1 (bh ) where b is the base, h is the height 2 Area of a triangle A Area of a circle A = πr 2 where r is the radius Circumference of a circle C = 2πr Volume of a cuboid V lwh where l is the length, w is the width, h is the height Volume of a cylinder V r 2h Volume of a prism V Ah where A is the area of cross-section Volume of a sphere V= 4 3 πr 3 Area of the curved surface of a cylinder A 2 rh Vectors A AV θ Trigonometric relationships AH A cos AV A sin tan θ = AH sin θ cos θ sin2 θ + cos2 θ = 1 2 Physics data booklet Uncertainties If: y = a ± b If: y If: y Δy: absolute/raw uncertainty in y y: value of y then: ∆y = ∆a + ∆b ab c ∆y ∆a ∆b ∆c = + + then: y a b c an then: Δa: absolute/raw uncertainty in a a: value of a Δb: absolute/raw uncertainty in b b: value of b ∆y ∆a = n y a Δc: absolute/raw uncertainty in c c: value of c Fundamental constants Quantity Symbol Approximate value Acceleration of free fall g 9 .8 m s Gravitational constant G 6.67 × 10−11 Nm2 kg−2 Avogadro constant NA 6.02 × 1023 mol−1 Gas constant R 8.31JK 1 mol Boltzmann constant kB 1.38 × 10−23 JK −1 Permittivity of free space (Earth’s surface) 1 5.67 × 10−8 W m−2 K −4 Stefan–Boltzmann constant Coulomb constant 2 k 0 8.99 × 109 Nm2 C−2 8.85 × 10−12 C2 N−1 m−2 4π × 10−7 T mA −1 Permeability of free space Speed of light in vacuum c 3.00 × 108 m s−1 Planck constant h 6.63 × 10−34 Js Elementary charge e 1.60 × 10−19 C Electron rest mass me 9.110 × 10−31 kg = 0.000549 u = 0.511MeV c −2 Proton rest mass mp 1.673 × 10−27 kg = 1.007276 u = 938 MeV c −2 Neutron rest mass mn 1.675 × 10−27 kg = 1.008665 u = 940 MeV c −2 (Unified) atomic mass unit u 1.661× 10−27 kg = 931.5 MeV c −2 Solar constant S 1.36 × 103 W m−2 Fermi radius R0 1.20 × 10−15 m Physics data booklet 3 Metric (SI) multipliers Prefix Abbreviation Value peta P 1015 tera T 1012 giga G 109 mega M 106 kilo k 103 hecto h 102 deca da 101 deci d 10 1 centi c 10 2 milli m 10 3 micro μ 10 6 nano n 10 9 pico p 10 12 femto f 10 15 Unit conversions 1 radian (rad) ≡ 180° π Temperature (K ) = temperature ( °C ) + 273 1 light year ( ly ) = 9.46 × 1015 m 1 parsec (pc ) = 3.26 ly 1 astronomical unit ( AU) = 1.50 × 1011 m 1 kilowatt-hour ( kWh ) = 3.60 × 106 J h c = 1.99 × 10−25 Jm = 1.24 × 10 −6 eV m 4 Physics data booklet Electrical circuit symbols Cell Battery Switch Voltmeter Ammeter Resistor Variable resistor Light-dependent resistor (LDR) Thermistor Potentiometer Lamp Light emitting diode (LED) Heating element Motor Earth (ground) Electromagnetic spectrum energy 10 16 10 14 rays 10 12 10 10 X-rays 10 8 UV 10 6 IR 10 4 10 2 microwaves 100 102 10 4 106 108 wavelength / m radio waves V I B G Y O R 400 700 wavelength / nm Physics data booklet 5 A. Space, time and motion Standard level and higher level A.1 Kinematics s= s: displacement u: initial velocity v: final velocity a: acceleration t: time u +v t 2 v = u + at s = ut + (Use + or - to include direction) 1 2 at 2 v 2 = u 2 + 2as A.2 Forces and momentum Ff ≤ µsFN FH: elastic force from helical spring (Hooke’s Law) k: spring’s constant x: extension or compression of spring Ff = µdFN Fb: buoyant force (upthrust) ρ: density of fluid V: volume submerged in fluid g: acceleration of free fall p: momentum m: mass v: velocity F: resultant/net force m: mass a: acceleration Δp: change in momentum (Δp=mv-mu) Δt: time taken A.3 Work, energy and power FH = −k x Fd = 6πη rv Fb = ρVg Fg p Fg: gravitational force (weight close to Earth’s surface) m: mass g:acceleration of free fall mg mv J = F ∆t F = ma = Fd: drag force (resistive force from fluids) η: coefficient of viscosity r: radius v: speed ∆p ∆t J: impulse F: force Δt: time taken a= v2 4π 2 r = ω 2r = 2 r T v= 2π r = ωr T W = Fs cos θ Ek: kinetic energy m: mass v: speed p: momentum EH: elastic potential energy stored in helical spring k: spring’s constant Δx : extension or compression of the spring P: Power ΔW: work done / energy transferred Δt: time taken F: average force v: average speed η: efficiency 6 Ff: frictional force (friction) μs: coefficient of static friction μd: coefficient of dynamic friction FN: normal contact force 1 p2 mv 2 ! 