Physics Notes

AQA A-Level Physics Notes /u/BaronPaprika May 2018

Contents Overview

3

1

Particles 1.1 Fundamental forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Balancing interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 4 4 5

2

Electricity 2.1 Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 6

3

Quantum Phenomena 3.1 The photoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Wave-particle duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 6 7

4

Mechanics 4.1 Newton’s laws of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 SUVAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 8

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Materials 5.1 Core vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Hooke’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Young’s modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 8 8 9

6

Waves 6.1 Core vocabulary . . . . 6.2 Phase difference . . . . 6.3 Polarisation . . . . . . 6.4 Stationary waves . . . 6.4.1 Superposition . 6.4.2 Stationary wave 6.5 Refraction . . . . . . .

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AQA A-Level Physics Notes

6.6 6.7

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6.5.1 Refractive index . . . . . . . . 6.5.2 Snell’s Law . . . . . . . . . . 6.5.3 Total internal reflection . . . . 6.5.4 Fibre optics . . . . . . . . . . Interference . . . . . . . . . . . . . . 6.6.1 Young’s double slit experiment Diffraction . . . . . . . . . . . . . . . 6.7.1 Single slit diffraction . . . . . 6.7.2 Diffraction gratings . . . . . .

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CONTENTS

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Further Mechanics 15 7.1 Circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 7.2 Simple harmonic motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 7.3 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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Thermal Physics 17 8.1 Thermal energy transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 8.2 Ideal gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 8.2.1 Kinetic theory of gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

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Fields 9.1 Gravitational fields . . . . . . . . 9.1.1 Gravitational potential . . 9.1.2 Orbits . . . . . . . . . . . 9.1.3 Escape velocity . . . . . . 9.2 Electric fields . . . . . . . . . . . 9.2.1 Electrical potential . . . . 9.3 Capacitance . . . . . . . . . . . . 9.3.1 Dielectrics . . . . . . . . . 9.3.2 Energy stored . . . . . . . 9.3.3 Charging and discharging . 9.4 Magnetic fields . . . . . . . . . . 9.4.1 Moving particles . . . . . 9.4.2 Magnetic flux . . . . . . . 9.4.3 Electromagnetic induction 9.4.4 Alternating current . . . . 9.4.5 Transformers . . . . . . .

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19 19 20 20 21 21 22 22 23 23 23 23 24 24 25 25 26

10 Nuclear Physics 10.1 Atomic structure . . . . . . . . . 10.1.1 Rutherford scattering . . . 10.2 Radiation . . . . . . . . . . . . . 10.2.1 Uses of radiation . . . . . 10.2.2 Background radiation . . . 10.2.3 Gamma radiation intensity

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26 26 26 27 27 27 28

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AQA A-Level Physics Notes

10.2.4 Safety . . . . . 10.3 Radioactive decay . . . 10.4 Nuclear radius . . . . . 10.5 Mass defect . . . . . . 10.5.1 Nuclear fusion . 10.6 Nuclear reactors . . . . 10.6.1 Structure . . . 10.6.2 Induced fission 10.6.3 Safety . . . . .

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OVERVIEW

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28 28 29 29 30 31 31 31 32

11 Astrophysics 11.1 Telescopes . . . . . . . . . . . . . . . . 11.1.1 Refracting telescopes . . . . . . 11.1.2 Reflecting telescopes . . . . . . 11.1.3 Resolving power . . . . . . . . . 11.1.4 Charge-coupled devices . . . . . 11.1.5 Non-optical telescopes . . . . . 11.2 Star classification . . . . . . . . . . . . 11.2.1 Luminosity . . . . . . . . . . . 11.2.2 Apparent magnitude . . . . . . 11.2.3 Parsecs . . . . . . . . . . . . . 11.2.4 Absolute magnitude . . . . . . . 11.2.5 Black body radiation . . . . . . 11.2.6 Stefan’s law . . . . . . . . . . . 11.2.7 Spectral classes . . . . . . . . . 11.2.8 Stellar evolution . . . . . . . . . 11.2.9 The Hertzsprung-Russel diagram 11.2.10 Stellar death . . . . . . . . . . 11.3 Cosmology . . . . . . . . . . . . . . . . 11.3.1 Doppler effect . . . . . . . . . . 11.3.2 Quasars . . . . . . . . . . . . . 11.3.3 Hubble’s law . . . . . . . . . . 11.3.4 The Big Bang . . . . . . . . . . 11.3.5 Exoplanets . . . . . . . . . . .

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32 32 32 33 33 33 34 34 34 34 34 35 35 35 35 36 36 37 38 38 38 39 39 39

12 Appendix 40 12.1 Data and units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 12.2 Damping graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Overview Paper 1 Paper 2 Paper 3

Mon. Fri. Thu.

June 4 June 8 June 14

Sections 1-7 Sections 8-10 and Paper 1 Section 11 and practical

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AQA A-Level Physics Notes

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1

PARTICLES

Formulae marked with a (?) are not given in the formula book, but can be easily derived. Formulae marked with (??) must be memorised. The full specification is available at http://filestore.aqa.org.uk/resources/physics/specifications/AQA-7407-7408-SP-2015.PDF

1

Particles Particles (classification) Hadrons Leptons Bosons Baryons Mesons electron photon (γ) proton π-meson muon W± neutron K-meson tau Z0 neutrino gluon graviton Higgs

Hadrons π+ , π− , π0 ¯0 K+ , K− , K0 , K proton neutron Λ0 Σ+ , Σ− , Σ0 Ξ− , Ξ0 ∆++ , ∆− Ω−

Composition u¯ d, ¯ ud, (u¯ u, d¯ d) u¯s, ¯ us, d¯s, ¯ ds uud udd uds uus, dds, uds dss, uss uuu, ddd sss

· Fermions have half-integer spin. This class of particles includes leptons and baryons. – Leptons are defined by their non-observance of the strong nuclear force. · Bosons are defined by their non-observance of the Pauli exclusion principle, meaning two of them can occupy the same space at once. They also have integer spin, so all mesons and some nuclei (those with even mass numbers) are bosons. · Hadrons are composed of quarks. Baryons contain three quarks and mesons contain a quarkantiquark pair.

1.1

Fundamental forces · Electromagnetic force – carried by photons, represented by γ. Responsible for the interactions between charged particles. · Strong nuclear force – carried by gluons and π-mesons. Holds quarks together to form larger hadrons (mediated by gluons), and holds hadrons together to form nuclei (mediated by pions). Very repulsive below 0.5 fm and strongly attractive between 0.5 and 3.0 fm. Leptons do not interact with the strong nuclear force. · Weak nuclear force – carried by W± and Z0 . Regulates flavour change in quarks. Has a limited range because the mass of its carrier particles causes them to decay after they travel a short distance. · Gravity – unknown carrier. Responsible for the interactions between objects with mass. Has an infinite range, but is the weakest of the fundamental forces.

1.2

Interactions

In most interactions, the exchange particle is a virtual particle – it briefly appears and disappears, just long enough to mediate the interaction.

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β − decay

Electron repulsion e

−

e

−

2

Electron capture

β−

p

W−

γ

e+

e−

W+

e−

Annihilation

νe

n

ν¯e

γ

e−

ELECTRICITY

n

e−

p

γ

Some important interactions, as they relate to atomic nuclei: Electron capture Beta plus decay Beta minus decay Alpha decay

p + e − → n + νe p → n + β + + νe n → p + β − + ν¯e

A ZX A ZX A ZX A ZX

→A Z−1 →A Z−1 →A Z+1 →A−4 Z−2

Y Y Y Y (+42 He2+ )

Gamma emission does not alter the nucleus, as photons have no mass and no charge. Sufficiently highly energetic photons can turn into particle-antiparticle pairs in a phenomenon called pair production. Free neutrons are unstable, with a lifetime of around 880 s, and decay into protons (which are stable) through β − decay. 1.2.1

Balancing interactions

Charge, lepton number, lepton generation, and baryon number must all be balanced between the two sides of an interaction. Strangeness must also be balanced unless the interaction is weak. Quarks all have a baryon number of 13 and strange quarks have a strangeness of −1. Strange quarks are created by the strong interaction, but because they are always created in quark-antiquark pairs, strangeness is conserved.

