G485 eBook OCR PHYSICS A

OCR Physics A

G485 Forces, Particles &

Frontiers of Physics Ebook

Table of Contents G485 Forces, Particles & Frontiers of Physics Syllabus Electric Fields Electric Field (Introduction)

Uniform Electric Fields Radial Electric Fields Comparing Electric and Gravitational Fields

Magnetic Fields

Magnetic Field Patterns Magnetic Force on a Straight Wire Magnetic Force on a Moving Charge Velocity Selector Mass Spectrometer

Electromagnetism

Magnetic Flux and Flux Linkage Electromagnetic Induction Lenzs Law A.C. Generator Transformers

Capacitors Capacitors and Capacitance Capacitors: Charging and Discharging

Capacitors in Series and Parallel Energy Stored in Capacitors Uses of Capacitors Capacitors Discharging through Resistor: Exponential Decay

Nuclear Atom

Rutherfords Alpha Scattering Experiment Elementary Particles: Leptons, Hadrons and Quarks Strong Nuclear Force

Radioactivity

Radioactive Transformations Radioactive Properties Radioactive Decay (Activity and the Decay Constant) Radioactive Decay (Exponential Decay) Radioactivity Decay Comparison with Capacitor Discharge Uses of Radioactive Isotopes: Smoke Alarms and Carbon Dating

Nuclear Physics

Einsteins Mass-Energy Equation: E = mc2 Nuclear Binding Energy Fission and Fusion

Nuclear Fission Reactor Medical Imaging

X-ray Production Intensity of an X-ray Beam X-rays Interaction with Matter X-ray Imaging Contrast Medium Image Intensifiers Computed Axial Tomography (CAT) Scan Ultrasound Production Acoustic Impedance and Impedance Matching Ultrasound Imaging: A and B Scans Doppler Ultrasound Radionuclide Imaging Gamma Camera Medical Tracers Positron Emission Tomography (PET) Principles of Magnetic Resonance MRI Scanner Magnetic Resonance Imaging (MRI)

Cosmology

Cosmological Distances Stellar Evolution Doppler Effect Hubbles Law Age of the Universe Olbers Paradox Cosmological Principle Origin of the Universe: Big Bang Theory and Evidence Evolution of the Universe Future of the Universe

G485 Forces, Particles & Frontiers of Physics Glossary Physics Data G481 Mechanics Formulae G482 Electrons, Waves & Photons Formulae G484 Newtonian World Formulae G485 Forces, Particles & Frontiers of Physics Formulae

G485 Forces, Particles & Frontiers of Physics Syllabus 5.1.1 Electric fields (a) State that electric fields are created by electric charges; (b) Define electric field strength as force per unit positive charge; (c) Describe how electric field lines represent an electric field;

(d) Select and use Coulombs law in the form2

04 r

QqF

;

(e) Select and apply 2

04 r

QE

for the electric field strength of a point charge;

(f) Select and use E = V/d for the magnitude of the uniform electric field strength between charged parallel plates; (g) Explain the effect of a uniform electric field on the motion of charged particles; (h) Describe the similarities and differences between the gravitational fields of point masses and the electric fields of point charges. 5.1.2 Magnetic fields (a) Describe the magnetic field patterns of a long straight current-carrying conductor and a long solenoid; (b) State and use Flemings left-hand rule to determine the force on current conductor placed at right angles to a magnetic field; (c) Select and use the equations F = BIL and F = BILsin; (d) Define magnetic flux density and the tesla; (e) Select and use the equation F = BQv for the force on a charged particle travelling at right angles to a uniform magnetic field; (f) Analyse the circular orbits of charged particles moving in a plane perpendicular to a uniform magnetic field by relating the magnetic force to the centripetal acceleration it causes; (g) Analyse the motion of charged particles in both electric and magnetic fields; (h) Explain the use of deflection of charged particles in the magnetic and electric fields of a mass spectrometer (HSW 6a). 5.1.3 Electromagnetism (a) Define magnetic flux; (b) Define the weber. (c) Select and use the equation for magnetic flux =BAcos; (d) Define magnetic flux linkage; (e) State and use Faradays law of electromagnetic induction; (f) State and use Lenzs law; (g) Select and use the equation: induced e.m.f. = rate of change of magnetic flux linkage; (h) Describe the function of a simple ac generator; (i) Describe the function of a simple transformer; (j) Select and use the turns-ratio equation for a transformer; (k) Describe the function of step-up and step-down transformers. 5.2.1 Capacitors (a) Define capacitance and the farad; (b) Select and use the equation Q = CV; (c) State and use the equation for the total capacitance of two or more capacitors in series; (d) State and use the equation for the total capacitance of two or more capacitors in parallel;

(e) Solve circuit problems with capacitors involving series and parallel circuits; (f) Explain that the area under a potential difference against charge graph is equal to energy stored by a capacitor; (g) Select and use the equations W = QV and W = CV2 for a charged capacitor; (h) Sketch graphs that show the variation with time of potential difference, charge and current for a capacitor discharging through a resistor; (i) Define the time constant of a circuit;

(j) Select and use time constant = RC;

(k) Analyse the discharge of capacitor using equations of the form RCtexx /0 ;

(l) Explain exponential decays as having a constant-ratio property; (m) Describe the uses of capacitors for the storage of energy in applications such as flash photography, lasers used in nuclear fusion and as back-up power supplies for computers (HSW 6a). 5.3.1 The nuclear atom (a) Describe qualitatively the alpha-particle scattering experiment and the evidence this provides for the existence, charge and small size of the nucleus (HSW 1, 4c); (b) Describe the basic atomic structure of the atom and the relative sizes of the atom and the nucleus; (c) Select and use Coulombs law to determine the force of repulsion, and Newtons law of gravitation to determine the force of attraction, between two protons at nuclear separations and hence the need for a short range, attractive force between nucleons (HSW 1, 2, 4); (d) Describe how the strong nuclear force between nucleons is attractive and very short-ranged; (e) Estimate the density of nuclear matter; (f) Define proton and nucleon number;

(g) State and use the notation XAZ for the representation of nuclides;

(h) Define and use the term isotopes; (i) Use nuclear decay equations to represent simple nuclear reactions; (j) State the quantities conserved in a nuclear decay. 5.3.2 Fundamental particles (a) Explain that since protons and neutrons contain charged constituents called quarks they are, therefore, not fundamental particles; (b) Describe a simple quark model of hadrons in terms of up, down and strange quarks and their respective antiquarks, taking into account their charge, baryon number and strangeness; (c) Describe how the quark model may be extended to include the properties of charm, topness and bottomness; (d) Describe the properties of neutrons and protons in terms of a simple quark model; (e) Describe how there is a weak interaction between quarks and that this is responsible for decay; (f) State that there are two types of decay; (g) Describe the two types of decay in terms of a simple quark model; (h) State that (electron) neutrinos and (electron) antineutrinos are produced during + and - decays, respectively; (i) State that a - particle is an electron and a + particle is a positron; (j) State that electrons and neutrinos are members of a group of particles known as leptons.

5.3.3 Radioactivity (a) Describe the spontaneous and random nature of radioactive decay of unstable nuclei; (b) Describe the nature, penetration and range of -particles, -particles and -rays; (c) Define and use the quantities activity and decay constant; (d) Select and apply the equation for activity A = N; (e) Select and apply the equations A = A0e-t and N = N0e-t where A is the activity and N is the number of undecayed nuclei; (f) Define and apply the term half-life; (g) Select and use the equation t = 0.693; (h) Compare and contrast decay of radioactive nuclei and decay of charge on a capacitor in a CR circuit (HSW 5b); (i) Describe the use of radioactive isotopes in smoke alarms (HSW 6a); (j) Describe the technique of radioactive dating (i.e. carbon-dating). 5.3.4 Nuclear fission and fusion (a) Select and use Einsteins massenergy equation E = mc2; (b) Define binding energy and binding energy per nucleon; (c) Use and interpret the binding energy per nucleon against nucleon number graph; (d) Determine the binding energy of nuclei using E = mc2 and masses of nuclei; (e) Describe the process of induced nuclear fission; (f) Describe and explain the process of nuclear chain reaction; (g) Describe the basic construction of a fission reactor and explain the role of the fuel rods, control rods and the moderator (HSW 6a and 7c); (h) Describe the use of nuclear fission as an energy source (HSW 4 and 7c); (i) Describe the peaceful and destructive uses of nuclear fission (HSW 4 and 7c); (j) Describe the environmental effects of nuclear waste (HSW 4, 6a and b, 7c); (k) Describe the process of nuclear fusion; (l) Describe the conditions in the core of stars that make fusion possible; (m) Calculate the energy released in simple nuclear reactions. 5.4.1 X-Rays (a) Describe the nature of X-rays; (b) Describe in simple terms how X-rays are produced; (c) Describe how X-rays interact with matter (limited to photoelectric effect, Compton Effect and pair production); (d) Define intensity as the power per unit cross-sectional area;

