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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. The IB respects the principles of intellectual property and makes strenuous efforts to identify and obtain permission before publication from rights holders of all copyright material used. The IB is grateful for permissions received for material used in this publication and will be pleased to correct any errors or omissions at the earliest opportunity. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior written permission of the IB, or as expressly permitted by law or by the IB’s own rules and policy. See http://www.ibo.org/copyright. IB merchandise and p ublications can b e purchased through th e IB store at http://store.ibo.org. Email: [email protected]

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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

Symbol

Approximate value

Acceleration of free fall (Earth’s surface)

g

9.81 m s−2

Gravitational constant

G

6.67 × 10−11 N m2 kg−2

Avogadro’s constant Gas constant Boltzmann’s constant

𝑁A

R 𝑘B

Stefan–Boltzmann constant

σ

Coulomb constant

k

Permittivity of free space

𝜀0

Permeability of free space

𝜇0

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

Speed of light in vacuum

c

3.00 × 108 m s−1

Planck’s constant

h

6.63 × 10−34 J s

Elementary charge

e

Electron rest mass

𝑚e

1.60 × 10−19 C

Proton rest mass Neutron rest mass

𝑚p 𝑚n

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

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

1 Physics data booklet

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

2 Physics data booklet

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

3 Physics data booklet

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

A

4 Physics data booklet

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 If: 𝑦 = 𝑎 ± 𝑏

Sub-topic 1.3 – Vectors and scalars

then: 𝛥𝑦 = 𝛥𝑎 + 𝛥𝑏

𝑎𝑏 𝑐 𝛥𝑦 𝛥𝑎 𝛥𝑏 𝛥𝑐 then: = + + 𝑐 𝑦 𝑎 𝑏 If: 𝑦 =

If: 𝑦 = 𝑎𝑛 then:

∆𝑎 ∆𝑦 = �𝑛 � 𝑎 𝑦

𝐴H = 𝐴 cos 𝜃

𝐴V = 𝐴 sin 𝜃

Sub-topic 2.1 – Motion

Sub-topic 2.2 – Forces

𝑠 = 𝑢𝑡 + 𝑎𝑡

𝐹f ≤ 𝜇s 𝑅

𝑣 = 𝑢 + 𝑎𝑡 1 2

2

𝑣 2 = 𝑢2 + 2𝑎𝑠

𝑠=

(𝑣 + 𝑢)𝑡 2

𝐹 = 𝑚𝑎

𝐹f = 𝜇d 𝑅

Sub-topic 2.3 – Work, energy and power

Sub-topic 2.4 – Momentum and impulse

𝐸K = 2𝑚𝑣 2

𝐹=

𝑊 = 𝐹𝑠 𝑐𝑜𝑠𝜃 1

𝐸P = 𝑘∆𝑥2 2 1

Ef�iciency =

∆𝑝 ∆𝑡

𝐸K =

∆𝐸P = 𝑚𝑔∆ℎ

power = 𝐹𝑣

𝑝 = 𝑚𝑣

useful work out

𝑝2 2𝑚

Impulse = 𝐹∆𝑡 = ∆𝑝

total work in useful power out = total power in

5 Physics data booklet

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 = 𝑘B 𝑇 = 2 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

𝑅 𝑇 𝑁A

Sub-topic 4.4 – Wave behaviour 𝑛1 sin 𝜃2 𝑣2 = = sin 𝜃1 𝑣1 𝑛2 𝑠=

𝜆𝐷 𝑑

Constructive interference: path difference = 𝑛𝜆 Destructive interference:

𝐼 ∝ 𝑥 −2

1

path difference = (𝑛 + )𝜆 2

𝐼 = 𝐼0 𝑐𝑜𝑠 2 𝜃 Sub-topic 5.1 – Electric fields ∆𝑞 𝐼= ∆𝑡 𝑞1 𝑞2 𝐹=𝑘 2 𝑟 1 𝑘= 4𝜋𝜀0 𝑉=