2 2m Ek ! ∆Ep = mg ∆h EH = 1 k ∆x 2 2 a: centripetal acceleration v: linear speed r: radius T: period ω: angular speed W: work done by force F F: force s: displacement of point of action of force F θ: angle between direction of F and direction of s ΔΕp: change in gravitational potential energy m: mass g: acceleration of free fall Δh: change in height P= ∆W = Fv ∆t η= useful work out useful power out = total work in total power in Physics data booklet Additional higher level A.4 Rigid body mechanics τ = Fr sin θ ∆θ = Δθ: angular displacement ωf: final angular velocity ωi: initial angular velocity a: angular acceleration t: time (use + and - to include direction ) I: moment of inertia m: mass r: distance from point or axis of rotation 2 (Note: Σ means sum i.e Σmr 2= m1r12 + m2r2 + m3r32 + …) L: angular momentum I: moment of inertia ω: angular velocity ΔL: change in angular momentum I: moment of inertia ω: angular velocity A.5 Galilean and special relativity x’:position of an event in an inertial frame of reference moving with relative speed v to the original frame of reference x:position of the same event in the original frame of reference v: relative speed between the two inertial frames of reference t’: time of an event in an inertial frame of reference moving with relative speed v to the original frame of reference t: time of the same event in the original frame of reference u’: velocity of body in n inertial frame of reference moving with relative speed v to the original frame of reference τ: torque F: force r: distance from axis to point of action of F θ: angle between direction of F and direction of r ωf + ωi t 2 ωf = ωi + α t 1 ∆θ = ωit + α t 2 2 ωf2 = ωi2 + 2α∆θ I = Σmr 2 τ = Iα L = Iω ∆L = τ∆t τ: resultant/net torque I: moment of inertia a: angular acceleration ΔL: change in angular momentum τ: resultant/net torque Δt: time taken ∆L = ∆( I ω ) Ek = 1 2 L2 Iω = 2 2I x ' = x − vt t' t u' = u −v x ' = γ ( x − vt ) where γ = u' = u −v uv 1− 2 c ( ∆s )2 = (c ∆t )2 − ∆x 2 c: speed of light in vacuum (constant) ∆t = γ∆t0 Δto: proper time (time interval measured by same clock) v2 c2  vx  t ' = γ t − 2  c   γ: the Lorrentz factor γ: the Lorrentz factor 1 1− u: velocity of the same body in the original frame of reference Δt: time interval between two observed events (2 different clocks) Ek: rotational kinetic energy I: moment of inertia ω: angular speed L: angular momentum L= L0 γ tan θ = v c Δs: space-time interval between two events c: speed of ligh Δt: time interval Δx: distance between the events L: observed length Lo: proper length γ: the Lorrentz factor θ: angle of worldline from the vertical axis in a space-time diagram v: speed of the body Physics data booklet 7 B. The particulate nature of matter Standard level and higher level B.1 Thermal energy transfers Ek: average random kinetic energy of particles kB: Boltzmann constant T: temperature ΔQ: amount of heat (energy) transfer Δt: time taken k: thermal conductivity of material A: surface area of the surface that emits heat ΔT: temperature difference between hot and cold sides Δx: thickness (distance between hot and cold sides) ρ= Ek m V ρ: density m: mass V: volume 3 kBT 2 Q: heat (energy transferred) m: mass c: specific heat capacity L: specific latent heat ΔΤ: change in temperature Q = mc ∆T Q mL ∆Q ∆T = kA ∆t ∆x L: luminosity (total power output) σ: Steffan-Boltzmann constant A: surface area of body T: temperature L = σ AT 4 b: brightness (intensity) L: luminosity d: distance from the source b= L 4π d 2 λmax: peak wavelength T: temperature λmaxT = 2.9 × 10−3 mK B.2 Greenhouse effect σ: Steffan-Boltzmann constant T : temperature emissivity = albedo B.3 Gas laws n: number of moles N: number of particles (atoms or molecules) NA: Avogadro constant P: pressure ρ: density of gas v: root mean square speed of particles (r.m.s speed) 8 P F A n N NA power radiated per unit area σT 4 total scattered power total incident power P: pressure F: force A: area PV T constant PV nRT P= 1 2 ρv 3 U 3 nRT 2 NkBT P: pressure V: volume T: temperature N:number of moles R: gas constant N: number of particles KB: Boltzmann constant U: internal energy of gas 3 NkBT 2 Physics data booklet B.5 Current and circuits V: potential difference W: work done q: charge ρ: resistivity R: resistance A: cross-sectional area L: length P: power I: current V: potential difference R: resistance I= ∆q ∆t V W q R V I ρ= RA L P IV R: resistance V: potential difference I: current I R 2 V2 R Series circuits Parallel circuits ... I = I1 + I 2 + ... I ε: electromotive force (emf) I: current R: resistance of connected circuit r: internal resistance I:current Δq: amount of charge passing through a surface Δt: time taken I1 I2 V = V1 + V2 + ... V Rs = R1 + R2 + ... 