(e) Select and use the equation I = I0 e-x to show how the intensity I of a collimated X-ray beam varies with thickness x of medium; (f) Describe the use of X-rays in imaging internal body structures including the use of image intensifiers and of contrast media (HSW 3, 4c and 6); (g) Explain how soft tissues like the intestines can be imaged using barium meal; (h) Describe the operation of a computerised axial tomography (CAT) scanner; (i) Describe the advantages of a CAT scan compared with an X-ray image (HSW 4c, 6). 5.4.2 Diagnosis methods in medicine (a) Describe the use of medical tracers like technetium-99m to diagnose the function of organs; (b) Describe the main components of a gamma camera; (c) Describe the principles of positron emission tomography (PET);

(d) Outline the principles of magnetic resonance, with reference to precession of nuclei, Larmor frequency, resonance and relaxation times; (e) Describe the main components of an MRI scanner; (f) Outline the use of MRI (magnetic resonance imaging) to obtain diagnostic information about internal organs (HSW 3, 4c and 6a); (g) Describe the advantages and disadvantages of MRI (HSW 4c & 6a); (h) Describe the need for non-invasive techniques in diagnosis (HSW 6a); (i) Explain what is meant by the Doppler effect; (j) Explain qualitatively how the Doppler effect can be used to determine the speed of blood. 5.4.3 Ultrasound (a) Describe the properties of ultrasound; (b) Describe the piezoelectric effect; (c) Explain how ultrasound transducers emit and receive high-frequency sound; (d) Describe the principles of ultrasound scanning; (e) Describe the difference between A-scan and B-scan; (f) Calculate the acoustic impedance using the equation Z = c;

(g) Calculate the fraction of reflected intensity using the equation 2

12

212

0 )(

)(

ZZ

ZZ

I

Ir

;

(h) Describe the importance of impedance matching; (i) Explain why a gel is required for effective ultrasound imaging techniques. 5.5.1 Structure of the universe (a) Describe the principal contents of the universe, including stars, galaxies and radiation; (b) Describe the solar system in terms of the Sun, planets, planetary satellites and comets; (c) Describe the formation of a star, such as our Sun, from interstellar dust and gas; (d) Describe the Suns probable evolution into a red giant and white dwarf; (e) Describe how a star much more massive than our Sun will evolve into a super red giant and then either a neutron star or black hole; (f) Define distances measured in astronomical units (AU), parsecs (pc) and light-years (ly); (g) State the approximate magnitudes in metres, of the parsec and light-year; (h) State Olbers paradox; (i) Interpret Olbers paradox to explain why it suggests that the model of an infinite, static universe is incorrect (HSW 7);

(j) Select and use the equation

c

v;

(k) Describe and interpret Hubbles red shift observations; (l) State and interpret Hubbles law (HSW 1 & 2); (m) Convert the Hubble constant H0 from its conventional units (km s-1 Mpc-1) to SI (s-1); (n) State the cosmological principle; (o) Describe and explain the significance of the 3K microwave background radiation (HSW 1). 5.5.2 The evolution of the universe (a) Explain that the standard (hot big bang) model of the universe implies a finite age for the universe (HSW 1, 2, 7); (b) Select and use the expression age of universe 1/H0; (c) Describe qualitatively the evolution of universe 10-43 s after the big bang to the present; (d) Explain that the universe may be open, flat or closed, depending on its density (HSW 7);

(e) Explain that the ultimate fate of the universe depends on its density; (f) Define the term critical density;

(g) Select and use the expression for critical density of the universe G

H

8

3 200 ;

(h) Explain that it is currently believed that the density of the universe is close to, and possibly exactly equal to, the critical density needed for a flat cosmology (HSW 7).

Back to Table of Contents

Electric Fields

Electric Fields

Due to Objects with charge. Charge creates an electric field in the space around it.

Definition Electric field strength at a point is defined as the force exerted per unit positive charge placed at that point.

Equation E = F/Q

Unit N C-1

Direction Direction a positive charge moves i.e. from positive to negative.

Electric field lines leave/meet charged surfaces at 90o.

Uniform Field Radial Field

e.g. charged parallel plates point charges / charged spheres


d

- +

+

Uniform Electric Field

Uniform Electric Field

Example Between charged parallel plates

Diagram

Equation E = V d

Units V m-1 (N C-1)

Application Charged parallel plates can be used to deflect charged objects.

In the direction of the E field charge Q will accelerate towards the negative plate.

In the perpendicular direction to the E field the charge will move at a constant speed. Why? No force is acting in that direction!


d

0 V V

-

+

u +Q

Radial Electric Fields

Electric Field Strength Coulombs Law

What is it about?

Radial field strength due to point charges or charged spheres

Electrostatic force between two point charges

Diagram

Description Electric field strength is directly proportional to the charge Q of the object and is inversely proportional to the square of distance r away.

The force between two charges Q1 and Q2 separated by distance r is directly proportional to the product of the charges and is inversely proportional to the square of the distance r.

Proportionality constant

k = 1/(40)

where 0 is the permittivity of free space

0 = 8.85 1012 F m1

Equation E =kQ r2

F = kQ1Q2 r2

Further information

Charges Q1 and Q2 exert equal/opposite forces on each other thus obeying Newtons third law of motion.

Like charges force F is positive repulsive.

Unlike charges force F is negative attractive.


r

Q2

Q1 F F

+Q

Comparing Electric and Gravitational Fields

Electric Gravitational

Due to objects with charge mass

Definition of field strength at a point

force per unit charge (E = F/Q)

force per unit mass (g = F/m)

Unit of field strength N C-1 N kg-1

Direction of field strength

+ve to -ve towards centre of mass

Formula for field strength for a point object

2r

kQE 2r

GMg

Name of Law and formula for force between point objects

Coulombs law

221

r

QkQFE

Newtons law of gravitation

221

r

MGMFg

Proportionality constant

04

1

k

where 0 is the permittivity of free space = 8.85 10-12 F m-1

Universal gravitational constant, G = 6.67 10-11 N m2 kg-2

Nature of force i.e. attractive and/or repulsive

Attractive (-ve) for unlike charge or repulsive (+ve) for like charges

Always attractive (hence negative sign)

Uniform Fields

Charged parallel plates

E = V/d

Near Earths surface g = 9.8 N kg-1


Q

E

+

-

v g

m v

Magnetic Fields Patterns

Long straight wire Solenoid

Diagram of

magnetic

field pattern

Direction of

magnetic

field

Right-hand grip rule:

Like a bar magnet

The direction of the current gives the pole ends.


current outwards

current inwards

-

I

B

Magnetic Force on a Straight Wire

The magnetic force exerted on a current-carrying conductor placed at right angles to a

magnetic field of flux density B is given by

Unit of flux density is tesla, T.

When the current elemperpendicular to the field contributes to the magnetic force:

Flemings Left Hand Rule Direction of Magnetic Force

If the current direction is at right angles to a magnetic field then the force exerted on current carrying conductor is perpendicular to both the current and field directions.

Remember FBI.


I Ll

B

field decreases above conductor

N

S

F

field increases below conductor

N

S

Current going out of board

F = BIL

F = BILsin

Force F

field (flux density B)

conventional current I

Charged Particle Moving in a Magnetic Field

A charged particle Q moving at velocity v right angles to a magnetic field of flux density B with experience a magnetic force.

The force is given by

The charged particle follows a circular path. Explain why. Velocity is always perpendicular to the magnetic field. The magnetic force is always perpendicular to the motion (velocity) from FLHR.

BQv = mv2/r Orbital Frequency

The frequency is independent of the radius and the velocity.

Uses: particle accelerators (cyclotrons and synchrotons) accelerate particles moving in circles; mass spectrometers (separate atoms according to their specific mass (Q/m).


F = BQv

r

mvBQv

2

r

mvBQ

rfT

rv

2

2

B inwards

+ v

F

r

frvBQ

2

m

BQf

2

Velocity Selector

Charged particles can be deflected by both magnetic and electric fields.

In a velocity selector the B and E fields are at right angles to each other.

So that charges moving at a certain velocity can travel through undeflected when the magnetic force FB is equal / opposite to the electric force FE.

Mass Spectrometer

It makes use of electric and magnetic fields to identify ions and determine their abundance.

Only ions with velocity v = E/B pass through the velocity selector undeflected.