2 2.1

Electricity Resistivity

Whereas resistance is specific to a component, resistivity applies to a material. It is defined by the equation

ρ=

RA `

ρ = resistivity in Ω m−1 , R = resistance in Ω, A = area in m2 , ` = length in m.

Temperature can also affects resistivity. For some materials, such as the ones used in thermistors, resistance increases with temperature, but for some materials, it decreases. Superconductors are materials that have zero resistivity below a certain critical temperature, and can be used to transfer power without loss or make very strong magnets. 5

AQA A-Level Physics Notes

2.2

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3

QUANTUM PHENOMENA

Circuits thermistor

diode

varaible resistor

capacitor fuse

inductor

lamp

Circuit analysis is similar to GCSE. Important things to remember: · For resistors in parallel, R1 = r11 + r12 + r13 + · · · . For two resistors, this can be expressed as r1 r2 R= . r1 + r2 · In series circuits, all components have the same current, but a different voltage, across them. In a parallel circuit, voltage is the same across each branch, but current varies. · The potential divider: in a circuit with multiple resistors, the voltage across each is proportional to its share of the total resistance – for a circuit with total resistance R, a component with resistance r has Rr of the total voltage across it. · Batteries have an internal resistance that is hard to measure. They can be modelled as a perfect battery with an attached perfect resistor.

ε = I(R + r )

ε = emf in V, I = current in A, R = external resistance in Ω, r = internal resistance in Ω. ε

r

R

3 3.1

Quantum Phenomena The photoelectric effect

When light of a sufficiently high frequency is shone on a metal surface, it is able to liberate electrons from that surface. The following observations can be made about the effect: 1. For a given metal, no electrons are emitted unless the light is of a sufficiently high frequency, 2. The maximum kinetic energy of the emitted electrons is determined by the maximum frequency of the light, 3. The rate of electron emission is proportional to the intensity of the light.

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4

MECHANICS

The wave theory of light cannot explain observations 1 and 2 – if light were a wave, the emission of electrons and their kinetic energy should be determined by the intensity of the light. This is explained by the particle model of light, where an electron can only be liberated if the energy of an incident photon exceeds its work function, φ: h = Planck’s constant, f = frequency of incident electron in m, φ = work function of metal in J.

hf > φ

After the photoelectron is liberated, its maximum energy and frequency are determined by hf = φ + Ek

(max)

h = Planck’s constant, f = frequency of incident electron in m, φ = work function in J, Ek (max) = maximum kinetic energy of emitted electrons in J.

If photoelectron energy and incident frequency are plotted against each other, a number of things can be calculated from the graph – the gradient is Planck’s constant, the x-intercept is the threshold frequency, and the y -intercept is the work function.

3.2

Wave-particle duality

Diffraction and interference suggest that light is a wave, but the photoelectric effect suggest that it’s a particle. It was therefore predicted that things which are classically thought of as particles (ie. electrons) could exhibit wave-like properties as well. This is expressed by h λ= mv

4

λ = de Broglie wavelength in m, h = Planck’s constant m = particle mass in kg, v = particle velocity in m s−1

Mechanics

4.1

Newton’s laws of motion

1. Objects in motion stay in motion, and objects at rest stay at rest, unless acted on by an external force. 2. The vector sum of the forces on an object are equal to its mass multiplied by its vector acceleration. 3. When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

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4.2

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MATERIALS

SUVAT Variable excluded s u v a t

5

Equation v = u + at s = v t − 21 at 2 s = ut + 12 at 2 s = 21 (u + v ) · t v 2 = u 2 + 2as

Materials

5.1

Core vocabulary · · · · · · · · ·

5.2

5

Brittle – fractures before undergoing plastic deformation. Ductile – can be drawn into wires. Hard – is resistant to being scratched or indented. Malleable – can be beaten into thin sheets. Elastic – returns to its original shape after being deformed. Plastic – does not return to its original shape after being deformed. Strong – withstands large static loads without failing. Tough – withstands large dynamic loads without failing. Stiff – resistant to deformation by tension or compression (i.e. Young’s modulus is high).

Hooke’s law

Hooke’s Law states that the extension of a spring is proportional to the force applied, as long as it does not exceed its limit of proportionality (P). F = k∆L

F = force in N, k = spring constant in N m−1 , ∆L = exention in m.

Once the extension passes the elastic limit (E), deformation becomes permanent. F is the point at which the material fractures.

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E

6

WAVES

F P

∆L / m

F /N The gradient of the graph (up to P) is equal to k, and the area of the shaded region is equal to the energy required to produce the corresponding extension, and conversely, the amount of energy stored by the extended material. E = 21 F ∆L

E = energy stored in J, F = force in N, ∆L = extension in m.

E = 12 k∆L2

E = energy stored in J, k = spring constant in N m−1 , ∆L = extension in m.

When springs are in series, they have an effective spring constant of K = k1 + k2 , and when in parallel, 1 1 1 = + . K k1 k2

5.3

Young’s modulus

Young’s modulus is a property of a material that quantifies its stiffness. σ=

F A

σ = tensile stress in Pa, ε = tensile strain, ∆L F = force in N, ∆L = extension in m, ε= L A = area in m2 . L = length in m. E = Young’s modulus in Pa, σ E= σ = tensile stress in Pa, ε ε = tensile strain.

The Young’s modulus of a material and the amount of work done per unit volume can also be found as the gradient of and area under a graph of stress against strain.

6 6.1

Waves Core vocabulary · Progressive waves – waves whose oscillations travel and do not stay about a fixed point; this type of wave transfers energy.

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WAVES

· Transverse waves – waves whose oscillations are perpendicular to the direction of energy transfer. (e.g. light). · Longitudinal waves – waves whose oscillations are parallel to the direction of energy transfer. (e.g. sound). · Coherent – waves are coherent if they have the same frequency and wavelength, and have a constant phase difference.

6.2

Phase difference

Two points are in phase if they are a whole wave apart – their oscillations will be in time with each other. Two points are in antiphase if they are half a wavelength apart, and oscillate perfectly out of time. φ=

6.3

2πx λ

φ = phase difference in rad, x = point separation in m, λ = wavelength in m.

(??)

Polarisation

When the oscillations of a transverse wave are confined to one plane, the wave is said to be polarised. The fact that light can be polarised was used to prove that EM waves are transverse. Applications: Used for polarised sunglasses, which reduce glare from sunlight reflected off water. Radio and TV broadcasts also use this to reduce interference.

6.4 6.4.1

Stationary waves Superposition

When two waves of similar natures meet, the resultant wave depends on their amplitudes and relative phase difference: · If they are in phase, constructive interference occurs, · If they are 180°/π rad out of phase and have the same amplitude, destructive interference occurs, and they cancel out. The Principle of Superposition states that the resultant displacement caused by two waves arriving at a point is the sum of the two displacements caused by each wave at that instant. 6.4.2

Stationary wave formation

Stationary waves form when two waves of the same frequency travel in opposite directions, forming nodes (points of zero displacement) and antinodes (points of maximum displacement).

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6

WAVES

` λ = 2` First harmonic, Fundamental frequency

λ=` Second harmonic, First overtone

λ = 23 ` Third harmonic, Second overtone

λ = 12 ` Fourth harmonic, Third overtone

In general, a standing wave with n antinodes and of length ` has λ = n2 `. The fundamental frequency is given by f0 =

c 2`

f0 = fundamental frequency in Hz, c = wave speed in m s−1 , ` = wave length in m.

(?)