𝐸=

𝑊 𝑞 𝐹 𝑞

𝐼 = 𝑛𝐴𝑣𝑞

Sub-topic 5.3 – Electric cells 𝜀 = 𝐼(𝑅 + 𝑟)

Sub-topic 5.2 – Heating effect of electric currents Kirchhoff’s circuit laws: Σ𝑉 = 0 (loop)

𝑅=

Σ𝐼 = 0 (junction)

𝑉 𝐼

𝑃 = 𝑉𝐼 = 𝐼2 𝑅 =

𝑉2 𝑅

𝑅total = 𝑅1 + 𝑅2 + ⋯ 1

𝑅total

𝜌=

=

𝑅𝐴 𝐿

1 1 +⋯ + 𝑅1 𝑅2

Sub-topic 5.4 – Magnetic effects of electric currents 𝐹 = 𝑞𝑣𝐵 sin 𝜃

𝐹 = 𝐵𝐼𝐿 sin 𝜃

6 Physics data booklet

Sub-topic 6.1 – Circular motion

Sub-topic 6.2 – Newton’s law of gravitation

𝑣 = 𝜔𝑟 𝑎=

𝐹=

𝑣 2 4𝜋2𝑟 = 2 𝑇 𝑟

𝑀𝑚 𝑟2

𝑔=𝐺

𝑀 𝑟2

𝑔=

𝑚𝑣 2 = 𝑚𝜔 2𝑟 𝑟

Sub-topic 7.1 – Discrete energy and radioactivity 𝐸 = ℎ𝑓

𝜆=

𝐹=𝐺

ℎ𝑐 𝐸

𝐹 𝑚

Sub-topic 7.2 – Nuclear reactions ∆𝐸 = ∆𝑚 𝑐 2

Sub-topic 7.3 – The structure of matter

Charge 2 𝑒 3

1 − 𝑒 3

Quarks

Baryon number

u

c

t

d

s

b

1

1 3

Particles mediating

Sub-topic 8.1 – Energy sources Power =

energy time

Power = 𝐴𝜌𝑣 3 2 1

Leptons

–1

e

µ

τ

0

υe

υµ

υτ

3

All quarks have a strangeness number of 0 except the strange quark that has a strangeness number of –1

Particles experiencing

Charge

All leptons have a lepton number of 1 and antileptons have a lepton number of –1

Gravitational

Weak

Electromagnetic

Strong

All

Quarks, leptons

Charged

Quarks, gluons

Graviton

W+, W–, Z0

γ

Gluons

Sub-topic 8.2 – Thermal energy transfer 𝑃 = 𝑒𝜎𝐴𝑇 4

𝜆max(metres) = 𝐼=

power 𝐴

albedo =

2.90 × 10 −3 𝑇(kelvin)

total scattered power total incident power

7 Physics data booklet

Equations—AHL Sub-topic 9.1 – Simple harmonic motion

Sub-topic 9.2 – Single-slit diffraction

𝜔=

𝜃=

2𝜋 𝑇 𝑎 = −𝜔2 𝑥

𝑥 = 𝑥0 sin 𝜔𝑡 ; 𝑥 = 𝑥0 cos 𝜔𝑡

𝑣 = 𝜔𝑥0 cos 𝜔𝑡 ; 𝑣 = −𝜔𝑥0 sin 𝜔𝑡 𝑣 = ±𝜔�(𝑥02 − 𝑥2 ) 𝐸K = 𝐸T =

𝑚𝜔 2 (𝑥0 2 2

1

1 𝑚𝜔 2 𝑥0 2 2

−𝑥

Pendulum: 𝑇 = 2𝜋�

2)

𝜆 𝑏

Sub-topic 9.3 – Interference 𝑛𝜆 = 𝑑 sin 𝜃

Constructive interference: 2𝑑𝑛 = (𝑚 + 2) 𝜆 Destructive interference:

