Physics - Ben Lewisblewis.me/Physics/A2/Physics_A2.pdf · 1 Physics A2: Unit 4 8 As the car drives, because for the car to continue moving in a circle, the cen-tripetal force must

Physics

by Ben Lewis

1

2

Contents

1 Physics A2: Unit 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1 Circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Key de�nitions in circular motion . . . . . . . . . . . . . . 6

1.1.2 Example: Conical pendulum . . . . . . . . . . . . . . . . 6

1.1.3 Example: Vertical circle . . . . . . . . . . . . . . . . . . . 7

1.1.4 Example: Car on a bridge . . . . . . . . . . . . . . . . . . 7

1.2 Simple harmonic motion (SHM) . . . . . . . . . . . . . . . . . . . 8

1.2.1 Key de�nitions in SHM . . . . . . . . . . . . . . . . . . . 11

1.2.2 Energy interchange in SHM . . . . . . . . . . . . . . . . . 11

1.2.3 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.4 Resonance curve . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.5 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.6 Resonance and damping . . . . . . . . . . . . . . . . . . . 13

1.3 Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.1 Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.2 Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.3 Conservation of Momentum . . . . . . . . . . . . . . . . . 15

1.3.4 Newton's laws . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.5 Momentum of a photon . . . . . . . . . . . . . . . . . . . 16

1.3.6 Electromagnetic Radiation Pressure . . . . . . . . . . . . 16

1.4 Thermal physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4.1 Internal enegy . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4.2 Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4.3 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4.4 The �rst law of thermodynamics . . . . . . . . . . . . . . 19

1.4.5 Speci�c heat capacity . . . . . . . . . . . . . . . . . . . . 19

1.5 Electric �elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.5.1 Coulomb's law . . . . . . . . . . . . . . . . . . . . . . . 20

1.5.2 Electric �eld strength . . . . . . . . . . . . . . . . . . . . 20

1.5.3 Uniform electric �elds . . . . . . . . . . . . . . . . . . . . 21

1.5.4 Uses of uniform electric �elds . . . . . . . . . . . . . . . . 21

1.5.5 Electric potential . . . . . . . . . . . . . . . . . . . . . . . 22

1.6 Gravitational �elds . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.6.1 Newton's law of gravitation . . . . . . . . . . . . . . . . 22

1.6.2 Gravitational �elds . . . . . . . . . . . . . . . . . . . . . . 22

3

1.6.3 Gravitational potential . . . . . . . . . . . . . . . . . . . . 23

1.7 Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Physics A2: Unit 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1 Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 B-�elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.1 Magnetic (B) Field strength . . . . . . . . . . . . . . . . . 26

2.2.2 Force from a magnetic �eld . . . . . . . . . . . . . . . . . 26

2.2.3 The Hall Probe . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 Ion beams and particle accelerators . . . . . . . . . . . . . . . . . 29

2.3.1 Linear accelerator (Linac) . . . . . . . . . . . . . . . . . . 30

2.3.2 Cyclotron . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.3 Synchrotron . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4 Electromagnetic induction . . . . . . . . . . . . . . . . . . . . . . 32

2.4.1 Magnetic �ux . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.2 Flux linkage . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4.3 Faraday's Law and Lenz's law . . . . . . . . . . . . . . 33

2.4.4 Fleming's right and left hand rules . . . . . . . . . . . . 33

2.4.5 Alternating current and the root mean square (rms) . . . 35

2.4.6 The oscilloscope . . . . . . . . . . . . . . . . . . . . . . . 35

2.5 Radioactivity and radioisotopes . . . . . . . . . . . . . . . . . . . 36

2.5.1 Background radiation . . . . . . . . . . . . . . . . . . . . 36

2.5.2 Calculating radioactivity . . . . . . . . . . . . . . . . . . . 37

2.5.3 Uses of radioisotopes . . . . . . . . . . . . . . . . . . . . . 38

2.6 Nuclear energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.6.1 Mass-energy equivalence . . . . . . . . . . . . . . . . . . . 38

2.6.2 Stability and binding energy . . . . . . . . . . . . . . . . 38

2.6.3 The nuclear �ssion reactor . . . . . . . . . . . . . . . . . . 39

3 Physics A2: Unit 5: Option D: Medical measurement and medicalimaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1 X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.1 Computerised axial tomography (CAT or CT) scans . . . 42

3.1.2 X-ray attenuation . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Ultrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.1 Acoustical impedance . . . . . . . . . . . . . . . . . . . . 43

3.3 Doppler probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Magnetic resonance imaging (MRI) . . . . . . . . . . . . . . . . . 44

3.5 Comparison of X-rays, ultrasound and MRI . . . . . . . . . . . . 45

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3.6 Electrocardiograms (ECG) . . . . . . . . . . . . . . . . . . . . . . 45

