Physics Data Booklet

Physics data booklet First assessment 2016 Diploma Programme Physics data booklet Published February 2014 Published on behalf of the International Baccalaureate Organization, a not-for-profit educational foundation of 15 Route des Morillons, 1218 Le Grand-Saconnex, Geneva, Switzerland by the International Baccalaureate Organization (UK) Ltd Peterson House, Malthouse Avenue, Cardiff Gate Cardiff, Wales CF23 8GL United Kingdom Website: www.ibo.org © International Baccalaureate Organization 2014 The International Baccalaureate Organization (known as the IB) offers four high-quality and challenging educational programmes for a worldwide community of schools, aiming to create a better, more peaceful world. This publication is one of a range of materials produced to support these programmes. The IB may use a variety of sources in its work and checks information to verify accuracy and authenticity, particularly when using community-based knowledge sources such as Wikipedia. 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Email: sales@ibo.org International Baccalaureate, Baccalauréat International and Bachillerato Internacional are registered trademarks of the International Baccalaureate Organization. 4082 Contents Fundamental constants 1 Metric (SI) multipliers 2 Unit conversions 3 Electrical circuit symbols 4 Equations—Core 5 Equations—AHL 8 Equations—Options 10 Physics data booklet Fundamental constants Quantity Acceleration of free fall (Earth’s surface) Gravitational constant Symbol g G Avogadro’s constant 𝑁A Boltzmann’s constant 𝑘B Coulomb constant k Gas constant Stefan–Boltzmann constant Permittivity of free space R σ 𝜀0 Permeability of free space 𝜇0 Planck’s constant h Electron rest mass 𝑚e Speed of light in vacuum Elementary charge Proton rest mass