They then follow a circular path in the uniform B field.

r

mvvBQ

2

E

rB

BE

Br

v

Br

Q

m 2

/

If the magnetic flux density for the velocity

selector B1 is different to that B2 deflecting the ions in a circular path then

E

rBB

Q

m 21

Ions deflect according to their mass m/Q ratio.

rQm /

The detector is moved to different radii to detect the different ions.

Measuring the number of ions arriving per unit time determines their abundance.


+ B into board

Q

v

FB FE

Q

v

E

circular motion parabolic

path

QEBQv BEv /

Magnetic Flux, Flux Density and Flux Linkage

Magnetic Quantity Description Definition Unit

Flux density (B) It is a measure of the number of field lines (flux) passing perpendicularly per unit area.

B = F/IL Tesla

Flux () The number of magnetic field lines passing perpendicularly through area A.

Product of flux density and area of coils perpendicular to flux.

Weber

Flux linkage (N) The total flux linking N turns of wire through area A.

Product of magnetic flux and number of turns of coils it passes through.

Weber

Electromagnetic Induction

An e.m.f. is induced when a wire cuts through magnetic field lines. Why?

Consider a wire whose length AB is moving down cutting through magnetic field B.

Charges +Q in the wire are moving downwards with the wire.

Applying FLHR, the charges experience a force along the wire.

Charges +Q move towards B and -Q charges move towards A.

This separation of charge gives rise to a voltage between the ends of the wire.

Hence an e.m.f. (electromotive force) is induced.

If the wire forms a complete circuit, then an induced current flows.

The direction of the induced current is given by Flemings right hand rule.

An e.m.f. is also induced when the flux linking the coil of wire i.e. flux linkage changes.

Laws of Electromagnetic Induction

Faradays law the induced e.m.f. is directly proportional to the rate of change of flux linkage

(N) or the rate at which the magnetic field lines are cut.

Lenzs law the direction of the induced e.m.f. is such that the induced current that flows opposes the change in magnetic flux linkage that is producing it.

The two laws can be expressed as


B

A

N S A

E.m.f. = -(N)

t

= BA

Motion (F)

Induced I

B

Lenzs Law

The direction of the induced e.m.f. is such that the induced current that flows opposes the change in magnetic flux linkage which is producing it.

Electromagnetic induction, KE electrical.

There must be a loss in KE in order to produce electricity.

If motion/KE remained the same then electrical energy would be created from nothing.

This is impossible because it violates the law of conservation of energy.

Induced currents flow so that they produce a force that opposes the motion (reduces the KE) that is being used to create them.

Energy has to be constantly put into the system in order to maintain KE and to produce the electrical energy.

Proof

From FLHR the direction of the current induced in the wire, carrying charges Q, which is moving downwards is towards B.

The now induced current-carrying wire 90o in the magnetic field will experience a force.

From FLHR the direction of the force acting on the wire is upwards.

How does this force affect the motion of the wire? Decelerates because the force is acting in the opposite direction to the motion.

Hence the induced current is flowing in a direction so as to produce a force that opposes the

motion that is creating it Lenzs law. Application

When the North pole of a magnet moves towards the coils of wire, the change in flux linkage induces an e.m.f. in the coils.

Induced current flows in the coils.

The coils become an electromagnet.

The current flows in a direction in order to produce a force that opposes the motion i.e. side A becomes a N pole in order to repel the magnet.

Thus reducing the KE of the magnet in order to produce electricity.


B

A

N S A

wire moving downwards

I

AC Alternator

Electricity can be generated by rotating a coil of wire in a magnetic field.

The change of flux linkage in the coils results in an induced voltage.

Every half cycle current direction reverses a.c. output.

Flux linkage N is when the magnetic field lines pass perpendicularly through the plane of the coil.

where is the angle between the magnetic field and the normal of the plane of the coil.

When = 0 then the magnetic field is perpendicular to plane of coil.

When flux linkage is maximum i.e. = 0 plane of coil is perpendicular to B field:

Rate of change of flux linkage is zero.

Induced e.m.f. is zero.

When flux linkage is zero i.e. = 90o plane of coil is parallel to B field:

Rate of change of flux linkage is maximum.

Induced e.m.f. is maximum.


time

flux linkage

time

induced e.m.f.

N = NBAcos t

NfmE

)(...

N S

Transformers

A transformer makes use of electromagnetic induction to increase / decrease voltage.

Two coils, the primary and secondary, are wound on an iron core.

Alternating current through the primary coil creates an alternating magnetic field (changing

magnetic flux) which links to the secondary coil.

Hence an e.m.f. is induced in the secondary coil.

N.B. The iron core links all the magnetic flux from the primary coil to the secondary coil.

In a step-up transformer the Ns > Np, so flux linkage in the secondary greater than in the primary so

the induced voltage Vs > Vp.

It the reverse in a step-down transformer.

The ratio of e.m.f.s in the secondary and primary coils is given by

If the transformer is 100 % efficient then there is no power loss.


s

p

s

p

V

V

N

N

sspp VIVI

Capacitors

Parallel metal plates which are separated by an insulating material (called a dielectric).

N.B. The positive plate connects to the positive terminal.

Capacitors store charge.

Plate A stores positive charge.

Plate B stores negative charge.

Plates A and B store equal amounts of charge.

Charge does not flow between the plates.

When the voltage across the capacitor is the same at that from the battery, charging stops.

Capacitance

Equal increases in charge Q results in equal increases in voltage V across the plates.

The proportionality constant is capacitance.

Capacitance of a capacitor is defined as the amount of charge stored per unit voltage across it.

Unit is Farad F


+

+Q -Q

A B

V

V

e

e

C = Q/V

1 F = 1 C V-1

Q V

Q = CV

100

A

470 F 100

6 V

Capacitors Charging and Discharging

When the switch is closed the capacitor is charging.

When the switch is open the capacitor is discharging through the resistor. 1. What direction does the current flow in the circuit when (a) charging? positive to negative terminal of battery (b) discharging? positive to negative terminal of capacitor (c)How can you tell this from the ammeter? Ammeter reading shows change in direction (+/-) 2. Does charge flow between the plates? Explain your

answer. No, because there is an insulating medium between them

During charging an equal amount of positive and negative charges is stored on the opposite plates of the capacitor.

When the voltage across the capacitor is equal to the voltage across the battery, charging process stops. 3. To the right is a sketch of a current-time graph during

the charging process. (a) Explain why the current decreases to zero. As charge is stored on the capacitor plates they repel further the charge being stored so current decreases (b) Sketch a current-time graph during the discharging process.

(c) The stored charge is found from the area under a current-time graph. What can you say about the areas for the charging and discharging processes? Equal


I

t

charging

discharging

Capacitors in Series and Parallel

In Series In Parallel

Circuit diagram with two capacitors C1 and C2

Equivalent circuit (combined capacitors)

Charge stored on capacitors

Same Q Shared/divided Q = Q1 + Q2

Voltage across capacitors

Shared/divided V =V1 + V2

Same V

Q = CV expression 21 C

QCQ

CQ CV = C1V + C2V

Total capacitance 21

111CCC

C = C1 + C2

Further information

Overall capacitance decreases! Overall capacitance increases!


V

C Q

V

Q

V

C Q

V

Q

V

C1 C2 Q Q

V1 V2

V

C1

C2

Q1 Q1

Q2 Q2

V

V

Q Q

Energy Stored in a Capacitor

It is found from the area under a voltage against charge graph.

W = QV

= CV2 = Q2/C

However the energy supplied by the battery to charge the capacitor is QV.

Half the energy from the battery is used do work against the electrostatic repulsion to store further charge on the plates.

Capacitors Batteries

Releases their stored energy quickly (provided the circuit is of a low resistance).

Slower release of energy.

No internal resistance. Have internal resistance which results in energy dissipation.

Uses of Capacitors

Flash Photography Lasers in Nuclear Fusion Back-Up Power Supplies for Computers

A cameras flashgun is powered by a capacitor.

The capacitor is charged from the cameras battery via a system that increases the voltage.

When a picture is taken, the energy is released very quickly through the lamp producing a bright flash.

It takes a few seconds to recharge the capacitor, so the flash cannot be used straightaway.

The capacitors are charged to thousands of volts.

Their energy is released to power lasers whose brilliant flash forces hydrogen nuclei together causing them to fuse to form helium nuclei.

The fusion of the nuclei releases energy.

Capacitors with high capacitances but relatively small volume are used as back-up power supplies.

They are charged up when the computer is in use.

If the battery or mains supply fails, they provide the electrical power needed to save data and shut down the computer safely.


Q

V

Capacitor Discharging through a Resistor

Discharging through Resistor R

Diagram

Charge Q on capacitor at time t

Q = Q0 e-t/RC - exponential decay

Graph

Time constant

( = RC)

Time taken to reach 37 % of initial value.