Additionally, for a string, 1 f0 = 2`

6.5

s

T µ

f0 = fundamental frequency in Hz, ` = wave length in m, T = tension in the string in N, µ = mass per unit length of string in kg m−1 .

Refraction

Definition: a change of direction caused by a speed change crossing between media. 6.5.1

Refractive index

Absolute refractive index (for an EM wave):

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AQA A-Level Physics Notes

n=

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6

WAVES

n = refractive index (dimensionless), c = speed of light in vacuum, cs = speed of light in medium in m s−1 .

c cs

Specific refractive index: n=

n2 n1

n = refractive index, n2 = absolute r.e. of medium 2, n1 = absolute r.e. of medium 1.

where the wave is passing from medium 1 to medium 2. 6.5.2

Snell’s Law

When passing from medium 1 to medium 2, n1 sin θ1 = n2 sin θ2 where n are refractive indices, and θ are the angles between the path of the ray and the normal to the surface. When n1 > n2 , the beam bends away from the normal, and then n1 < n2 , the beam bends toward the normal.

θ2 n2 n1 θ1

6.5.3

Total internal reflection

If the angle of incidence is sufficiently large and the wave is moving from a more dense to a less dense medium (i.e. n1 > n2 ), total internal reflection can occur if the angle of refraction (θ2 ) would be more than 90°. The angle this occurs at is called the critical angle, is denoted by i or θc , and is given by sin i =

6.5.4

n1 n2

i = critical angle, n1 and n2 are refractive indices.

Fibre optics

Fibre optic cables consist of an optically dense core surrounded by a less optically dense cladding. Light travels through the core; the cladding reflects light from within the core and protects the core from scratches. Advantages of using fibre optics over conventional wires: · Immune to electromagnetic interference, · Do not heat up, 12

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· Transmit signals more efficiently and need fewer repeaters, · Do not corrode, · Allow a higher bandwidth and transmit more data. Issues with using fibre optics: · Absorption reduces the strength of the signal, requiring repeaters to be used at certain intervals. · Dispersion – Modal dispersion occurs when multiple rays of light enter the fibre at different angles, and take different times to reach the destination. Single-mode fibres only allow light to take one path, and do not suffer from this effect. – Material dispersion occurs because different wavelengths of light travel at different speeds in the fibre. This can be counteracted by using monochromatic light Both of these effects cause pulse broadening, where signals become spread out in the fibre. This can cause binary digits to overlap and become muddled.

6.6

Interference

Waves can interfere with each other when two similar waves exist at the same point in space. For this to happen, the waves must be coherent or have a constant phase difference. The pattern produced depends on the phase difference. 6.6.1

Young’s double slit experiment

In this experiment, light, (preferably a laser, as its light is coherent and monochromatic) is shone through two slits onto a screen. Bright maxima occur on the screen where the beams of light from the slits are in phase (constructive interference) and dark minima are produced when the beams are out of phase (destructive interference).

Subsidiary maxima occur within the outer maxima shown above. The equation governing this pattern is λD w= s

w = distance between maxima in m, λ = wavelength in m, D = distance to screen in m, s = separation of slits in m.

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WAVES

Diffraction

Diffraction occurs when a wave passes through a gap or around an object. The greatest diffraction occurs when the width of a slit is the same as the wavelength of the light diffracting through it. 6.7.1

Single slit diffraction

When light passes through a single slit and is projected onto a screen, it forms a pattern similar to a double slit pattern. Its key features are: · The central maximum is twice as wide as the others, · The secondary maxima rapidly decrease in intensity as they get further from the center, · Minima occur when the path difference between two waves arriving at the screen is a half multiple of wavelength. · Maxima appear approximately between minima. A single slit can be used as a light source of coherent light for a dual-slit experiment. 6.7.2

Diffraction gratings

A diffraction grating is a series of tightly packed narrow slits that separates light into its constituent wavelengths; they exhibit the same subsidiary maxima as double slits.

d sin θ = nλ

d = distance between adjacent slits in m, θ = angle of order, n = number of order, λ = wavelength in m.

This formula can be derived as shown:

λ d

θ θ

Maxima occur when the path difference between two slits is exactly λ; the formula is self-evident from the diagram above and some trigonometry. Spectrometers use diffraction to analyse the wavelengths present in light.

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FURTHER MECHANICS

Further Mechanics Circular motion v m F

ω

r

θ

When an object moves at a constant velocity in a circle, because it is constantly changing direction, a force must be acting on it. The angular speed of an object is given by

ω=

v = 2πf r

ω = angular speed in rad s−1 , v = linear speed in m s−1 , r = circle radius in m, f = frequency in Hz.

The centripetal (towards the centre) acceleration is given by 2

a=

v = ω2 r r

a = acceleration in m s−2 , v = linear speed in m s−1 , ω = angular speed in rad s−1 , r = circle radius in m.

The force required to hold the object in motion is given by

F =

mv 2 = mω 2 r r

F = force in N, m = mass in kg, v = linear speed in m s−1 , r = circle radius in m, ω = angular speed in rad s−1 .

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Simple harmonic motion θ

T A

x ω

v

Simple harmonic motion (SHM) is very similar to circular motion; it describes the motion of pendulums and springs. It is defined by 2

a = −ω x

a = acceleration toward center in m s−2 , ω = angular speed in rad s−1 , x = displacement in m.

This equation can be solved to give

x = A cos ωt

x = displacement in m, A = max. displacement in m, ω = angular speed in rad s−1 , t = time in s.

√

v = ±ω A2 − x 2

v = linear speed in m s−1 , ω = angular speed in rad s−1 , A = max. displacement in m, x = displacement in m.

From these, the equations for maximum speed and acceleration can be easily derived: vmax = ωA

vmax = max. speed in m s−1 , ω = angular speed in rad s−1 , A = max. displacement in m.

2

amax = ω A

amax = max. acceleration in m s−2 , ω = angular speed in rad s−1 , A = max. displacement in m.

The restoring force can also be found for a spring, r m T = 2π k

7.3

T = tension in N, m = object mass in kg, k = spring constant in Nm−1 .

for a pendulum, r T = tension in N, ` ` = string length in m, T = 2π g g = gravity in Nkg−1 .

Resonance

The way an oscillator interacts with its surroundings varies. · In damped oscillation, energy is lost to the surroundings. – Under light and heavy damping, the amplitude of the oscillation slowly decays to zero. – Under critical damping, no oscillation occurs, and the amplitude decays to zero immediately, without crossing the zero displacement line. 16

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– Over-damping is the same as critical damping, but slower. See section 12.2 for graphs. · In free oscillation, there is no energy transfer with the surroundings. This occurs most notably at the system’s resonant frequency. · There can also be a driving force adding energy to the system. The system’s response to this force is greatest when the driving frequency and the resonant frequency are the same. Resonance is used by radios (their internal circuitry resonates at the same frequency as the broadcast signal), and it is the effect responsible for shattering glass with sound. Damping can be used to prevent certain types of damage – some buildings use dampers to avoid wind damage. Lightly damped systems respond very strongly (have a large amplitude) at their resonant frequency; this decreases with more damping.

8 8.1

Thermal Physics Thermal energy transfer

The thermal energy of a system is defined as the sum of the kinetic and potential energies of its constituent particles. Their kinetic energies are distributed in a bell curve, with a peak at the average energy of the system. At absolute zero, the particles have no energy, so clustering occurs with systems that have very low energy, and the peak of their energy distribution is higher. The temperature of a system can be changed according to the following equation:

Q = mc∆θ

Q = energy in J, m = mass in kg, c = specific heat capacity in J K−1 , ∆θ = change in temperature in K.

When a material changes state, work must be done to change its atomic structure. A certain amount of energy must enter the system for this to happen, but the system’s temperature will not change – the potential energies, not the kinetic energies, or the molecules are changing. Q = ml

Q = energy in J, m = mass in kg, l = specific latent heat in J kg−1 .