Neutron rest mass Unified atomic mass unit Solar constant Fermi radius Physics data booklet c e 𝑚p 𝑚n u S 𝑅0 Approximate value 9.81 m s−2 6.67 × 10−11 N m2 kg−2 6.02 × 1023 mol−1 8.31 J K−1 mol−1 1.38 × 10−23 J K−1 5.67 × 10−8 W m−2 K−4 8.99 × 109 N m2 C−2 8.85 × 10−12 C2 N−1 m−2 4π × 10−7 T m A−1 3.00 × 108 m s−1 6.63 × 10−34 J s 1.60 × 10−19 C 9.110 × 10−31 kg = 0.000549 u = 0.511 MeV c−2 1.673 × 10−27 kg =1.007276 u = 938 MeV c−2 1.675 × 10−27 kg =1.008665 u = 940 MeV c−2 1.661 × 10−27 kg = 931.5 MeV c−2 1.36 × 103 W m−2 1.20 × 10−15 m 1 Metric (SI) multipliers Prefix Abbreviation Value peta P 1015 G 109 tera T mega M hecto h giga kilo k deca da centi c deci 1012 106 103 102 101 d 10–1 milli m 10–3 nano n micro pico femto Physics data booklet µ p f 10–2 10–6 10–9 10–12 10–15 2 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 hc = 1.99 × 10-25 J m = 1.24 × 10–6 eV m Physics data booklet 3 Electrical circuit symbols cell battery ac supply switch voltmeter V ammeter resistor variable resistor lamp potentiometer light-dependent resistor (LDR) thermistor transformer heating element diode capacitor Physics data booklet A 4 Equations—Core Note: All equations relate to the magnitude of the quantities only. Vector notation has not been used. Sub-topic 1.2 – Uncertainties and errors Sub-topic 1.3 – Vectors and scalars If: 𝑦 = 𝑎 ± 𝑏 then: 𝛥𝑦 = 𝛥𝑎 + 𝛥𝑏 𝑎𝑏 𝑐 𝛥𝑦 𝛥𝑎 𝛥𝑏 𝛥𝑐 then: = + + 𝑦 𝑎 𝑏 𝑐 If: 𝑦 = If: 𝑦 = 𝑎𝑛 then: ∆𝑦 ∆𝑎 = �𝑛 � 𝑦 𝑎 Sub-topic 2.1 – Motion 𝐴H = 𝐴 cos 𝜃 𝐴V = 𝐴 sin 𝜃 Sub-topic 2.2 – Forces 𝑣 = 𝑢 + 𝑎𝑡 𝐹 = 𝑚𝑎 𝑣 2 = 𝑢2 + 2𝑎𝑠 𝐹f = 𝜇d 𝑅 1 2 𝑠 = 𝑢𝑡 + 𝑎𝑡 2 𝑠= (𝑣 + 𝑢)𝑡 2 𝐹f ≤ 𝜇s 𝑅 Sub-topic 2.3 – Work, energy and power Sub-topic 2.4 – Momentum and impulse 𝑊 = 𝐹𝑠 𝑐𝑜𝑠𝜃 𝑝 = 𝑚𝑣 1 𝐸K = 2 𝑚𝑣 2 1 𝐸P = 2 𝑘∆𝑥 2 ∆𝐸P = 𝑚𝑔∆ℎ power = 𝐹𝑣 useful work out total work in useful power out = total