Ib Physics Hl Formula Booklet 353vz

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 these programmes. The IB may use a variety of sources in its work and checks information to 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 publications can be purchased through the IB store at http://store.ibo.org. Email: [email protected]

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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