The value of l is different for different materials and state changes. Changes between solid and liquid (melting and freezing) are referred to as fusion, and changes between liquid and gas (boiling and condensing) are referred to as vaporisation.

8.2

Ideal gases

The temperature of a gas is linked to its pressure an volume by the ideal gas equation

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THERMAL PHYSICS

for moles,

for molecules, p = pressure in Pa, p = pressure in Pa, V = volume in m3 , V = volume in m3 , pV = nRT n = number of moles in mol, pV = NkT N = number of molecules, R = molar gas constant, k = Boltzmann constant in J K−1 , T = temperature in K. T = temperature in K. (Note: k = R/NA , where NA is Avogadro’s number.) This is an amalgamation of three laws, · Boyle’s Law, that pV is constant at constant T, · Charles’ Law, that V /T is constant at constant p, · the pressure law, that p/T is constant at constant V. For a gas to change volume at a constant pressure, work must be done (and usually ends up as heat). This energy is equal to the pressure multiplied by the change in volume (W = p∆V ), and can therefore be taken as the area underneath a graph of volume against pressure. 8.2.1

Kinetic theory of gases

The equations above arose from experimental evidence, but the kinetic theory of gases was derived from theory, using mathematics, before sufficiently powerful measurements were available. · Ancient Greek and Roman philosophers such as Democritus had ideas about gases, some of which turned out to be close to the truth. · Robert Boyle was the first to discover his law in 1662; Jacques Charles was second, in 1787. The pressure law was formulated by Guillaume Amontons in 1699 and Joseph Louis Gay-Lussac in 1809. · Daniel Bernoulli laid the foundations for kinetic theory by using it to explain Boyle’s law. · Robert Brown discovered Brownian motion in 1827, which supports the kinetic theory. · Kinetic theory wasn’t accepted until the 20th century, when Einstein used it to make predictions about Brownian motion. Brownian motion strongly supports the kinetic theory, because fastmoving particles in a fluid explain what Brown observed very well.

`

Q m

u

A

Imagine a cuboid container with volume V and sides of length ` containing N particles each of mass m. · Say particle Q moves directly towards wall A with velocity u. Its momentum approaching the wall is mu. It strikes wall A. Assuming the collisions are elastic, it rebounds and heads in the opposite direction with momentum −mu. Its change in momentum is then 2mu. · Assuming Q does not collide with any other particles, the time between its collisions with A is 2`/u, and the number of collisions per second is u/2`.

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· Its rate of change of momentum is therefore −2mu × (u/2`). · According to Newton’s second law, rate of change of momentum is equal to force, so the force exerted on Q by A is −mu 2 /`. · There are many particles in the box, each with a different value of u. The total force exerted on
m 2 wall A is given by F = u + u22 + · · · . ` 1
· From this, the mean squared speed can be defined as u¯2 = u12 + u22 + · · · /N. · Substituting into the equation for force, F = Nmu¯2 /`. Nmu¯2 . · The pressure on A is then given by the force divided by its area: p = V · The particles are actually moving in all three dimensions. Their overall speed can be found using Pythagoras as c 2 = u 2 + v 2 + w 2 , where u, v , and w are their velocities in the x, y , and z dimensions. · For all particles c¯2 = u¯2 + v¯2 + w¯ 2 , and because the velocities are randomly distributed, u¯2 ≈ v¯2 ≈ w¯ 2 , so c¯2 = 3u¯2 . · Substituting into the equation for pressure above,

2 pV = 13 Nmcrms

p = pressure in Pa, V = volume in m3 , N = number of molecules, m = molecular mass in kg, crms = root mean squared speed in m s−1 .

A large number of assumptions must be made for this theory to hold: 1. All molecules are equal, and their velocities are distributed randomly. 2. Collisions between the molecules and the container are perfectly elastic. 3. The molecules have negligible size compared to the container. 4. The molecules obey Newton’s laws. 5. The duration of collisions is negligible compared to the duration between collisions. To calculate the average kinetic energy of each molecule, the equation above and the ideal gas equation can be equated and rearranged to the form 12 mcr2ms as below. for molecules,

Ek =

9 9.1

3 nRT 2 N

Ek = kinetic energy in J, n = number of moles in mol, R = molar gas constant, T = temperature in K, N = number of molecules.

for temperature, Ek = kinetic energy in J, Ek = 32 kT k = Boltzmann constant in J K−1 , T = temperature in K.

Fields Gravitational fields

Gravity is the attractive field that acts between all objects with mass. It is defined by the equation

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Gm1 m2 F = r2

9

FIELDS

F = force in N, G = the gravitational constant, m = masses in kg, r = separation in m.

Gravity is much weaker than the other fundamental forces – between protons, it is weaker than the electrostatic force by a factor of around 1036 . F Gravitational field strength defines the force per unit mass (it can also be defined as g = m ) felt in an object’s field: GM g= 2 r

9.1.1

g = gravitational field strength in N kg−1, G = the gravitational constant, M = object mass in kg, r = separation in m.

Gravitational potential

The gravitational potential at a point is the amount of gravitational potential energy a unit mass at that point would have. It has a value of zero at an infinite distance from the mass, and is negative because work would have to be done to reach this zero point.

V =−

GM r

V = gravitational potential in J kg−1 , G = the gravitational constant, M = mass in kg, r = separation in m.

Work must be done to change an object’s gravitational potential energy: ∆W = m∆V

∆W = work done in J, m = mass in kg, ∆V = change in gravitational potential in J kg−1 .

Objects can, however, move without changing gravitational potential if they stay a constant distance from the mass exerting the field. In this case, they move along an equipotential surface, which has the same gravitational potential across it, and no work must be done (against gravity) to move between points on it. g can also be defined as the gradient of gravitational potential with respect to distance, and is sometimes referred to as potential gradient. ∆V g=− ∆r 9.1.2

g = gravitational field strength in N kg−1 , ∆V = change in gravitational potential in J kg−1 , ∆r = change in distance in m.

Orbits

Characteristics of an object’s orbit can be found by equating the equation for force by gravity and the equation for force in circular motion. This reveals that T 2 ∝ r 3 , or that the time period of a satellite is unaffected by its mass.

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FIELDS

In a circular orbit, satellites remain on the same equipotential at all times, and never change speed, and so do no net work. Satellites in elliptical orbits exchange potential energy for kinetic energy and back again, but also do no net work. Synchronous satellites orbit with the same period as their planet. Geostationary satellites are synchronous satellites in the same plane as the equator, and so remain above the same point on the Earth at all times. For the Earth’s time period and mass, geostationary satellites orbit 42,000 km from the center of the Earth, or 36,000 km above the Earth’s surface. Low-orbit satellites orbit under 2,000 km above the surface, and have a number of advantages. They are cheaper to launch, require less powerful communications equipment, and can be used to take detailed pictures of the Earth. However, their low orbit means that they move very fast, and large numbers of them are needed for constant coverage. Their orbits usually pass over the north and south poles, so they cover all parts of the Earth. 9.1.3

Escape velocity

It can be derived from the equations for kinetic energy and gravitational potential energy that the escape velocity, or the velocity required to fully escape a planet’s gravitational pull, is given by r v=

9.2

2GM r

v = escape velocity in m s−1 , G = the gravitational constant, M = planet mass in kg, r = distance in m.

(?)

Electric fields

Electric fields are very similar to gravitational fields. They both follow the inverse-square law, and both allow spherical force-exerting objects to be treated as points. Electric forces, however, can be both attractive or repulsive, whereas gravity only attracts. Coulomb’s law states that: 1 Q1 Q2 F = 4πε0 r 2

F = force in N, ε0 = permittivity of free space, Q = charge in C, r = separation in m.

Permittivity is a measure of how much a medium supports the formation of an electric field; its value in a vacuum is ε0 . Electric field strength is defined the same way as it is for gravity:

E=

1 Q 4πε0 r 2

E = electric field strength in N C−1 , ε0 = permittivity of free space, Q = charge in C, r = separation in m.