Voltage across

capacitor VC ( Q)

V = V0 e-t/RC exponential decay

Voltage across resistor VR

Same as capacitor (parallel) - exponential decay

Graph of VC and VR

Current I (= VR/R)

I = I0 e-t/RC - exponential decay


R

I I

C

Q

t

V

t

Rutherfords Alpha Particle Scattering Experiment

It produced the first evidence for the present model of the atom.

It replaced the generally accepted plum pudding model by Thomson.

Experiment

A thin gold foil is bombarded by high-

energy -particles.

The number of -particles deflected at

various angles was counted. Results

Angle / deg 5 15 30 45 60 75 105 120 135 150

Number scattered 8289000 120570 7800 1435 477 211 70 52 43 33

Most of the -particles passed straight through. A small number were deflected at very large angles. 1 in 8000 were back-scattered. Conclusions

The atom is mostly empty space because most of the -particles pass straight through (or experience very little deflection from the nucleus).

The nucleus is a very small compared to size of atom because only a few -particles are deflected through very large distances. These are the particles coming close to the nucleus.

Nucleus is positively charged because positively charged -particles are being repelled (deflected) by nucleus.


Elementary Particles

Fundamental particles

Non-fundamental particles

Cannot be broken into smaller particles Can be broken into smaller particles

Leptons Hadrons

electron family e- electron e+ positron

e electron neutrino

e electron anti-neutrino

consists of quarks

Mesons Baryons

Consist of two quarks

Consist of three quarks

Pions Kaons K

Neutrons n, n Protons p, p

Baryons B = 1 Anti-baryons B = -1

All particles have anti-particles same mass but opposite charge Q, baryon B number and strangeness. Quarks Believed to be fundamental particles.

Type of quark Charge (e) Baryon number (B) Strangeness (S)

up, u 2/3 1/3 0

anti-up, u -2/3 -1/3 0

down, d -1/3 1/3 0

anti-down, d 1/3 -1/3 0

Strange, s -1/3 1/3 -1

Anti-strange, s 1/3 -1/3 1

Other quarks: top, bottom and charm and their anti-quarks, anti-top, anti-bottom and anti-charm respectively.

Conservation Rules For any reaction to occur charge Q, baryon number B and strangeness must be conserved.

eepn

Q 0 + 1 = (-1) + 0

B 1 + 1 = 0 + 0 S 0 + 0 = 0 + 0

This reaction cannot take place because Q and B are not conserved.


Nuclear Forces

Protons in a nucleus exert both electrostatic and gravitational forces on each other.

Are the forces balanced?

Electrostatic Force Gravitational Force

Due to charge mass

Nature repulsive attractive

Size

20

2

4 r

QFe

= 230 N 2

2

r

GMFg = 1.86 10-34 N

So how are the protons confined in the nucleus?

Strong Nuclear Force

There must be another force acting in the nucleus, stronger than the electrostatic force.

This force holds the nucleons together but is repulsive at distances less than 0.5 fm.

The strong nuclear force is limited to a very short range (up to 3 fm).

It is believed to arise from the interaction between quarks.


p p

Fn Fn Fe Fe

1 fm

p p

Fg Fg Fe Fe

1 fm

Radioactive Transformations

Atoms that lie on the curve are stable.

Atoms that do not lie on the curve are radioactive.

Radioactive atoms decay emitting , and/or radiation.

In a radioactive transformation Z and A numbers are conserved.

decay decay decay -decay Nature Helium nucleus Fast moving (highly

energetic) electron Fast moving (highly energetic) positron. A positron is an anti-electron i.e. a negatively charged electron.

An electromagnetic wave

Why does this decay occur?

The nucleus is too heavy. The nucleus is neutron rich i.e. has too many neutrons than protons.

The nucleus is proton rich i.e. has too many protons than neutrons.

The nucleus has excess energy.

What is happening inside the nucleus?

Nucleus is losing two protons and two neutrons.

In the nucleus the neutron transforms into a proton, electron and antineutrino.

n p + e- +

In the nucleus the proton transforms into a neutron, positron and a neutrino.

p n + e+ + v

There is no change to the structure of the nucleus loses energy as a photon.

Nuclear equation

energyYX AZA

Z

4

2

4

2 energyvYX AZ

A

Z

0

0

0

11 energyvYXAZA

Z

0

0

0

11 0

0

* XX AZA

Z

Example energy42234

90

238

92 ThU energy0

1

12

6

12

5 CB energy0

1

14

7

14

8 NO 0060

27

*60

27 CoCo


Properties of Alpha, Beta and Gamma Radiation

Alpha Radiation Beta Radiation Gamma Radiation

Nature Helium nucleus Fast moving electrons/positrons

Electromagnetic wave

Ionising power Very strong Weak Very weak

Range in air Few centimetres About a metre Several kilometres (obeys an inverse-square law with

distance)

Penetrating power (stopped by)

Sheet of Paper 1 cm thick aluminium 10 cm thick lead

Speed About 5 - 10 % c About 90 % c c

Energy range About 4 to 10 MeV About 0.025 to 3.2

MeV

E = hf

where h = 6.63 10-34 J s and f is the frequency

Additional information

Alpha particles emitted from a particular source all have the same energy.

Beta particles emitted by a particular source have an almost continuous energy spectrum (i.e. range of energies)

Gamma rays emitted by a particular source can have only certain sharply-defined energies. For example, cobalt-60 emits 1.2 and 1.3

MeV gamma rays.

1 MeV = 1.6 10-13 J

Speed of electromagnetic waves in vacuum, c = 3.0 108 m s-1 Deflection in an electric field

Source


Radioactive Decay

Background Radiation

It is naturally occurring radiation that is always present in the surroundings.

When measuring the activity of a radioactive substance the background radiation must be taken into account because it always contributes to the measurement. An average reading of background radiation should be taken (without the source being present). This value should then be subtracted from the measured count rate from the source in order to eliminate its effect.

Radioactive decay

Radioactive decay is spontaneous it cannot be induced - changing external conditions (e.g. temperature or pressure) will not cause or prevent the decay of a radioactive nucleus.

Radioactive decay is a random process do not know when a particular nucleus will decay.

Activity

Activity is the number of radioactive decays per unit time.

It is measured in Becquerel (Bq).

It is given by

A = N

where is decay constant (s-1) i.e. the probability of a nucleus decaying per unit time or the fraction of nuclei that decay per unit time;

and N is number of undecayed nuclei present.

Activity decreases with time. Explain why.

As time increases, the number of undecayed nuclei present decreases (i.e. N ) and because A

N activity decreases.


Radioactivity Decay (Exponential Decay) Activity

Activity is the number of radioactive decays per unit time.

It is given by

Ndt

dNA

Rearranging and integrating gives

where N is number of undecayed nuclei remaining at time t

N0 is initial number of undecayed nuclei

It is an exponential decay i.e. constant ratio property (for equal increments in time the same fraction decays).

As A = N then

Taking logs Half-life

It is the mean time taken for half the number of nuclei to decay.

Time % left

t 50

2 t 25

3 t 12.5

For a fast rate of decay ( is large) t is short.


t = ln2

N = N0 e-t

A = A0 e-t

ln A = ln A0 -t

Radioactivity Decay Comparison with Capacitor Discharge

Radioactive Decay Capacitor Discharge

Decaying radioactive nuclei charge stored

Rate of decay Activity rate of decay of nuclei

NtN

A

Current rate of decay of charge

RCQ

tQ

I

Exponential equations teNN 0

teAA 0

RCteQQ /0

RCteII /0

Half-life

2ln21 t

2ln21 RCt

Time constant /1 RC


Uses of Radioactive Isotopes Smoke Alarms

Smoke alarms contain americium-241 which emits alpha particles.

The alpha particles ionise the air between two metal plates.

This allows current to flow. When smoke enters the alarm, the ions stick to the smoke particles.

Current flow is reduced.

When the current drops below a certain threshold, the alarm is triggered.

without smoke with smoke inside Carbon Dating

Carbon dating is based on the assumption that living things take in and give out carbon-12 as well as radioactive carbon-14, the percentage of which is assumed constant.

After death, no more carbon is taken in and the carbon-14 present decays by - emission. The ratio of carbon-14 to non radioactive carbon-12 decreases with time.

The ratio of carbon-14 to carbon-12 nuclei for the sample to be dated is determined (x).

The current ratio of carbon-14 to carbon-12 nuclei is determined (x0).

The age of the sample is found using x = x0e-t.

Limitations: (a) assumption that ratio of carbon-14-carbon-12 is constant in living things; (b) that the activity of sample may be comparable (so indistinguishable) to that of background radiation.


Einsteins Mass-Energy Relationship

E = mc2

Einstein showed that mass and energy coexist i.e. any change in mass results in a change in energy and vice versa.