power in Ef�iciency = Physics data booklet 𝐹= ∆𝑝 ∆𝑡 𝐸K = 𝑝2 2𝑚 Impulse = 𝐹∆𝑡 = ∆𝑝 5 Sub-topic 3.1 – Thermal concepts 𝑄 = 𝑚𝑐∆𝑇 𝑄 = 𝑚𝐿 Sub-topic 3.2 – Modelling a gas 𝐹 𝐴 𝑁 𝑛= 𝑁A 𝑝= 𝑝𝑉 = 𝑛𝑅𝑇 3 3 𝐸�K = 2 𝑘B 𝑇 = 2 Sub-topic 4.1 – Oscillations 𝑇= 1 𝑓 Sub-topic 4.2 – Travelling waves 𝑐 = 𝑓𝜆 Sub-topic 4.3 – Wave characteristics 𝐼 ∝ 𝐴2 𝐼 ∝ 𝑥 −2 𝐼 = 𝐼0 𝑐𝑜𝑠 2 𝜃 𝑅 𝑇 𝑁A Sub-topic 4.4 – Wave behaviour 𝑛1 sin 𝜃2 𝑣2 = = 𝑛2 sin 𝜃1 𝑣1 𝑠= 𝜆𝐷 𝑑 Constructive interference: path difference = 𝑛𝜆 Destructive interference: 1 path difference = (𝑛 + 2)𝜆 Sub-topic 5.1 – Electric fields Sub-topic 5.2 – Heating effect of electric currents ∆𝑞 ∆𝑡 𝑞1 𝑞2 𝐹=𝑘 2 𝑟 1 𝑘= 4𝜋𝜀0 Kirchhoff’s circuit laws: 𝐼= 𝑉= 𝐸= 𝑊 𝑞 𝐹 𝑞 𝐼 = 𝑛𝐴𝑣𝑞 Σ𝑉 = 0 (loop) 𝑅= Σ𝐼 = 0 (junction) 𝑉 𝐼 𝑃 = 𝑉𝐼 = 𝐼 2 𝑅 = 𝑅total = 𝑅1 + 𝑅2 + ⋯ 1 𝑅total = 𝑅𝐴 𝐿 1 1 + +⋯ 𝑅1 𝑅2 Sub-topic 5.3 – Electric cells 𝜌= 𝜀 = 𝐼(𝑅 + 𝑟) 𝐹 = 𝑞𝑣𝐵 sin 𝜃 Physics data booklet 𝑉2 𝑅 Sub-topic 5.4 – Magnetic effects of electric currents 𝐹 = 𝐵𝐼𝐿 sin 𝜃 6 Sub-topic 6.1 – Circular motion Sub-topic 6.2 – Newton’s law of gravitation 𝑣 = 𝜔𝑟 𝑎= 𝐹= 𝑣 2 4𝜋 2 𝑟 = 2 𝑟 𝑇 𝐹=𝐺 𝑀𝑚 𝑟2 𝑔=𝐺 𝑀 𝑟2 𝑔= 𝑚𝑣 2 = 𝑚𝜔2 𝑟 𝑟 𝐹 𝑚 Sub-topic 7.1 – Discrete energy and radioactivity Sub-topic 7.2 – Nuclear reactions 𝐸 = ℎ𝑓 ∆𝐸 = ∆𝑚 𝑐 2 𝜆= ℎ𝑐 𝐸 Sub-topic 7.3 – The structure of matter Charge 2 𝑒 3 Quarks u 1 − 𝑒 3 d c s Baryon number t b Charge –1 1 3 0 1 3 Particles mediating e υe µ τ υµ υτ All leptons have a lepton number of 1 and antileptons have a lepton number of –1 All quarks have a strangeness number of 0 except the strange quark that has a strangeness number of –1 Particles experiencing Leptons Gravitational Weak Electromagnetic Strong All Quarks, leptons Charged Quarks, gluons Graviton W+, W–, Z0 γ Sub-topic 8.1 – Energy sources Sub-topic 8.2 – Thermal energy transfer energy Power = time 𝑃 = 𝑒𝜎𝐴𝑇 4 1 Power = 2 