The field strength in a uniform field (such as one between two plates, shown below) is given by the equation V E= d

E = electric field strength in N C−1 , V = potential difference between plates in V, d = distance between plates in m. 21

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FIELDS

+ 400 V d 0V Similarly to gravitational fields, the work done to move a charge is defined by ∆W = Q∆V

∆W = work done in J, Q = charge in C, ∆V = change in electrical potential in V.

When a charged particle enters an electric field, the field exerts a force on it, changing its path, and causing it to move in a parabola. As such, electric fields can be used to separate charged particles from uncharged ones, for applications such as spectrometers. 9.2.1

Electrical potential

As with gravitational fields, electrical potential is given by 1 Q V = 4πε0 r

V = electrical potential in V, ε0 = permittivity of free space, Q = charge in C, r = distance in m.

E, V, and r can be related graphically:

V

E E ∆V r

r ∆V E= ∆r

9.3

E = electic field strength in N C−1 , V = electrical potential in V, r = distance in m.

Z ∆V =

E dr

E = electric field strength in N C−1 , r = distance in m, ∆V = change in electrical potential in V.

Capacitance

A capacitor is an electrical component consisting of two parallel conducting plates and either a gap or a dielectric in between. Capacitance is defined as the ability of an object to store electric charge. C=

Q V

C = capacitance in F, Q = charge in C, V = voltage in V.

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FIELDS

Dielectrics

A dielectric can be used to increase capacitance. It consists of polar molecules, which, when an electric field is passed across the capacitor, create their own opposing electric field. This reduces the voltage across the plates, and increases the capacitance. This effect is governed by two equations:

C=

9.3.2

Aε0 εr d

C = capacitance in F, A = plate area in m2 , ε0 = permittivity of free space, εr = dielectric constant, d = plate separation in m.

εr =

ε1 ε0

εr = relative permittivity, ε1 = permittivity of material ε0 = permittivity of free space.

Energy stored

Charge accumulates on the plates of a capacitor, which gives them the ability to store energy. Graphically, this is represented by the area under a graph of charge against voltage, or: 1 Q2 E = 21 QV = 12 CV 2 = 2 C

9.3.3

E = energy in J, Q = charge in C, V = voltage in V, C = capacitance in F.

Charging and discharging

Capacitors charge and discharge exponentially: discharging:

t

Q = Q0 e − RC

charging: Q = charge in C, Q0 = maximum charge in C, t = time in s, R = resistance in Ω, C = capacitance in F.

  t Q = Q0 1 − e − RC

Q = charge in C, Q0 = maximum charge in C, t = time in s, R = resistance in Ω, C = capacitance in F.

Q can be substituted for V. The equation for I is the same for charging and discharging, and has the same form as the Q/V discharging formula. RC, sometimes τ, is called the time constant, and is the time taken for the capacitor to charge/discharge to around 37% of its ultimate value. Conversely, the time taken for a capacitor to charge/discharge to half of its ultimate value is around 0.69RC.

9.4

Magnetic fields

Magnetic fields are closely related to electric fields, and exert forces on objects that are themselves magnetic.

F = BI`

F = force in N, B = magnetic flux density in T, I = current in A, ` = length in m. 23

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FIELDS

They can be related to gravitational and electrical fields by defining B as force per unit current-length F (B = I` ). The tesla is the unit used to measure magnetic flux or field density, and is defined as the strength of a magnetic field that exerts 1 N of force on a 1 m long wire carrying 1 A. The force, the current, and the magnetic field are all at right angles to each other: F B I Keep in mind that this rule applies for conventional current, and is reversed for flow of charge. 9.4.1

Moving particles

When a charged particle moves in a magnetic field, it experiences a force at right angles to its motion according to the following equation:

F = BQv

F = force in N, B = magnetic flux density in T, Q = charge in C, v = velocity in m s−1 .

This is the condition for circular motion, so a charged particle in a magnetic field will tend to move in a circle. Combining the equation above and the equation for circular motion gives

r=

mv BQ

r = radius in m, m = mass in kg, v = velocity in m s−1 , B = magnetic flux density in T, Q = charge in C.

(?)

A cyclotron is a type of particle accelerator that uses this phenomenon. It consists of two large semicircular electrodes; particles enter the cyclotron, and move in a circular path around one electrode. They accelerate due to the potential difference between the electrodes, move faster through the second electrode, and the process repeats. Their radius increases as their velocity increases, so the fastestmoving particles at the edge are siphoned off. 9.4.2

Magnetic flux

Magnetic flux, measured in webers, is a measure of total magnetic flux: Φ = BA

Φ = magnetic flux in Wb, B = magnetic flux density in T, A = area in m2 .

A current is induced in a coil of wire in a magnetic field. A measure of this is flux linkage:

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FIELDS

NΦ = flux linkage in Wb, B = magnetic flux density in T, A = area in m2 , N = number of turns.

NΦ = BAN

NΦ is referred to as flux linkage. Magnetic flux is only felt at its fullest when the coil and field are at right angles to each other, so for a coil rotating in a field:

NΦ = BAN cos θ

9.4.3

NΦ = flux linkage in Wb, B = magnetic flux density in T, A = area in m2 , N = number of turns, θ = angle to field.

Electromagnetic induction

According to Faraday’s law, the emf induced in a coil is proportional to the rate of change of flux linkage, i.e. more emf is induced if the coil moves faster: ∆Φ ε=N ∆t

ε = induced emf in V, N = number of turns, ∆Φ = change in flux linkage in wb, ∆t = time in s.

For a coil rotating in a magnetic field:

ε = BANω sin ωt

ε = induced emf in V, B = magnetic flux density in T, A = area in m2 , N = number of turns, ω = angular speed in rad s−1 , t = time in s.

Lenz’s law states that the induced emf will always oppose the direction of the movement that caused it:

9.4.4

Alternating current

For a sinusoidal alternating current, its stated voltage is usually its peak voltage, or maximum displacement. However, for most of its cycle, the voltage does not have this value. A better measure of voltage and current is using root mean squared values: V0 Vr ms = √ 2

Vr ms = rms voltage in V, V0 = peak voltage in V. 25

I0 Ir ms = √ 2

Ir ms = rms current in A, I0 = peak current in V.

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NUCLEAR PHYSICS

Other relationships still hold: Pr ms = Vr ms Ir ms

9.4.5

Pr ms = rms power in W, Vr ms = rms voltage in V, Ir ms = rms current in A.

Transformers

Transformers use Faraday’s law to exchange voltage and current, according to the following relationship: Ns Vs = Np Vp

N = number of turns V = voltage in V.

However, this equation only holds for a perfect transformer. Eddy currents in real transformers are looping currents caused by magnetic flux in the transformer’s core. They generate heat and an opposing magnetic field, which reduces overall magnetic field strength. Transformer efficiency can be calculated as the ratio of input power to output power: e=

e = efficiency, I = current in A, V = voltage in V.

Is Vs × 100% Ip Vp

Transformers are used to transmit power efficiently – power losses are proportional to the square of current, so this can be reduced by exchanging a high current for a high voltage.