E = m c2

Nuclear Binding Energy

Consider the constituents of the helium nucleus.

Separated Bound in nucleus

The mass of the nucleus of helium-4 is 6.646782 10-27 kg.

Compare the mass of this nucleus with the total mass of its constituents (mp = 1.672623 10-27 kg

and mn = 1.674929 10-27 kg).

Total mass of the nucleons inside a nucleus is less than the mass of the same nucleons when separated.

The difference in mass m is known as the mass defect.

The energy equivalent of this mass defect is called the binding energy.

It is the energy released when nucleons bind together or it is the energy needed to separate the nucleons from the nucleus.

It results from the strong nuclear force which acts to bind the nucleons.

Energy has to be supplied to remove a nucleon from the nucleus. Why?

In order to overcome the strong nuclear force.


p n

n p n

p p n

BE

A

A

Average BE per nucleon

BE = m c2

Nuclear Fission and Fusion

Fission Fusion

Description of process

It is the splitting up of a heavy nucleus to form two lighter nuclei (known as fission fragments).

It is the combining of two lighter nuclei to form a heavier nucleus.

Conditions The neutron must be at the right speed to make the heavy nucleus unstable so that the electrostatic repulsion splits the fragments apart.

High temperatures are needed to provide enough KE to overcome the electrostatic repulsion of the nuclei for the strong nuclear force to take over.

Why is energy released?

Mass of products < mass of reactants BE of products > BE of reactants Loss in mass and so an increase in binding energy.

Elements that can undergo process

A > 56 A < 56

Examples of use

Nuclear power stations Atomic bombs

Energy processes in Stars Hydrogen bombs

Nuclear Fission Reactor

Key Parts Function

Fuel rods (uranium)

Contains the fissile material (uranium) that releases energy in the reactor core.

Control rods (boron)

It ensures the chain reaction takes place at a steady rate by absorbing surplus neutrons so that exactly the right number is free to react. Controlled chain reaction: the control rods are inserted into the reactor so as to allow (on average) one neutron from previous reaction to cause subsequent fission. To slow down the reaction or stop the chain reaction, they are lowered into the core. To speed up the reaction, they are partially withdrawn.

Moderator (carbon)

Only slow (thermal) neutrons can produce the fission of uranium-235. A moderator is used to slow down the fast (energetic) neutrons that are released in the process.


A

B.E. per nucleon

Fission

56Fe

Fusion

X-rays

X-rays are very short wavelength (10-12 10-7 m) electromagnetic waves.

They are produced from the large deceleration of high-energy electrons when they hit a metal surface.

X-ray Tube and Generator

The cathode filament is heated by the a.c. current flowing through it.

Electrons are produced at the cathode by the process of thermionic emission.

A metal cup surrounding the filament focuses the electrons, which are accelerated through a high voltage, to a tungsten target embedded in a copper anode.

X-rays are produced.

Only 1 % of the electron energy is converted to X-rays.

The rest is lost as heat resulting in a high heat concentration at the anode.

But copper with a high thermal conductivity and specific heat capacity conducts the heat away.

The oil within the tube housing cools the copper and is in turn cooled by water in a heat exchanger.

For further reduction in heat, the anode is rotated rapidly so that the target area is constantly changing.

The X-rays emitted at the anode is non-isotropic i.e. they are emitted all around the tube.

A lead shielding around the tube housing absorbs this leakage radiation. Intensity of an X-ray Beam

The intensity of an X-ray beam from a point source obeys the inverse square law with distance.

The intensity of a collimated (parallel) X-ray beam remains constant.

However due to absorption processes that occur in matter the intensity I of a collimated X-ray beam varies exponentially with thickness x:

xeII 0

where - linear attenuation coefficient (units m-1)

The distance for halving the intensity is called half value thickness.

2ln2/1 x


KE of electrons X-ray photon + Heat

The Interaction of X-rays with Matter

X-rays are absorbed as they pass through matter due to three mains processes:

Process Energy range of X-ray photons / MeV

Description Diagram Conservation of Energy

Photoelectric effect

< 0.1 An X-ray photon is absorbed by an electron, transferring all its energy to the electron, which then escapes the atom.

hf = + mv2

where (= hf0) work function The X-ray photon must have a frequency above the threshold frequency i.e. f > f

0.

Compton scattering

0.5 5.0 An X-ray photon collides with an atomic electron, the electron gains some of the photon energy and the photon is scattered with a longer wavelength.

hf = + mv2 + hf The electron gains energy from the photon. The scattered photon has a longer wavelength / lower frequency i.e. f < f.

Pair production

> 1.02 An X-ray photon strikes an atomic nucleus and disappears to create an electron-positron pair.

hf = 2(mc2 + mv2) The photon energy is converted into mass energy for the electron-positron pair. Any remaining energy is converted into KE.


X-ray Imaging

Materials with a high proton number (density) absorb X-rays more than those with a low proton number.

Bone contains calcium (Z = 20) and so absorbs X-rays easily.

Soft tissue, made up largely of carbon (Z = 6), hydrogen (Z = 1) and oxygen (Z = 8) does not absorb X-rays as much.

An 'X-ray is a 'shadow image'.

The patient is placed between the X-ray machine and the film.

Parts of the body which absorb radiation e.g. bone cast an X-ray shadow which shows up white on the film. Air (e.g. lungs) appears black; soft tissues appear grey.

Contrast Medium

Several types of tissue have very similar proton numbers so they absorb the X-rays by similar amounts.

There is little contrast (difference) between them on the X-ray image.

A contrast medium, e.g. barium or iodine, is used to distinguish between them.

It has a very high proton number and so is radio-opaque i.e. it absorbs X-rays more than the surrounding tissue.

Barium sulphate which is swallowed barium meal to show the outline / shape of the gastro-intestinal tract (intestines and stomach) as white on the film.

Iodine containing contrast medium which is injected into the blood stream to show the outline the blood vessels in the X-ray image.

Image Intensifiers

Photographic film is not very sensitive to X-rays; most pass straight through.

A high dose of X-rays is needed to produce a bright enough image.

But high doses of radiation are harmful to patients.

Image intensifiers are used to reduce exposure times.

Film is more sensitive to light than to X-rays.

Double-sided film is sandwiched between two image intensifier sheets coated with phosphor.

The phosphor atoms absorb X-ray photons and are excited into a higher energy state.

They fluoresce emitting many visible light photons.

So the phosphor converts an X-ray photon into many visible light photons, which are then absorbed by the film.

A metal plate at the back of the cassette stops radiation from penetrating out of the screen.

In digital systems, phosphor atoms convert X-ray photons into many visible light photons.

These photons release electrons (by the photoelectric effect) which are then accelerated onto a fluorescent screen where they are converted into visible light flashes.

The image on this screen can be viewed on a television or stored electronically. Back to Table of Contents

Computerised Axial Tomography (CAT) Scans

Computerised Axial Tomography uses X-rays to produce cross-sectional images of the body.

Conventional X-ray images show all depths in the body superimposed on each other.

X-rays cannot be focused onto one chosen plane in the body.

However in CAT, X-rays pass through the same section of the body from different directions producing a cross-sectional image i.e. a slice.

In modern scanners the X-ray tube produces a thin fan-shaped beam that is rotated round the patient.

There is a ring of thousands of fixed detectors surrounding the patient.

The diagram on the right shows a cross-section of a CAT scanner for the X-ray source and detectors in two different positions.

The shadow cast by the body depends on the direction of the beam.

A computer processes all the data from one full rotation to construct a cross-sectional image / slice of the section of the body of interest.

Between rotations the patient is moved bit by bit through the machine and a slice is computed at each new position.

These slices can be combined to produce a 3D image.

Sections of the image can be viewed at many different angles from the computer.

CAT scanners can detect very small differences in X-ray absorption providing excellent soft tissue contrast e.g. detailed images of the brain, chest, abdominal or pelvic organs e.g. lungs, liver, kidneys, bladder; and intestines.

The patient has to remain very still and hold their breath, to ensure a sharp image.

Advantages Disadvantages

Provides more detailed information than conventional X-rays particularly soft tissues contrast.

3D images are possible any angle / section / plane can be viewed.

Significantly higher radiation doses but the dose is less than it used to be with the use of increased sensitivity of sensors.

Much more expensive than conventional X-rays

Requires a co-operative or sedated patient.


Ultrasound

Ultrasound is high frequency sound wave beyond human hearing i.e. > 20 kHz.

The Piezoelectric Effect

When high frequency alternating voltage i.e. the signal is applied across the crystal of an ultrasound transducer, the crystal expands and contracts oscillating at the forcing frequency of the signal and ultrasound waves are transmitted.

To maximise the effect, the frequency of the voltage must match the natural frequency of the crystal so that resonance can occur.