1 1 1 = + + ... Rp R1 R2 V1 V2 ... ε = I(R + r ) Additional higher level B.4 Thermodynamics W: work done by gas P: pressure ΔV: change in volume ΔS: change in entropy ΔQ: amount of thermal energy (heat) that flows into a body T: temperature S: entropy kB: Boltzmann constant Ω: number of possible micro states of the system Q = ∆U + W W = P ∆V ∆U = ∆S = Physics data booklet 3 3 nR ∆T = NkB ∆T 2 2 ∆Q T ΔU: change in internal energy of a gas n: number of moles R: Gas constant ΔΤ: change in temperature N: number of atoms kB: Boltzmann constant S = kB ln Ω PV η: efficiency ηCarnot: efficiency of a Carnot cycle Tc: temperature of cold gas Th: temperature of hot gas Q: amount of thermal energy (heat) ΔU: change in internal energy W: work done by the gas η= 5 3 constant useful work input energy ηCarnot = 1 − (Model for adiabatic processes) P: Pressure of monatomic ideal gas V: Volume of monatomic ideal gas Tc Th 9 C. Wave behaviour Standard level and higher level C.1 Simple harmonic motion Τ: period f : frequency ω: angular frequency T: period of simple pendulum l: length g: acceleration of free fall (constant) a = −ω 2 x T = a: acceleration ω: angular frequency ×: displacement from equilibrium position 1 2π = f ω T = 2π m k T = 2π l g T: period of a mass-spring system m: mass k: spring’s constant λ T C.2 Wave model v= fλ = C.3 Wave phenomena n1 sin θ 2 v 2 = = n2 sin θ1 v1 n1: refractive index of medium 1 n2: refractive index of medium 2 θ1: angle of incidence θ2: angle of refraction v1: speed of wave in medium 1 v2: speed of wave in medium 2 s: distance between adjacent maxima λ: wavelength D: distance between slits and screen d: distance between slits C.5 Doppler effect v: wave speed f : frequency λ: wavelength Constructive interference: path difference = nλ Destructive interference: 1 path difference = (n + )λ 2 λD d ∆f ∆λ v = ≈ f λ c s= Additional higher level C.1 Simple harmonic motion x: displacement from equilibrium position xo: amplitude ω: angular frequency t: time φ: initial phase v: velocity ET: total energy of simple harmonic oscillator Ep: potential energy of simple harmonic oscillator m: mass 10 n= 0, 1, 2, 3, … λ: wavelength Δf : change/shift in frequency f : frequency of emitted wave Δλ: change/shift in wavelength λ: wavelength of emitted wave v: relative speed between source and observer c: speed of light (constant) x = x0 sin(ωt + φ ) v = ω x0 cos(ωt + φ ) v = ±ω x0 2 − x 2 ET = 1 mω 2 x02 2 Ep = 1 mω 2 x 2 2 Physics data booklet ΄ θ: angle at which first diffraction minimum appears λ: wavelength b: slit width C.3 Wave phenomena θ= λ b nλ = d sin θ C.5 Doppler effect n: order (1, 2, 3, … ) λ: wavelength d: distance between slits of diffraction grating θ: angle at which this order minimum will appear  v    v ± us  Moving source: f ′ = f   v ± uo  Moving observer: f ′ = f    v  Physics data booklet f ΄ :observed frequency f : emitted frequency v: wave speed uo: speed of observer us: speed of source 11 F: gravitational force G: gravitational constant m1: mass of body 1 m2: mass of body 2 r: distance between the centres of the 2 bodies D. Fields Standard level and higher level D.1 Gravitational fields D.2 Electric and magnetic fields F: electric field force between two charged particles k: Coulomb’s constant εο: permittivity of free space (constant) q1: charge of particle 1 q2: charge of particle 2 D.3 Motion in electromagnetic fields F: magnetic force on moving charged particle q: charge of particle v: speed of particle B: magnetic field strength θ: angle between magnetic field lines and direction of speed F G m1m2 r2 g F m G F k E F q E V d M r2 q1q2 1 where k = 2 4πε 0 r Ep: gravitational potential energy G: gravitational constant m1: mass of body 1 m2: mass of body 2 r: distance