2𝑑𝑛 = 𝑚𝜆

𝑙 𝑔

Mass–spring: 𝑇 = 2𝜋�

𝑚 𝑘

Sub-topic 9.4 – Resolution

Sub-topic 9.5 – Doppler effect

𝜃 = 1.22

Moving source: 𝑓 ′ = 𝑓 �

𝑅=

𝜆 𝑏

𝜆 = 𝑚𝑁 Δ𝜆

Sub-topic 10.1 – Describing fields 𝑊 = 𝑞∆𝑉𝑒

1

𝑣 � 𝑣 ± 𝑢s

Moving observer: 𝑓 ′ = 𝑓 �

∆𝑓 ∆𝜆 𝑣 ≈ = 𝜆 𝑓 𝑐

𝑣±𝑢0 𝑣

�

Sub-topic 10.2 – Fields at work

𝑊 = 𝑚∆𝑉𝑔

𝑉𝑔 = − 𝑔=−

𝐺𝑀 𝑟

Δ𝑉𝑔 Δ𝑟

𝐸P = 𝑚𝑉𝑔 = − 𝐹G = 𝐺 𝑣esc = �

𝑉𝑒 =

𝑘𝑞 𝑟

𝐸=− 𝐺𝑀𝑚 𝑟

𝑚1 𝑚2 𝑟2

Δ𝑉𝑒 Δ𝑟

𝐸P = 𝑞𝑉e = 𝐹E = 𝑘

𝑘𝑞1 𝑞2 𝑟

𝑞1 𝑞2 𝑟2

2𝐺𝑀 𝑟

𝑣orbit = �

𝐺𝑀 𝑟

8 Physics data booklet

Sub-topic 11.1 – Electromagnetic induction

Sub-topic 11.3 – Capacitance 𝑞 𝐶= 𝑉

𝛷 = 𝐵𝐴 cos 𝜃 𝜀 = −𝑁

𝜀 = 𝐵𝑣𝑙

∆𝛷 Δ𝑡

𝜀 = 𝐵𝑣𝑙𝑁

Sub-topic 11.2 transmission 𝐼rms =

𝑉rms =

𝐶parallel = 𝐶1 + 𝐶2 + ⋯ 1

𝐶series

–

Power

generation

and

𝐼0

√2 𝑉0

√2 𝑉0 𝑉rms 𝑅= = 𝐼rms 𝐼0 𝑃max = 𝐼0 𝑉0

1 𝑃� = 2 𝐼0 𝑉0

𝐶=𝜀

= 𝐴 𝑑

1 1 + +⋯ 𝐶1 𝐶2

𝐸 = 𝐶𝑉 2 2 1

𝜏 = 𝑅𝐶

𝑡

𝑞 = 𝑞0 𝑒 −𝜏

𝐼 = 𝐼0 𝑒

𝑡 − 𝜏

𝑉 = 𝑉0 𝑒

𝑡 − 𝜏

𝜀 p 𝑁p 𝐼s = = 𝜀s 𝑁s 𝐼p Sub-topic 12.1 – The interaction of matter with radiation

Sub-topic 12.2 – Nuclear physics

𝐸max = ℎ𝑓 − 𝛷

𝑁 = 𝑁0 𝑒 −𝜆𝑡

𝐸 = ℎ𝑓

13.6 𝑒𝑉 𝑛2 𝑛ℎ 𝑚𝑣𝑟 = 2𝜋

𝐸=−

𝑃(𝑟) = |ψ|2 Δ𝑉

𝑅 = 𝑅0 𝐴1/3

𝐴 = 𝜆𝑁0 𝑒 −𝜆𝑡

sin 𝜃 ≈

𝜆 𝐷

ℎ 4𝜋 ℎ Δ𝐸Δ𝑡 ≥ 4𝜋 Δ𝑥Δ𝑝 ≥

9 Physics data booklet

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

�1 − 𝑣 2 𝑐

2

𝑥 ′ = 𝛾(𝑥 − 𝑣𝑡) ; ∆𝑥 ′ = 𝛾(∆𝑥 − 𝑣∆𝑡 ) 𝑡 ′ = 𝛾(𝑡 −

𝑢′ =

𝑣𝑥 𝑐2

𝑢−𝑣 𝑢𝑣 1 − 𝑐2

) ; ∆𝑡 ′ = 𝛾(∆𝑡 −

𝑣∆𝑥 𝑐2

)