The process also works in reverse i.e. the same crystal can also act as an ultrasound receiver - ultrasound waves arriving at the crystal causes it to expand and contract that an alternating voltage is induced across the crystal.

A short pulse of ultrasound is transmitted into the patients body which gives enough time for the reflected pulse to be detected by the same receiver.

Acoustic Impedance

It is used to determine the fraction of sound intensity transmitted at a boundary.

It is defined by:

Z = c

where c - speed of sound in medium and density of medium

Units of Z: kg m-2 s-1

The fraction of reflected intensity using the equation

212

212

0 )(

)(

ZZ

ZZ

I

Ir

When Z2 = Z1, Ir = 0, i.e. no reflection.

When Z2 > Z1, Ir I0 i.e. most energy is

reflected.

Impedance Matching

There is a layer of air between the ultrasound transducer and the body.

The large difference in acoustic impedance between air

and skin most of the ultrasound is reflected.

A coupling medium (gel), with an acoustic impedance

similar to (matching) skin, is used most of the ultrasound is transmitted into the body.


Medical Uses of Ultrasound

Amplitude A-Scan Brightness B-Scan

An A-scan uses a single pulse to determine the depth and nature of reflecting surfaces.

The ultrasound partially reflects when it hits a barrier.

The time taken to receive the echo indicates the depth of barrier.

The relative amplitudes indicate the nature of the reflecting surfaces.

A B-scan builds up a 2D image.

Many elements (sensors) are used together or a single probe is moved through different positions/ angles.

The position of the dots represents the position of the reflecting surface.

The brightness of the dots represents the intensity of the reflection.

Doppler Ultrasound

The change in frequency/wavelength of the waves from a moving source is called the Doppler shift.

The Doppler shift (f) can be used to determine the speed v of blood cells.

f v

An ultrasound wave is emitted inside the body.

With a stationary blood cell there is no Doppler shift.

Blood cells moving away from the probe reflect the incident wave with a lower frequency.

In fact, there are two Doppler shifts. The first one for the incident wave and the second for the reflected wave.

Blood cells moving towards the probe reflect the incident wave with a higher frequency.

Doppler imaging uses both normal ultrasound imaging and Doppler imaging e.g. to image blood flow.


Nuclear Medicine Radionuclide Imaging

A radionuclide is 'attached' to a convenient chemical compound (medical tracer) and administered (injected or ingested) into the body.

It then travels through the body, leaving a trail of its radiation, thus allowing its path to be traced. Often, it will concentrate in a particular organ of the body, allowing detailed imaging of that structure.

To enable such a radioactive tracer to be detected outside the body, the radiation it emits

needs to be relatively penetrating, making -emitters suitable.

Gamma cameras detect the radiation, which are then used to build up an image of the distribution of the radionuclide in the body.


Assess body function rather than just structure

Whole body scanning is possible

Monitors behaviour following treatment

Poor resolution compared to X-rays

Radiation risk

Invasive procedure

Disposal of radioactive waste

Relatively high costs production and administration of radionuclide

Gamma Camera

Components Function

collimator Consists of a series of long, narrow lead tubes. It defines exactly where the

-ray originated. Only -rays perpendicular to the plane to the collimator (or parallel to the lead tubes) can pass through it.

scintillator Converts the incoming -photons into thousands of light photons.

photomultiplier tubes (PMTs)

Convert the flashes into amplified electrical pulses.

computer Analyses and processes the electrical information in order to form an image.


Medical Tracer

The medical tracers must be sterile, non-toxic, of precisely known chemical composition, compatible with the body and behave just like the substance being investigated, so that it accurately reflects body behaviour.

90 % of medical tracers use of 99m technetium (99mTc). Technetium-99m

short half-life of 6 hours

emits only -rays (safer for patient)

-rays are of a convenient energy for the gamma camera

easily attached to different compounds to form radioactive tracers

easily produced in hospitals using a generator Some of uses of technetium-99m are shown below.

Organ Investigations

blood Estimates total body plasma and blood count

bone Bone metabolism and localisation of bone disease e.g. cancer

heart and circulation Labelled red blood cells used to monitor cardiac output, blood volume and circulation. Identifies presence of blood clots (thrombosis) through build up of a tracer at those points.

kidneys and bladder Assesses blood and urine flow

liver Liver disease and disorders of its blood supply

Positron Emission Tomography

The radionuclide used in a PET scan emits positrons i.e. a +-emitter e.g. fluorine-18

The positrons annihilate with any electron present and two gamma ray photons are emitted.

It is these gamma photons that are detected.

In a PET scanner a ring of gamma detectors surrounds the patient.

They detect pairs of gamma-photons coming from inside the patient and travelling in opposite directions.

The difference in time at which they arrive at the detectors is measured.

From this the position from where they were emitted can be determined.

Gradually, a 3D image of the distribution of the radionuclide in the patient is constructed.

Any abnormal functioning can be discovered.


Magnetic Resonance Imaging

Principles of Magnetic Resonance Nuclei spin around an axis, like a spinning

top or gyroscope. This movement of the axis of a spinning

object around another axis is called precession.

In an external magnetic field, the nuclei align along the direction of the field.

In an external magnetic field, the nuclei align along and precess about the magnetic field direction.

The angular frequency of precession of nuclei in an external field B is called the Larmor

frequency (L).

L B When a radio frequency (RF) pulse of same frequency as the Larmor frequency is applied

resonance occurs. The nuclei flip from a low energy state to a high energy state.

Absorption Relaxation

When the RF pulse is removed the nuclei relaxes back to the lower energy (equilibrium) state, emitting a RF signal as it does so.

The time taken for the nuclei to return back to their lower energy state is called the relaxation time.


MRI Scanner

Main Component Function

Superconducting magnet produces the strong external field

Gradient coils produce a magnetic field that varies with position

RF coil transmits radio wave pulses into the patient

RF receiver coil detects the signal emitted by the relaxing nuclei

Computer controls the gradient coils and RF pulses and processes the signals and displays the image

Principles of Magnetic Resonance Imaging Hydrogen bound in water in tissue has long relaxation times where as in fat - short relaxation

times. The different tissues can be differentiated by their relaxation times. Gradient coils are arranged such that they alter the field strength across the length, depths and

width of the patient. Thus the Larmor frequency of the nuclei within the patient will vary across of the body. Only a small volume of the body is at exactly the right field for resonance. The computer can precisely locate the source of the RF signal within the patients body. Several scans to cover the range of frequencies are needed to construct the image. Hence it can take a long time.


Non-ionising

Differentiates well between tissues (of similar density) e.g. excellent soft tissue contrast

Higher resolution than X-ray imaging

Slow scan times

Not portable / MRI larger

Cannot be used if patient has metal implants including heart pacemakers

Examination can be claustrophobic and noisy


Cosmological Distances

Measuring Unit Definition Conversion When is it used?

The Astronomical Unit (AU)

The average distance between the Earth and Sun.

1AU = 1.5 1011 m

(1 AU 1.5 1011 m)

Within the solar system

Light-year (ly)

The distance travelled by light in one year.

1 ly = 9.5 1015 m

(1 ly 1016 m)

Distance involving stars from Earth.

Parsec (pc) A star which has a parallax of 1 arc-second is said to be one parallax second or one parsec distant from Earth.

1 pc = 3.1 1016 m

(1 pc 3 1016 m)

Distance involving stars (pc) and galaxies (Mpc) from Earth

Parsec

Parallax p is the half angle through which a star's direction changes as the Earth moves from one extremity of its orbit to another.

The distance of stars can be determined from their parallax.

A star which has a parallax of 1 arc-second is said to be one parallax second or one parsec distant from Earth.

3600 arc-seconds = 1o

p = 1 arc-second = 1/3600o

Distance star from Earth:

tanp = 1AU/x x = 1 AU/tanp x = 1 pc = 1.5 1011 / tan(1/3600o) 1 pc = 3.1 1016 m

The further away a star is, the smaller its parallax.


S 2 AU 2p

x

sec)-(arc paral lax

1(pc) Distance

Stellar Evolution


For extremely large stars (>12 MS), the gravitational pressure causes the core to shrink to such a high density that not even light can escape its gravitational field.

Low mass stars e.g. Sun

Gas and dust

Protostar

Main-Sequence

star e.g. Sun

Nebula

Supernova Explosion

Neutron star

Black Hole

Continued gravitational collapse gives rise to the very high core temperatures (-107 K), which allows the fusion of hydrogen nuclei into helium nuclei

Red Giant

For very high mass stars (8 - 12 MS) a neutron core is left behind.

Outer layers are lost

Fusion stops, core contracts due to gravity and heats up.