10

Nuclear Physics

10.1

Atomic structure

· The Greek philosopher Democritus was the first person to propose atomic theory, in the 5th century bce. He believed they are identical, and gave them a name meaning “indivisible”. · In 1804, John Dalton proposed a similar idea based on chemistry. He believed that each element is a different kind of atom, and that they are indivisible. · In 1897, J. J. Thomsom discovered that electrons could be removed from atoms of any element, which contradicted Dalton’s theory. He modelled atoms after plum pudding, with the electrons distributed throughout. · Rutherford’s scattering experiment was the first to suggest that charges within a nucleus are clustered, which led to the proposal of the nuclear model of the atom. 10.1.1

Rutherford scattering

In 1909, Ernests Rutherford and Marsden conducted an experiment to test the “plum pudding” model. They fired a beam of alpha particles at a thin gold foil; the model predicted that there should be very minimal deflection due to the diffuse charges. Instead, most of the alpha particles were not deflect at all, whilst a small but significant portion were deflect at angles as large as 180 degrees. This suggested the following conclusions: 26

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· Since the alpha particles were able to go through the foil, most of the atom must be empty space. · At the center of the atom, there must be a very positively charged nucleus, capable of deflecting alpha particles. · The nucleus must be very small, as only a small number of particles were deflected. · Most of the atom’s mass must be in the nucleus, because it was able to deflect the particles. The closest approach of an alpha particle to a gold nucleus can be estimated by equating the electrical potential energy of the gold nucleus and alpha particle to the alpha particle’s initial kinetic energy: Ek =

1 79e · 2e 4πε0 r

Ek = initial kinetic energy in J, ε0 = permittivity of free space, r = closest approach in m.

(?)

The values 79e and 2e are the charges of the gold nucleus and the alpha particle respectively.

10.2

Radiation

Nuclei emit several types of radiation, including · Alpha (α) radiation consists of 42 He2+ nuclei ejected from larger nuclei at high speed. Alpha particles have a large charge and mass, so is very ionising, and can be stopped by paper or a few cm of air. It can be blocked by smoke, and is therefore used in smoke detectors. · Beta (β) is an electron or positron. Its smaller mass and charge give it a smaller ionisation potential, and β − particles can be stopped by around 3 cm of aluminium. β + particles annihilate on contact with electrons, and consequently have a very short range. Alpha and beta radiation’s strong ionisation make them very hazardous to humans if sources are ingested. · Gamma (γ) radiation consists of photons with wavelengths less than 10 pm; they are very weakly ionising and have a very long range, requiring several cm of lead to be stopped. The type of radiation being emitted by a radioactive source can be determined by seeing what material stops it. 10.2.1

Uses of radiation

Radiation can be used to determine the thickness of a known material. This is used in industrial settings to adjust the separation of rollers. As it is less ionising, gamma is the type of radiation predominantly used in medicine. Radionuclides with a short half life, usually bonded to an organic molecule, are injected into a patient; where the molecules gather can be measured using a detector such as a PET scanner. It is also used in cancer treatment to destroy tumours. This can, however, have side effects such as burns and infertility, and is a long-term hazard to medical staff. 10.2.2

Background radiation

Radiation is ever-present. Some sources of background radiation include: · Primarily radon gas, which is released from rocks. Rocks in the ground and in buildings are also slightly radioactive.

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· Cosmic rays, usually emitted from stars. They start out as high-energy protons, but produce radiation when they collide with other particles in the atmosphere. · Living things, which contain carbon-14 and potassium-40. Levels of these isotopes stay roughly constant while an organism is alive, but they cease to be replenished after it dies, so their rates of occurrence can be used to determine how long something has been dead. · Radiation from medical and industrial equipment; this is, in almost all situations, negligible. 10.2.3

Gamma radiation intensity

Gamma radiation is emitted in all directions by a source, so its intensity at any point it proportional to the surface area of the sphere with the source at its center and the observer on its surface. This is an inverse-square relationship: I∝

1 x2

I = intensity in W m−2 , x = distance from source in m.

This can easily be tested by measuring the count rate for a gamma source at various separations. 10.2.4

Safety

When handling radioactive sources, minimising exposure is important. This can be achieved by: · Pointing directional sources away from you at all times. · Handling the source with tongs, and never touch it. · Staying as far away as possible from it except for when necessary.

10.3

Radioactive decay energy 60 27 Co

1.49 MeV β

1.33 MeV γ 60 28 Ni

Radioactive nuclei decay in a random, unpredictable fashion, but in large enough numbers, their behaviour can be statistically quantified. As the chance of decaying is the same for all radioisotopes (of the same type), the rate of decay of a sample is proportional to the number of atoms in the sample, where the constant of proportionality is the specific isotope’s decay constant: ∆N = −λN ∆t

∆N ∆t

= activity in Bq, λ = decay constant s−1 , N = number of nuclei. 28

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Instead of the decay constant, half-life is used for most purposes. Half-life and the decay constant can be related by solving the equation above:

A = A0 e

−λt

A = activity in Bq, A0 = initial activity in Bq, λ = decay constant, t = time in s.

T1/2 =

ln 2 λ

T1/2 = half life in s, λ = decay constant in s−1 .

Calculations with the decay constant and half life can be done in any unit of time, as long as the units are consistent throughout the calculation. Different half-lives can be useful or dangerous in different ways. · Short half lives are useful in medicine. Technetium-99m has a half-life of 6 hours, so stops harming the body relatively quickly after it is injected, and decays into a much more stable isotope. · Radiocarbon dating uses carbon-14, with a half-life of around 5730 years, to date dead organisms. It is only able to produce accurate results for time periods similar to its half life. · Long-lived radioisotopes can be dangerous, because they remain radioactive for a long time, and must be stored. Several of these are produced as the by-products of nuclear fission. For a nucleus to be stable, it must have a very particular ratio of protons to neutrons. · Neutron-rich nuclei tend to undergo β + decay and neutron-deficient nuclei tend to undergo β − decay. · Excessively heavy nuclei of both types decay by α emission. · γ emission occurs when a nucleus has too much energy. This usually occurs after it has decayed through another process or captured an electron. The neutron requirement for stability increases faster than the number of protons increases: consider 12 238 6 C and 92 U.

10.4

Nuclear radius

Protons and neutrons bunch together in the nucleus with roughly uniform density. The radius of the nucleus ranges from about 2 fm for hydrogen to about 12 fm for uranium. The radius can be approximated using R = R0 A

10.5

1/3

R = radius in m, R0 = a constant (1.25 fm), A = atomic mass number.

Mass defect

A bound nucleus is in a lower energy state than its constituent parts if they were free. Therefore, when a nucleus forms, it releases some energy. Energy and mass are equivalent, and this means that nuclei are lighter than the sum of their parts. The difference between their predicted mass and their actual mass is called the mass defect, and it has the same value as the binding energy of the nucleus.

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H

4

6 238

8

56

0

U

Fe

50

100 150 Nucleon number

200

250

Nuclei are most stable when the greatest binding energy is holding them together. The isotope with the most binding energy (per nucleon) is iron-56. To become more stable, larger nuclei undergo fission, and smaller nuclei undergo fusion. 10.5.1

Nuclear fusion

In nuclear fission, two small nuclei join together to make a larger nucleus, such as two hydrogen atoms fusing to form a helium atom: 2 H +1 H →3 He + γ. This releases a large amount of energy, but to overcome the electrostatic repulsion between the two hydrogen nuclei, it requires a large amount of energy to initiate as well. This process generates energy in stars.

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Nuclear reactors

turbine

steam generator

control rods

fuel rods moderator/coolant shielding 10.6.1

Structure

· The key component of a nuclear reactor is the fuel rods. These contain pellets of enriched uranium-235. When the uranium nuclei split, they release neutrons. · The neutrons enter a moderator, which is usually water or salt. The moderator slows down the neutrons with elastic collisions, which increases their likelyhood of being captured by other uranium nuclei, causing them to split and release more neutrons. · In this way, a chain reaction is occurring inside the reactor. The speed of the reaction can be adjusted using the control rods, which absorb neutrons, and are commonly made of boron. · In an ideal situation, each decay causes exactly one further decay. This usually happens when there is a critical mass of uranium present. If there is less than the critical mass, the reaction eventually dies, and if there is more, the reactor becomes supercritical and explodes. · The heat generated by the fission is removed by a coolant, which is usually also the water being used as a moderator. In a steam generator, the coolant transfers its heat to other water, which boils to become steam, and drives a turbine. It is important that the coolant, which can become radioactive, never leaves the reactor. · The shielding around the reactor prevents radiation from escaping. 10.6.2