between the centres of bodies F = qvB sin θ F = B I L sin θ II F = µ0 1 2 2π r L Ep = −G m1m2 r Vg = −G M r g=− g: gravitational field strength ΔVg: change in the gravitational potential between two points Δr: distance between the two points W: work done to move a mass in a gravitational field m: mass of body that is moving ΔVg: change in the gravitational potential between two points ∆Vg ∆r W = m∆Vg 2GM r v esc Ve: electric potential at a point in an electric field k: Coulomb’s constant Q: charge creating the field r: distance between point and centre of charge E: electric field strength ΔV: electric potential difference between two points in the field Δr: distance between the points 12 F: magnetic force on current currying wire B: magnetic field strength I: current L: length of wire in the magnetic field θ: angle between magnetic field lines and current F: magnetic force between current currying wire L: length of wire μο: permeability of free space (constant) I1: current in wire 1 I2: current in wire 2 r: distance between wires Vg: gravitational potential at a point in a gravitational field G: gravitational constant M: mass of the body creating the field r: distance of the point from the centre of the body. vesc: speed needed to escape a gravitational field vorbital: orbital speed G: gravitational constant M: mass of body creating the gravitational field r: distance from the centre of that body GM r v orbital D.2 Electric and magnetic fields E: electric field strength F: electric field force q: charge E: electric field strength of a uniform electric field V: potential difference between two points (or metal plates) d: distance between the two points (or metal plates) Additional higher level D.1 Gravitational fields g: gravitational field strength F: gravitational force m: mass G: gravitational constant M: mass of the body that creates the gravitational field r: distance from the centre of that body q1q2 r Ep k Ve kQ r E=− ∆Ve ∆r W = q ∆Ve Ep: electric potential energy k: Coulomb’s constant q1: charge on body 1 q2: charge on body 2 r: distance between the centres of the bodies W: work done to move a charge in an electric field q: charge moved ΔVe: electric potential difference between the points Physics data booklet Φ: magnetic flux B: magnetic field strength A: area θ: angle between magnetic field lines and the perpendicular direction to the surface D.4 Induction ε: induced emf N: number of loops on coil ΔΦ: change in magnetic flux Δt: time taken Φ = BA cos θ ε = −N ∆Φ ∆t ε: emf induced across the ends of a straight conductor moving in a magnetic field ε = BvL B: magnetic field strength v: speed of conductor l: length of conductor in field E. Nuclear and quantum physics E: energy of a photon h: Planck’s constant f : frequency Standard level and higher level E.1 Structure of the atom E hf E.3 Radioactive decay E mc 2 E.5 Fusion and stars E: energy released m: mass ‘loss’ (change in mass) c: speed of light (constant) d(parsec) 1 p(arc-second) d: distance to star p: parallax angle Additional higher level E.1 Structure of the atom E: energy value of energy level n: quantum number of energy level (n= 1,2,3,..) (eV is just the unit, energy here is calculated in electrovolts) R 1 R0 A 3 E=− 13.6 eV n2 R: radius of atom Ro: Fermi radius (constant) A: atomic number (number of protons) mvr: angular momentum m: mass v: linear speed r: radius of circular path n: quantum number (n=1,2,3,4,…) h: Planck’s constant nh mvr = 2π E.2 Quantum physics Emax: maximum kinetic of energy of emitted electrons h: Planck’s constant f: frequency of incident radiation Φ: work function of metal surface E.3 Radioactive decay N: number of nuclei left after time t No: original number of nuclei in the sample (at t=0) λ: decay constant of material t: time A: activity (number of decays per second) T1/2: half-life Physics data booklet Emax = h f − Φ λ= h p λf − λi = ∆λ = h (1 − cos θ ) me c N = N0 e − λ t A = λ N = λ N0 e − λ t T1 = 2 ln 2 λ λ: wavelength h: Planck’s constant p: momentum λf: final wavelength λi: initial wavelength Δλ: change in wavelength h: Planck’s constant me: mass of electron (constant) c: speed of light in vacuum (constant) θ: scattering angle 13