Interplanetary nebula

Attracted by gravitational forces

Black dwarf

Star just cools

For large stars > 8 Ms, No more core fusion star collapses due to gravity. Gravity overcomes the Fermi pressure and the electrons are forced inside the nucleus. The electrons and protons combine to form neutrons. The intense pressure in the core causes the star to explode.

The Chandrasekhar limit is the upper limit of the mass of a white dwarf; it is 1.4Ms. There is not enough mass to overcome the Fermi pressure (repulsion) of the electrons that further collapse is not possible.

White dwarf

During the gravitational collapse of the cloud GPE is converted into KE of the gas molecules thus raising the temperature of the gas.

When the hydrogen runs out, the core contracts due to gravity. The resulting increase in core temperature allows the fusion of helium nuclei into carbon nuclei. The energy released overcomes gravity to make the stars outer layers expand and cool.

The Doppler Effect

As a result of the relative motion between a source and observer, the wavelength of the wave detected by the observer is different to the wavelength emitted from the source.

The Doppler equation is given by

cvc

v

'

where ' observed/detected wavelength

wavelength emitted by source v speed of object/observer c speed of wave

Source S emits waves of wavelength and observer O detects waves at wavelength '

Source and observer are stationary

Source is moving towards observer (or vice versa)

Source is moving away from observer (or vice

versa)

Diagram

waves

'

v '

v '

Comparing and

'

'

'


S O S O S O

Hubbles Law

This is another piece of evidence that supports an expanding universe.

Consider galaxies A, B, C & D that are initially 1 Mpc apart. After 1 hour they are 2 Mpc apart.

A B C D A B C D

Relative to Galaxy A, Distance moved (Mpc) Speed (Mpc/hr)

Galaxy B 1 1

Galaxy C 2 2

Galaxy D 3 3

What conclusion can you draw? speed distance away

Hubbles law is:

This can be represented graphically as:

The proportionality constant is H0, Hubble constant.

So Hubbles law equation is where v is measured in km s-1

d is measured in Mpc H0 is measured in km s-1 Mpc-1

H0 can be found from the gradient of the v-d graph.

As more and more data is being obtained, the more accurate the value of H0 that can be determined.

The value of H0 is important because from it the age of the universe can be found. Age of the Universe

Assuming a uniform rate of expansion (galaxies moving at a constant velocity), the age of the universe can be calculated from how long ago the galaxies were all together at a single point.

A galaxy has travelled a distance d at a constant velocity of H0d from a reference galaxy when at the beginning of the universe they originated at the same point.

The time taken to travel this distance represents the age of the universe and is found using

t = s/v


The speed with which a distant galaxy moves away from Earth (and other galaxies) is directly proportional to its distance away.

v = H0d

Age = d = 1/H0 H0d

distance

speed

Cosmology

Cosmology - the study of the nature, origin and evolution of the universe.

The universe is all matter and energy that exists.

Universe was first thought to be infinite, uniform and static. Olber's Paradox

Olber argued that for an infinite uniform universe filled with stars we should see a star in any direction.

Even though the luminosity decreases the number of stars increases with distance

sky should have a similar brightness to the Sun.

The night time sky should therefore be uniformly bright - not dark.

No day or night.

Interstellar matter (gas/dust) can diminish the light from distant stars enough to make the sky dark. However the light absorbed would heat up the matter considerably causing them to glow

uniformly bright nighttime sky.

So why is the sky dark at night?

The universe is finite. Beyond the universe, stars and galaxies do not exist. Many lines of sight

reach this darkness dark nighttime sky. Also light takes time to travel across space, so not all

the light has had time to reach Earth. Also a finite universe finite number of stars their total energy would be too little to make a bright nighttime sky.

Cosmological Principle

The cosmological principle states that the universe is homogeneous and isotropic.

Homogeneous evenly distributed i.e. uniform (true over large volumes).

Isotropic physical properties e.g. temperature are the same in all directions.

The laws of physics are universal i.e. the laws of physics on Earth are the same anywhere in the universe.


Origin of the Universe

Big Bang Theory

Proposed by LeMaitre in 1927 and revised by Gamow in 1946.

The universe (all its energy and matter) was concentrated at a single point (a singularity).

The resulting infinite pressure and temperature caused the universe (space-time) to expand. Evidence to Support the Big Bang Theory

Evidence Description

Cosmic microwave background radiation

It was predicted by Gamow and was accidently detected in 1965 by Penzias and Wilson.

It is low frequency radiation that is distributed uniformly in the universe and in all directions.

Its spectrum corresponds to a black-body temperature of 2.7 K the average temperature of the universe.

It is the radiation that was emitted during the Big Bang and has since cooled down.

The original short wavelength high energy radiation has stretched as space-time has expanded, so that its wavelength has increased greatly.

Red shift of galaxies & Hubbles law

Analysis of spectra of light from distant galaxies tells us that the galaxies are moving away from each other.

The speed with which a distant galaxy moves away from Earth (and

other galaxies) is directly proportional to its distance away universe had a zero size in a finite past.

High abundance of helium in the universe

The visible matter in the universe contains about 27 % helium.

Hydrogen burning in stars only accounts for 2-3% helium.

The remaining (primordial) helium was formed in first few minutes after the Big Bang.

Evolution of the universe

Consistent with experimental results.


Evolution of the Universe

Time after Big Bang

Temperature of Universe

Feature of Universe

0 Infinite A singularity infinitesimal (approaching zero) in size and infinitely dense.

All four (gravitational, electromagnetic, strong and weak) fundamental forces are united.

10-12 s 1015 K The universe was made up of -radiation and the fundamental particles of matter i.e. quarks and leptons (e.g. electrons) and their anti-particles.

An asymmetry (unevenness) in the annihilation of particles/anti-particles left the universe with an excess of matter over antimatter.

0.01 s 1012 K Quarks combined to form protons and neutrons.

3 min 107 K Protons and neutrons combined to form H21 , H31 and

He42 nuclei.

About 25 % of the mass was in the form of (primordial) helium.

105 years 104 K Nuclei and electrons combined to form hydrogen and helium atoms.

Universe became transparent before photons were absorbed by the free electrons, but now the atoms allow the radiation to travel freely.

The universe was matter-dominated rather than radiation-dominated.

Microwave background radiation fills the universe

~106 years 4000 K Gravity became a major force: density fluctuations allowed some parts of the gas cloud with a higher density to attract neighbouring gas together to form stars, galaxies and galactic clusters.

15109years

2.7 K Approximate age of the universe

Before 0.01 s, there is no direct experimental for temperatures above 1012 K (energies are too high to be achieved). We only have theories.


The Future of the Universe

The universe is expanding.

But gravity is slowing down the rate of expansion.

There are three possible futures for the universe depending on the density of the universe.

Critical (Flat) Universe Closed Universe Open Universe

Comparing KE and GPE KE = GPE GPE > KE KE > GPE

Density of universe = critical density > critical density < critical density

Is the universe dense enough to stop the expansion?

No Yes No

Description of future

Continues to expand towards but never reaching a definite

(finite) limit.

Eventually expansion stops

contracts to the Big Crunch

Continues to expand forever

The critical density is the density of the universe that will give rise to a flat universe is given by:

G

Hc

8

3 20

Determining the mean density of the Universe is proving difficult because (a) of the presence of dark matter and dark energy (96 % of the universe). (b) if neutrinos have mass (previously thought to be massless).

Current measurements:


flat universe c

G485 Forces, Particles & Frontiers of Physics Glossary Electric Fields

electric field Created in the space around electric charge where a force acts on charged particle.

electric field strength

Force exerted per unit positive charge at a point. (E = F/Q).

Coulombs law The force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between the charges.

Magnetic Fields

magnetic flux density (B)

It is the amount of force exerted per unit length of conductor carrying per unit current perpendicular to the direction of the magnetic field.

Tesla (T) 1 T is the amount of magnetic flux density that will produce a force of 1 N per metre length of conductor carrying a current of 1 A perpendicular to the direction of the magnetic field.

Flemings left-hand rule

It shows the direction of the force on a conductor carrying a current at right angles to a magnetic field. Force thumb, field first finger, current second finger, all at right angles to each other. Used with electric motors and forces acting on current carring condctors or charges moving in magnetic fields.

velocity selector A device using electric and magnetic fields superimposed at right angles to each other in order to select a specific velocity of charged particles to pass through the fields undeflected.

magnetic flux () The product of magnetic flux density and the area of coil perpendicular to the flux.

Weber (Wb) 1 Wb is the flux when a magnetic flux density of 1 T passes perpendicularly through an area of 1 m2.

magnetic flux linkage (N)

The product of the magnetic flux and the number of turns of wire it passes through. It is related to the total flux linking / passing through N number of turns of wire.