Induced fission

Sufficiently large nuclei (atomic numbers 83 and above) have a constant random chance of undergoing spontaneous fission. In a nuclear reactor, this is sped up by bombarding the nuclei with neutrons. This is only able to happen if the neutrons are travelling sufficiently slowly, which is why the moderator is required in nuclear reactors. The two daughter nuclei have a higher binding energy than the parent,

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and the difference is released as energy. In a nuclear reactor, the most common fission reaction is 1 0n

10.6.3

92 141 1 +235 92 U →36 Kr +56 Br + 30 n + γ

Safety

A large number of measures are taken to avoid accidents with nuclear fuel. · Fuel rods are handled remotely by robots to avoid humans being exposed to their radiation. · Shielding prevents radiation from spreading from the reactor, and can sometimes contain a meltdown. · The control rods are set up in a fail-safe system, so if the system operating them becomes damaged, they fall into the reactor under gravity and shut down the chain reaction. · Nuclear waste, before it can disposed of, must be cooled. This is usually done on-site by robots in pools of coolant. After it has cooled, it must be stored securely for several times its half-life. Some fission by-products are very long lived, and no permanent solution to this problem exists yet. One of the current best solutions involves turning the waste into a glass-like substance in a process called vitrification, storing this in casks, and burying the casks.

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Astrophysics

11.1

Telescopes

11.1.1

Refracting telescopes fo

fe

h

A standard refracting telescopes uses two converging lenses with focal lengths fo for the objective lens and fe for the eyepiece lens. Its overall magnification can be calculated as M=

fo θi = fe θo

M = magnification, f = focal distances in m, θ = angles of object and image.

Refracting telescopes, however, have a number of important issues: · Chromatic aberration: Dispersion causes different wavelengths of light to refract different amounts when they enter the lenses. This causes refracting telescopes to form several separate images, each in a different colour. · Lenses with few enough defects to be used in telescopes are difficult to make. · The lenses must be supported from their edges, which can cause them to bend and distort the image. 32

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· To have a high magnification, lenses must have a long focal distance, and must be installed in a very long telescope. 11.1.2

Reflecting telescopes

Reflecting telescopes us a large concave parabolic mirror to form an image. In Cassegrain arrangement, a smaller objective mirror reflects the image into an eyepiece lens. High-quality mirrors are much easier to build than equivalent lenses, and can be supported from below, making reflecting telescopes a better choice in most circumstances. However, they suffer from spherical aberration, which occurs when the mirror is not parabolic enough, and causes light to converge on different points, distorting the image. 11.1.3

Resolving power

Resolving power, or the Rayleigh criterion, is a measure of how much detail a telescope can see. Specifically, it is defined as the minimum angular distance between two objects for the telescope to be able to distinguish them as two not one. λ θ≈ D

θ = minimum angular resolution in rad, λ = wavelength in m, D = aperture diameter in m.

The collecting power of a telescope is a measure of how much energy falls in it, and is proportional to its area, or radius squared. 11.1.4

Charge-coupled devices

Charge-coupled devices (CCDs) are used by telescopes (and most cameras) to capture images digitally. They differ from human eyes in an number of ways: Quantum efficiency Detectable spectrum Resolution Spatial resolution

Human eye 1% visible light 500 mpx. 100 µm

CCD 80% infrared, visible, UV 50 mpx. 10 µm

Additionally, human eyes do not require extra equipment, but CCDs produce digital images, which can be stored, copied, and edited. 33

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Non-optical telescopes

Radio telescopes are similar in structure to Cassegrain telescopes. They have a large collecting dish, but instead of an objective mirror and eyepiece lens, they have an antenna and an amplifier. Radio waves have much longer wavelengths than optical light, so they typically have a much worse Rayleigh criterion, and are less capable of capturing a clear image. To overcome this, radio telescopes are combined together in giant networks that function as a single telescope, and have much better resolutions. The atmosphere is opaque to most electromagnetic wavelengths, with the exception of visible light and radio waves. Infrared telescopes are usually situated in very dry areas at high altitude, because IR radiation is mostly absorbed by water vapour, whereas UV and X-ray telescopes must be attached to planes, balloons, or satellites. Infrared telescopes need to be kept very cold so their own radiation doesn’t interfere with the images they are trying to collect.

11.2

Star classification

11.2.1

Luminosity

Luminosity is a measure of the amount of energy emitted by an astronomical object per second; the Sun has a luminosity of about 426 W. Intensity is a measure of the power per unit area received by an observer (usually the Earth), and is measured in Wm−2 . Assuming that a star radiates energy in all directions equally, its intensity decreases according to the inverse-square law: P I= 4πd 2 11.2.2

I = intensity in W m−2 , P = power in W, d = distance in m.

(?)

Apparent magnitude

A star’s magnitude is a measure of how bright the star appears, and is therefore influenced by the star’s luminosity and distance from Earth. Hipparcos, a Greek astronomer, assigned stars numbers based on their apparent magnitude: 1 for the brightest, and 6 for the dimmest. In the 19th century, this was redefined as a logarithmic scale, so that a magnitude 1 star would be exactly 100 times brighter than a magnitude 6 star. Some stars were badly classified, and there are some very bright objects with apparent magnitudes that are less than one. A difference of 1 in apparent magnitude corresponds to a difference of 1001/5 , or 2.51 times. I2 ≈ 2.51m1 −m2 I1 11.2.3

I = intensities in W m−2 , m = apparent magnitudes.

(?)

Parsecs

A parsec is a unit of length in astronomy, defined as the distance at which 1 AU (astronomical unit, the mean distance between the Earth and the Sun, equal to around 150 million km) subtends an angle

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of 1 arcsecond (a sixtieth of a sixtieth of a degree). A parsec is approximately equal to 3.26 light years. 11.2.4

Absolute magnitude

The absolute magnitude of an object is defined as its apparent magnitude if it were 10 pc from Earth. m − M = 5 log

11.2.5

d 10

m = apparent magnitude, M = absolute magnitude, d = distance in pc.

Black body radiation

A black body is a theoretical object that absorbs all wavelengths of radiation and can emit all wavelengths of radiation. Stars can be modelled as black bodies. The wavelengths emitted by black bodies cluster to short wavelengths, and the higher the temperature of the body, the higher the peak. The temperature can be calculated from the peak wavelength: λmax T = 2.9 × 10−3 11.2.6

λmax = peak wavelength in m, T = temperature in K.

Stefan’s law

According to Stefan’s (or the Stefan-Boltzmann) law, the luminosity of a black body depends on its surface area and temperature:

P = σAT 4

P = power in W, σ = Stefan’s constant, A = surface area in m2 , T = temperature in K.

Luminosity can be calculated using the star’s intensity and distance from Earth. 11.2.7

Spectral classes

For historical reasons, star classification by temperature is out of order. Class O B A F G K M

Colour Temperature / K Absorption spectra blue 25 000 − 50 000 He+ , He, H blue 11 000 − 25 000 He, H blue-white 7 500 − 11 000 H, ionised metals white 6 000 − 7 500 ionised metals yellow-white 5 000 − 6 000 ionised and neutral metals orange 3 500 − 5 000 neutral metals red < 3 500 neutral metals, TiO As a mnemonic: “Oh Be A Fine Girl, Kiss Me”

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The Balmer series is a group of lines (frequencies) in hydrogen’s absorption spectrum. They are produced by electrons de-exciting to the first excitation (n = 2) state. Balmer lines are only produced at some temperatures – if the temperature is too high, most electrons will de-excite to n = 3, and if it is too low, most remain in the ground state. Balmer lines are most prominent in A-class stars, but also occur in the B class and weakly in the O and F classes. 11.2.8

Stellar evolution

· Stars are born when a cloud of gas becomes dense enough to initiate hydrogen fusion. The energy this releases exerts radiation pressure on the inside of the star, which prevents gravity from fully collapsing it. · Initially, stars have a core that is full of hydrogen. After all of the core hydrogen is fused into helium, the outward radiation pressure stops; the core contracts, and the outer layers expand, turning the star into a red giant. · The core continues to contract until it becomes dense enough for helium to start fusing, at which point the radiation pressure restores balance. The heat this generates allows hydrogen in the star’s shell to start fusing. · When the helium in the core runs out, the process repeats – carbon and oxygen are formed in the core, and helium in the shell is burned. 11.2.9

The Hertzsprung-Russel diagram

The lifecycle of a typical sun-like star is illustrated on the diagram below.

