Physics data booklet
First assessment 2016
Diploma Programme
Physics data booklet
Published February 2014
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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