∆𝑡 = 𝛾∆𝑡0 𝐿=

Sub-topic A.4 – Relativistic mechanics (HL only) 𝐸 = 𝛾𝑚0 𝑐 𝐸0 = 𝑚 0 𝑐

2

2

𝐸K = (𝛾 − 1)𝑚0𝑐 2 𝑝 = 𝛾𝑚0 𝑣

𝐸 2 = 𝑝 2 𝑐 2 + 𝑚0 2 𝑐 4

𝑞𝑉 = ∆𝐸K

𝐿0 𝛾

(𝑐𝑡 ′)2 − (𝑥 ′ )2 = (𝑐𝑡)2 − (𝑥)2

Sub-topic A.5 – General relativity (HL only) ∆𝑓 𝑔∆ℎ = 2 𝑐 𝑓 𝑅s =

∆𝑡 =

2𝐺𝑀 𝑐2 ∆𝑡0

�1 − 𝑅

s

𝑟

10 Physics data booklet

Sub-topic B.1 dynamics

–

Rigid

bodies

and

rotational

𝛤 = 𝐹𝑟 sin 𝜃 𝐼 = ∑𝑚𝑟 𝛤 = 𝐼𝛼

2

𝜔 = 2𝜋𝑓

𝜔f = 𝜔 i + 𝛼𝑡

𝜔f2 = 𝜔i2 + 2𝛼𝜃

𝜃 = 𝜔i 𝑡

𝐿 = 𝐼𝜔

𝐸Krot =

1 + 2

1 2

𝛼𝑡 2

𝐼𝜔 2

Sub-topic B.3 – Fluids and fluid dynamics (HL only) 𝐵 = 𝜌f 𝑉f 𝑔

𝑃 = 𝑃 0 + 𝜌f𝑔𝑑

𝐴𝑣 = constant

𝜌𝑣 2 + 𝜌𝑔𝑧 + 𝑝 = constant 2 1

Sub-topic B.2 – Thermodynamics 𝑄 = ∆𝑈 + 𝑊

𝑈 = 2 𝑛𝑅𝑇 3

∆𝑆 = 5

∆𝑄 𝑇

𝑝𝑉3 = constant (for monatomic gases) 𝑊 = 𝑝∆𝑉

𝜂=

useful work done energy input

𝜂Carnot = 1 −

𝑇 cold 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

𝐹D = 6𝜋𝜂𝑟𝑣 𝑣𝑟𝜌 𝑅= 𝜂

Sub-topic C.1 – Introduction to imaging

𝑀=

1 1 1 = + 𝑓 𝑣 𝑢 𝑃=

1 𝑓

𝑀near point

𝑓o 𝑓e

Sub-topic C.3 – Fibre optics

ℎi 𝑣 𝑚= =− 𝑢 ℎo 𝜃i 𝑀= 𝜃o

Sub-topic C.2 – Imaging instrumentation

𝐷 𝐷 = + 1 ; 𝑀in�inity = 𝑓 𝑓

𝑛=

1

sin 𝑐

attenuation = 10 log

𝐼 𝐼0

Sub-topic C.4 – Medical imaging (HL only) 𝐿I = 10 log 𝐼 = 𝐼0 𝑒 −𝜇𝑥

𝐼1 𝐼0

𝜇𝑥1 = ln2 2

𝑍 = 𝜌𝑐

11 Physics data booklet

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=

𝑧=

∆𝜆 𝑣 ≈ 𝜆0 𝑐 𝑅 −1 𝑅0

𝑣 = 𝐻0𝑑

𝑇≈

Sub-topic D.2 – Stellar characteristics and stellar evolution 𝜆max𝑇 = 2.9 × 10−3 m K 𝐿 ∝ 𝑀3.5

Sub-topic D.5 – Further cosmology (HL only) 𝑣=� 𝜌c =

4𝜋𝐺𝜌 𝑟 3 3𝐻 2 8𝜋𝐺

1

𝐻0

12 Physics data booklet...