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-10

supergiants

Absolute magnitude

-5

red giants

0 ma

in s

equ

enc

e

Sun

5

white dwarves

10

15

O

B

A

F

G

K

M

Spectral class

11.2.10

Stellar death

· Small stars, such as the Sun, will not have a heavy enough core to fuse carbon and oxygen. When radiation pressure stops balancing the star, the core contracts until it is balanced by electron degeneracy pressure, forming a white dwarf. The outer layers are ejected to form a planetary nebula. · Larger stars have denser cores, and burn their fuel faster. They can become supergiants, and fuse elements up to iron. When their cores become full of iron, which can’t be fused, they collapses and then rebound as supernovae, which are responsible for the creation of the remaining elements up to uranium. Supernovae are also a source of gamma ray bursts. – If a star is massive enough (1.4 to 3 solar masses), its collapsing mass can overcome electron degeneracy pressure, causing all of the electrons in the core to be captured by protons. This forms a neutron star, a sphere roughly 20 km across composed entirely of neutrons, and with a density of around 4 × 1017 kgm−3 . – In the largest stars (more than 3 solar masses), the neutrons will continue to collapse until they form a black hole, a body with an escape velocity greater than the speed of light. The region within which this occurs is bounded by the Schwarzschild radius: 2GM Rs = c2

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Supermassive black holes, with masses hundreds of times that of the sun, are found at the centres of many galaxies – Sagittarius A∗ is at the center of the Milky Way, and has a mass of around 4.2 × 106 Suns. · A type Ia supernova occurs when a white dwarf in a binary star system leeches matter from its partner. When it reaches about 1.4 solar masses it collapses into a supernova. All type Ia supernovae have roughly the same mass, so they all have roughly the same peak apparent magnitude, and as such, are referred to as standard candles. A graph of absolute magnitude against time shows a sharp peak with a maximum at around −19.5, which slowly decays to about −16 after 40 days. · In 1998, observation of type Ia supernovae in distant galaxies revealed that the expansion of the universe is accelerating; it had previously been assumed that gravity would be slowing this process down. A candidate explanation for this is the presence of dark matter and energy, which are hard to observe.

11.3

Cosmology

11.3.1

Doppler effect

The Doppler effect causes the wavelengths of waves to change when the source and observer are moving relative to each other. Objects moving away form us appear slightly red-shifted and objects moving towards us appear slightly blue-shifted due to this effect.

z=

∆f ∆λ v =− = f λ c

z = Doppler shift, f = frequency in Hz, λ = wavelength in m, v = recessional velocity in m s−1 , c = speed of light.

(v  c) In binary star systems, the movement of spectral lines caused by the Doppler effect can be used to determine the period of their orbit. Each star will produce its own version of each line, and when the two spectral lines are the same (ie. not shifted), it means the stars are not moving relative to the observer. This occurs twice in each full rotation. Stars detected by this method are referred to as spectroscopic binaries. 11.3.2

Quasars

Quasars were discovered in the 1950s, and were originally thought to be very bright stars (quasar is short for quasi-stellar object), but they have several characteristics that set them apart. They are very strong radio wave emitters, occasionally shoot out jets of gas, and show very redshifted Balmer absorption lines. They are the most distant observable objects, with a luminosity on the order of 1040 W and at distances of tens of thousands to tens of billions of light years. The current consensus is that quasars are active black holes at the centres of galaxies; the energy is emitted by the accretion disk of matter that surrounds them.

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Hubble’s law

Distant astronomical objects appear redshifted, indicating that the universe is expanding. Hubble’s law estimates the rate of expansion: v = Hd

v = recessional velocity in km s−1 , H = Hubble’s constant, d = distance in Mpc.

There is disagreement over the value of H – estimates typically range from 50 to 100. 11.3.4

The Big Bang

The expansion and cooling of the universe suggests that in the past it was smaller and hotter. This observation informs the Big Bang theory, that the universe started out as a very hot and dense singularity. · The abundance of helium is further evidence for this; in the early universe, it would have been hot enough for hydrogen fusion to occur, which is where the disproportionately large amount of helium in the universe came from. · Cosmic microwave background radiation is radiation that was emitted by the very hot early universe. Since then, it has been redshifted to microwaves from where it was emitted. 11.3.5

Exoplanets

Exoplanets are very difficult to detect because they emit no light of their own, are very small, and are always outshone by the star that they orbit. Other techniques must be used to detect them: · Radial velocity method: Planets do not perfectly orbit their stars; instead, both orbit their common center of mass. The red- and blue-shift of the star as it moves towards and away from the Earth can be used to calculate the planet’s minimum mass. · Transit method: When exoplanets pass between the Earth and their star, the intensity of the star dips. This can be used to calculate the planet’s radius.

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APPENDIX

Appendix Data and units Quantity speed of light in a vacuum permeability of free space permittivity of free space magnitude of the electronic charge the Planck constant gravitational constant Avogadro constant molar gas constant the Boltzmann constant the Stefan constant the Wien constant gravitational field strength absolute zero angstrom astronomical unit light year parsec

Symbol c µ0 ε0 e h G NA R k σ α g ˚ A AU ly pc

Hubble constant atomic mass unit

H u

electron rest mass

me

electron charge/mass ratio proton rest mass

e/me mp

proton charge/mass ratio neutron rest mass

e/mp mn

Body Sun Earth

Value 3.00 × 108 4π × 10−7 8.85 × 10−12 1.60 × 10−19 6.63 × 10−34 6.67 × 10−11 6.02 × 1023 8.31 1.38 × 10−23 5.67 × 10−8 2.90 × 10−3 9.81 -273.15 10−10 1.50 × 1011 9.46 × 1015 2.06 × 105 3.08 × 1016 3.26 (approx.) 65 1.661 × 10−27 931.5 9.11 × 10−31 5.5 × 10−4 1.76 × 1011 1.67(3) × 10−27 1.00728 9.58 × 107 1.67(5) × 10−27 1.00867

Astronomical data Mass / kg Mean radius / m 1.99 × 1030 6.96 × 108 24 5.97 × 10 6.37 × 106

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Units m s−1 H m−1 F m−1 C Js N m2 kg−1 mol−1 J K−1 mol−1 J K−1 W m−2 K−4 mK N kg−1 , m s−2 °C m m m AU m ly km s−1 Mpc−1 kg MeV kg u C kg−1 kg u C kg− 1 kg u

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Unit force energy power charge voltage capacitance resistance magnetic field strength

Units Symbol N J W C V F Ω T

Letter T G M k c m µ n p f

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Dimensions kg m s−2 kg m2 s−2 kg m2 s−3 sA kg m 2 s−3 A−1 kg−1 m−2 s−4 A2 kg m2 s−3 A−2 kg s−2 A−1

SI prefixes Name tera giga mega kilo centi milli micro nano pico femto

Order 12 9 6 3 −2 −3 −6 −9 −12 −15

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proton neutron

Particle masses Symbol Rest energy / MeV γ 0 νe , νµ negligible e± 0.510999 µ± 105.659 π± 139.576 0 π 134.972 K± 493.821 K0 497.762 p 938.257 n 939.551

SI base Name metre kilogram second ampere kelvin mole candela

units Measure length mass time electric current temperature amount luminous intensity

Name photon neutrino electron muon π-meson K-meson

Symbol m kg s A K mol cd

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APPENDIX

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Damping graphs Light damping

Heavy damping

Over-damping

Critical damping

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APPENDIX