Faradays law The magnitude of the induced e.m.f. is equal (or directly proportional) to the rate of change of magnetic flux linkage.

Lenzs law The induced e.m.f. / current acts in a direction so as to oppose the change in magnetic flux linkage that causes it.

Flemings right-hand rule

It shows the direction of the induced current when a conductor moves through a magnetic field. Force thumb, field first finger, induced current second finger, all at right angles to each other. Used in electromagnetic induction e.g generators.

a.c. generator

(alternator)

A generator that, via the use of slip rings, produces an alternating e.m.f. and so an alternating current from electromagnetic induction.

transformer, step-down

A device that has a greater number of turns of wire on the input side (primary coils) than the output side (secondary coils) and so reduces the e.m.f. and increases the electric current on the output side.

transformer, step-up

A device that has a greater number of turns of wire on the output side (secondary coils) than the input side (primary coils) and so increases the e.m.f. and reduces the electric current on the output side.

turns ratio The ratio of turns of wire on the input side to the output side of a transformer equals the ratio of the input e.m.f. to the output e.m.f.

Capacitors

capacitance The charge stored on a capacitor per unit voltage across it.

farad 1 F is the amount of capacitance when 1 C of charge stored on the plate per volt across the plates.

exponential decay It has a constant-ratio property: quantity decays by equal ratio in equal intervals of time. For example capacitor discharge and radioactive decay.

time constant (RC) The time taken for the charge stored on a capacitor to fall to 1/e (= 37%) of its original value. It is the product of resistance and capacitance of a circuit.

Radioactivity

radioactive decay The breakdown of an unstable radionuclide causing the emission of an alpha particle, beta particle or gamma photon from the nucleus.

random Radioactive decay is random; it is not possible to predict when a radioactive nucleus will decay.

spontaneous Radioactive decay is spontaneous; it cannot be induced - changing external conditions (e.g. temperature or pressure) will not cause or prevent the decay of a radioactive nucleus.

alpha particle A particle comprising of two protons and two neutrons emitted from the nucleus during alpha decay.

beta particle A high-speed (energy) electron (positron) emitted from the nucleus during beta decay.

beta minus decay The radioactive decay that causes emission of a - particle and an antineutrino from the nucleus when a neutron breaks down into a proton under the influence of the weak nuclear force.

beta plus decay The radioactive decay that causes emission of a + particle and a neutrino from the nucleus when a proton breaks down into a neutron under the influence of the weak nuclear force.

weak nuclear force The fundamental force responsible for beta decay. It occurs between leptons and during hadron decay.

activity (A) The number of radioactive decays per unit time.

becquerel (Bq) 1 Bq is 1 radioactive decay per second.

decay constant () The probability of a nucleus decaying per unit time or the fraction of nuclei that decay per unit time ( = A/N).

half-life The average time taken for the activity of a radioactive source (number of undecayed nuclei) to decrease to half its original value.

ionisation The process of adding or removing an electron from an atom. N.B. ionisation by radioactive particles only involves removal of electrons.

Nuclear Physics

nucleon A particle in the nucleus; a proton or neutron.

nucleon (mass) number

The total number of neutrons and protons inside a nucleus.

proton (atomic) number

The number of protons inside a nucleus.

isotope Atoms with same number of protons but different number of neutrons.

unified atomic mass unit

Unit of mass equal to 1/12 of the mass of a carbon-12 atom.

strong nuclear force

The fundamental force between nucleons that holds the nucleus together. It is attractive and has a very short range (~10-14 m).

binding energy The energy required to separate a nucleus into all its nucleons i.e. all the individual protons and neutrons. Or the energy released to form a nucleus from its nucleons.

binding energy per nucleon

The average energy required to remove a nucleon from the nucleus.

mass defect The difference between the total mass of the individual, separate nucleons and the mass of the nucleus.

nuclear fusion The process of two nuclei joining together and releasing energy from the increase in binding energy.

nuclear fission The process of splitting a large nucleus into two smaller nuclei, often with the emission of several neutrons. This releases energy because of the increase in binding energy of products compared with reactants.

fission products The particles and energy released when a nucleus undergoes fission.

chain reaction One reaction causing another, which causes another, etc E.g. a neutron

produced in nuclear fission can induce further fission reactions.

moderator A material e.g. carbon (graphite) used in a nuclear fission reactor to slow down the fast-moving neutrons from a fission reaction so that they have a greater chance of interacting with the fissile nuclei.

control rod Boron rod absorbs neutrons so as to reduce the rate of a nuclear fission chain reaction.

Particle Physics

fundamental particle

A particle that cannot be broken down into smaller components (e.g. leptons).

antiparticle A particle of antimatter that has the same rest mass but, if charged, the equal and opposite charge to its corresponding particle.

positron The antiparticle of the electron.

annihilation The process when a particle and antiparticle interact and their combined mass is all converted into energy via E = mc2.

lepton A fundamental particle. For example an electron or a neutrino.

neutrino A fundamental particle (lepton) with a very small mass and no charge.

quark A component of hadrons, possibly a fundamental particle. There are six types of quark; up, down, strange, charm, top and bottom.

up quark Type of quark with charge = 2/3e, baryon number = 1/3 and strangeness = 0

down quark Type of quark with charge = -1/3e, baryon number = 1/3 and strangeness = 0

strange quark Type of quark with charge = -e, baryon number = 1/3 and strangeness = -1

hadron A group of particles consisting of quarks (i.e. baryons and mesons).

baryon A group of particles consisting of three quarks (e.g. a proton or neutron).

neutron A nucleon comprising of 3 quarks (1 up and 2 down) and has no charge.

proton A nucleon comprising 3 quarks (2 up and 1 down) and has a charge of +e.

baryon number A property of baryons and quarks that is conserved in particle interactions.

strangeness The property of some quarks that is conserved in the strong interaction but not in weak interactions.

X-rays

X-rays An electromagnetic wave with wavelengths between 1012 m and 109 m.

intensity The energy incident per unit area per unit time; measured in W m2.

half-value thickness

The distance in a medium over which X-ray intensity is reduced to half its original value.

collimation Focussing an electromagnetic wave (e.g. X-rays) to provide a parallel beam.

photoelectric effect

An X-ray photon is absorbed by an electron in an atom resulting in the removal of the electron. The electron gains all the energy from the incident photon.

Compton effect The effect whereby X-ray photon ejects an electron from an atom. The X-ray photon is scattered and has a longer wavelength (lower energy). The ejected electron gains some energy from the X-ray photon.

pair production The process of creating a particleantiparticle pair from a high-energy photon. For example X-ray electron + positron. Example of conversion of energy into mass via E = mc2.

image intensifier A device used to convert X-ray photons into an increased number of visible light photons in order to enhance the X-ray image and reduce exposure times.

contrast medium Medium with a significantly higher density (proton number) compared to surroundings that easily absorbs X-rays e.g. barium is used to reveal outline / shape of soft tissues in an X-ray image.

computerised axial tomography (CAT)

A process using X-rays and computers to produce an image of a slice through the body by changing the relative positions of the X-rays and detectors.

Ultrasound

ultrasound High frequency (> 20 kHz) sound waves above the human hearing range.

acoustic impedance

The product of density of medium and its speed of sound in medium (Z =c).

impedance matching

The process of matching materials with similar acoustic impedances to maximise transmission of ultrasound through the materials (hence minimise reflection). For example a coupling gel used in ultrasound scanning.

piezoelectric effect The change in volume (expand / contract) of certain crystals when an alternating p.d. is applied across them results in the production of ultrasound. Alternatively the production of an e.m.f. when certain crystals are placed under stress when receiving ultrasound.

pulse repetition frequency

Number of ultrasound pulses per unit time in ultrasound imaging.

A scan A pulse of ultrasound is sent into the body and the echoes are detected and displayed on an oscilloscope as a voltage against time graph.

B scan A 2D image is built up, using many ultrasound elements, onto a screen positioning dots to represent the position of the reflecting surfaces and brightness determining the intensity of the reflection.

Doppler effect The change in wavelength (frequency) caused by the relative motion between the wave source and an observer.

Nuclear Medicine

medical tracer A radioactive substance that is ingested or injected into a patient.

gamma camera A detector of gamma photons emitted from a patient given a radioactive tracer. This is used to produce a real-time image of the path of the medical tracer through the body.

collimator Consists of a series of long, narrow lead tubes. It defines exactly where the -

ray originated. Only -rays perpendicular to the plane to the collimator (or parallel to the lead tubes) can pass through it.

scintillator Converts the -rays photons into thousands of light photons.

photomultiplier tubes (PMTs)

Converts light photons into amplified electrical pulses.

positron emission tomography (PET)

The detection of gamma photons produced when positrons annihilate with electrons inside the body to map o

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