𝐴𝜌𝑣 3 Physics data booklet 𝜆max (metres) = Gluons 2.90 × 10−3 𝑇(kelvin) power 𝐴 total scattered power albedo = total incident power 𝐼= 7 Equations—AHL Sub-topic 9.1 – Simple harmonic motion 𝜔= 2𝜋 𝑇 Sub-topic 9.2 – Single-slit diffraction 𝜃= 𝑎 = −𝜔2 𝑥 𝜆 𝑏 Sub-topic 9.3 – Interference 𝑥 = 𝑥0 sin 𝜔𝑡 ; 𝑥 = 𝑥0 cos 𝜔𝑡 𝑛𝜆 = 𝑑 sin 𝜃 𝑣 = ±𝜔�(𝑥0 2 − 𝑥 2 ) Destructive interference: 𝑣 = 𝜔𝑥0 cos 𝜔𝑡 ; 𝑣 = −𝜔𝑥0 sin 𝜔𝑡 𝐸K = 1 2 1 𝑚𝜔2 (𝑥0 2 𝐸T = 2 𝑚𝜔2 𝑥0 2 −𝑥 2) 1 Constructive interference: 2𝑑𝑛 = (𝑚 + 2) 𝜆 2𝑑𝑛 = 𝑚𝜆 𝑙 Pendulum: 𝑇 = 2𝜋� 𝑔 𝑚 Mass–spring: 𝑇 = 2𝜋� 𝑘 Sub-topic 9.4 – Resolution 𝜃 = 1.22 𝑅= 𝜆 𝑏 𝜆 = 𝑚𝑁 Δ𝜆 Sub-topic 10.1 – Describing fields Sub-topic 9.5 – Doppler effect 𝑣 Moving source: 𝑓 ′ = 𝑓 � � 𝑣 ± 𝑢s Moving observer: 𝑓 ′ = 𝑓 � ∆𝑓 ∆𝜆 𝑣 = ≈ 𝑓 𝜆 𝑐 𝑣±𝑢0 𝑣 � Sub-topic 10.2 – Fields at work 𝑊 = 𝑞∆𝑉𝑒 𝑊 = 𝑚∆𝑉𝑔 𝑉𝑔 = − 𝑔=− 𝐺𝑀 𝑟 Δ𝑉𝑔 Δ𝑟 𝐸P = 𝑚𝑉𝑔 = − 𝐹G = 𝐺 𝑣esc = � 2𝐺𝑀 𝑟 𝐺𝑀𝑚 𝑟 𝑚1 𝑚2 𝑟2 𝑉𝑒 = 𝑘𝑞 𝑟 𝐸=− Δ𝑉𝑒 Δ𝑟 𝐸P = 𝑞𝑉e = 𝐹E = 𝑘 𝑘𝑞1 𝑞2 𝑟 𝑞1 𝑞2 𝑟2 𝐺𝑀 𝑣orbit = � 𝑟 Physics data booklet 8 Sub-topic 11.1 – Electromagnetic induction Sub-topic 11.3 – Capacitance 𝛷 = 𝐵𝐴 cos 𝜃 𝐶= 𝜀 = −𝑁 𝜀 = 𝐵𝑣𝑙 ∆𝛷 Δ𝑡 𝜀 = 𝐵𝑣𝑙𝑁 Sub-topic 11.2 transmission 𝐼rms = 𝑉rms = 𝐶parallel = 𝐶1 + 𝐶2 + ⋯ 1 – Power generation and 𝐼0 √2 𝑉0 √2 𝑉0 𝑉rms 𝑅= = 𝐼0 𝐼rms 𝑃max = 𝐼0 𝑉0 1 𝑃� = 2 𝐼0 𝑉0 𝜀p 𝑁p 𝐼s = = 𝜀s 𝑁s 𝐼p 𝑞 𝑉 𝐶series 𝐶=𝜀 𝐸 = = 𝐴 𝑑 1 1 + +⋯ 𝐶1 𝐶2 1 𝐶𝑉 2 2 𝜏 = 𝑅𝐶 𝑡 𝑞 = 𝑞0 𝑒 −𝜏 𝑡 𝐼 = 𝐼0 𝑒 −𝜏 𝑡 𝑉 = 𝑉0 𝑒 −𝜏 Sub-topic 12.1 – The interaction of matter with radiation Sub-topic 12.2 – Nuclear physics 𝐸 = ℎ𝑓 𝑅 = 𝑅0 𝐴1/3 13.6 𝑒𝑉 𝑛2 𝑛ℎ 𝑚𝑣𝑟 = 2𝜋 𝐴 = 𝜆𝑁0 𝑒 −𝜆𝑡 𝐸max = ℎ𝑓 − 𝛷 𝐸=− 𝑃(𝑟) = |ψ|2 Δ𝑉 𝑁 = 𝑁0 𝑒 −𝜆𝑡 sin 𝜃 ≈ 𝜆 𝐷 ℎ 4𝜋 ℎ Δ𝐸Δ𝑡 ≥ 4𝜋 Δ𝑥Δ𝑝 ≥ Physics data booklet 9 Equations—Options Sub-topic A.1 – The beginnings of relativity ′ 𝑥 = 𝑥 − 𝑣𝑡 ′ 𝑢 =𝑢−𝑣 Sub-topic A.3 – Spacetime diagrams 𝑣 𝜃 = tan−1 � � 𝑐 Sub-topic A.2 – Lorentz transformations 𝛾= 1 2 �1 − 𝑣2 𝑐 𝑥 ′ = 𝛾(𝑥 − 𝑣𝑡) ; ∆𝑥 ′ = 𝛾(∆𝑥 − 𝑣∆𝑡) 𝑡 ′ = 𝛾(𝑡 − 𝑢′ = 𝑣𝑥 𝑐2 𝑢−𝑣 𝑢𝑣 1− 2 𝑐 ) ; ∆𝑡 ′ = 𝛾(∆𝑡 − 𝑣∆𝑥 𝑐2 ) ∆𝑡 = 𝛾∆𝑡0 𝐿= Sub-topic A.4 – Relativistic mechanics (HL only) 𝐸 = 𝛾𝑚0 𝑐 2 𝐸0 = 𝑚0 𝑐 2 𝐸K = (𝛾 − 1)𝑚0 𝑐 2 𝑝 = 𝛾𝑚0 𝑣 𝐸 2 = 𝑝2 𝑐 2 + 𝑚0 2 𝑐 4 𝑞𝑉 = ∆𝐸K Physics data booklet 𝐿0 𝛾 (𝑐𝑡 ′ )2 − (𝑥 ′ )2 = (𝑐𝑡)2 − (𝑥)2 Sub-topic A.5 – General relativity (HL only) ∆𝑓 𝑔∆ℎ = 2 𝑓 𝑐 𝑅s = ∆𝑡 = 2𝐺𝑀 𝑐2 ∆𝑡0 �1 − 𝑅s 𝑟 10 Sub-topic dynamics B.1 – Rigid bodies and rotational 𝛤 = 𝐹𝑟 sin 𝜃 𝑄 = ∆𝑈 + 𝑊 3 𝐼 = ∑𝑚𝑟 2 𝑈 = 2 𝑛𝑅𝑇 𝛤 = 𝐼𝛼 ∆𝑆 = 𝜔 = 2𝜋𝑓 𝜔f = 𝜔 i + 𝛼𝑡 𝜔f2 = 𝜔i2 + 2𝛼𝜃 1 2 𝜃 = 𝜔i 𝑡 + 𝛼𝑡 𝐿 = 𝐼𝜔 2 1 𝐸Krot = 2 𝐼𝜔2 Sub-topic B.3 – Fluids and fluid dynamics (HL only) 𝐵 = 𝜌f 𝑉f 𝑔 𝑃 = 𝑃0 + 𝜌f 𝑔𝑑 𝐴𝑣 = constant 1 2 Sub-topic B.2 – Thermodynamics 𝜌𝑣 2 + 𝜌𝑔𝑧 + 𝑝 = constant 𝐹D = 6𝜋𝜂𝑟𝑣 𝑣𝑟𝜌 𝑅= 𝜂 5 ∆𝑄 𝑇 𝑝𝑉 3 = constant (for monatomic gases) 𝑊 = 𝑝∆𝑉 𝜂= useful work done energy input 𝜂Carnot = 1 − 𝑇cold 𝑇hot Sub-topic B.4 – Forced vibrations and resonance (HL only) 𝑄 = 2𝜋 energy stored energy dissipated per cycle 𝑄 = 2𝜋 × resonant frequency × energy stored power loss Sub-topic C.1 – Introduction to imaging Sub-topic C.2 – Imaging instrumentation 1 1 1 = + 𝑓 𝑣 𝑢 𝑀= 𝑃= 1 𝑓 Sub-topic C.3 – Fibre optics ℎi 𝑣 𝑚= =− ℎo 𝑢 𝑀= 𝜃i 𝜃o 𝑀near point 𝑓o 𝑓e 𝑛= 1 sin 𝑐 attenuation = 10 log 𝐼 𝐼0 Sub-topic C.4 – Medical imaging (HL only) 𝐷 𝐷 = + 1 ; 𝑀in�inity = 𝑓 𝑓 𝐿I = 10 log 𝐼 = 𝐼0 𝑒 −𝜇𝑥 𝐼1 𝐼0 𝜇𝑥1 = ln2 2 𝑍 = 𝜌𝑐 Physics data booklet 11 Sub-topic D.1 – Stellar quantities 𝑑 (parsec) = 𝐿 = 𝜎𝐴𝑇 𝑏= 4 1 𝑝 (arc–second) 𝐿 4𝜋𝑑 2 Sub-topic D.3 – Cosmology 𝑧= ∆𝜆 𝑣 ≈ 𝜆0 𝑐 𝑧= 𝑅 −1 𝑅0 𝑇≈ 1 𝐻0 𝑣 = 𝐻0 𝑑 Physics data booklet Sub-topic D.2 – Stellar characteristics and stellar evolution 𝜆max 𝑇 = 2.9 × 10−3 m K 𝐿 ∝ 𝑀3.5 Sub-topic D.5 – Further cosmology (HL only) 4𝜋𝐺𝜌 𝑣=� 𝑟 3 𝜌c = 3𝐻 2 8𝜋𝐺 12

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