3.7 Nuclear radiation in medicine . . . . . . . . . . . . . . . . . . . . 47

3.7.1 Measuring radiation . . . . . . . . . . . . . . . . . . . . . 48

3.7.2 Radionuclides (Radioactive tracers) . . . . . . . . . . . . 48

3.7.3 Gamma camera . . . . . . . . . . . . . . . . . . . . . . . . 49

3.7.4 Positron emission tomography (PET) scan . . . . . . . . . 50

1 Physics A2: Unit 4 5

1 Physics A2: Unit 4

Note: Before starting A2 revision, it is always a good idea to top up AS knowl-edge since most of it will still be useful, but in particular I'd highly recommendchecking out the PH1 mechanics before looking at PH4 mechanics below.

1.1 Circular motion

In order to do some calculations in physics and maths, you often need to use theradian measure for angles. Radians are calculated by dividing the circumferenceof the segment by the radius (θ = s

r ), such that 2Π rad. is a full circle, so:

1 rads. = (180

Π)◦

and

1◦ =Π

180rads.

We can use angles to describe the rate at which something turns, known asangular velocity or angular frequency :

Angular velocity =Angle turned through (in radians)

Time taken

ω =θ

t

Angular frequency = 2Π× Frequency

ω = 2Πf

and linear velocity is related to angular velocity (or frequency) by

v = ωr

and here is the acceleration (derived by the mathematical technique of di�eren-tiation)

a = ω2r

and using F = ma,

F = mω2r or F =mv2

r


1.1.1 Key de�nitions in circular motion

Period,T for an oscillating body The time taken for one complete cycle

Amplitude,A of an oscillating object The maximum value of the object's dis-placement (from its equilibrium position)

1.1.2 Example: Conical pendulum

h

L

r

mg

T

Resolve vertically:

T cos θ = mg because its not accelerating up or down

Resolve horizontally:

T sin θ = mv2

r because the horizontal component provides Fc


1.1.3 Example: Vertical circle

At the top:

T +mg =mv2

r

At the bottom:

T −mg =mv2

r

1.1.4 Example: Car on a bridge


As the car drives, because for the car to continue moving in a circle, the cen-tripetal force must increase and so the reactionary force must decrease, sincethe smallest value for the centripetal force is 0, the car must have a maximumspeed when R = 0

Resolve:

mg − 0 =mv2

r

g =v2

r

v =√gr

1.2 Simple harmonic motion (SHM)

A body exhibits SHM if the time period (T) of an oscillation is not a�ected bymagnitude.

SHM is de�ned by WJEC in two ways:

1. SHM occurs when an object moves such that its acceleration is alwaysdirected toward a �xed point and proportional to its distance from the�xed point. (a = −ω2x)

2. The motion of a point whose displacement,x changes with time,t accordingto x = A sin(ωt + ε), where A, ω and ε are constants. [Variations of thiskind are said to be sinusoidal.]

Considering a mass-spring system of a small amplitude as an example of SHM,we can show that:


T = 2Π

√m

k

where T is the time period of oscillation, m is the mass and k is the springconstantDisplacement +A Equilibrium -A

Restoring force - maximum 0 + maximumAcceleration - maximum 0 + maximumVelocity 0 ± maximum 0

This results in an acceleration-displacement graph that looks like this:

acceleration/ms-1

MAX

-MAX

displacement/x

Straight line gradient = direct proportionality

Negative gradient = when displacement is positive, acceleration and restoringforce are negative and vice versa

∴ a ∝ −x


a = −kx

and experimental evidence shows that k = ω2, where ω is the angular velocity,which we found earlier to be ω = 2Πf , so:

a = −ω2x

and since the speed is constant, the maximum acceleration must be achieved atamplitude:

amax = ω2A

If we look at the graph for displacement against time:

x/m

t/s

A

-A

The uniform acceleration means that the graph must be sinusoidal (sine/cosineshaped), and experiments show that:

x = A sin(ωt+ ε)

where ε shifts the curve dependent on where it starts with no displacement orat negative or positive amplitude.

(note: mathematically, this is found by solving the 2nd di�erential equation:dvdt + kx = 0 from F = ma = mdv

dt and F = −kx)

If we look at the graph for velocity against time:

v/ms-1

t/s

Aω

-Aω


The sinusoidal shape occurs again for velocity, however, since amplitude occursat 0ms−1, the equation must be:

v = Aω cos(ωt+ ε)

(you can also show this by di�erentiation)

Using some substitution of x into v we �nd:

v = ±ω(√A2 − x2)

and so we can see that x = 0 when v = max:

vmax = ±ωA

Note: Since the mass-spring system is a typical SHM system, all ofthese rules apply to all SHM systems.

1.2.1 Key de�nitions in SHM

Phase The phase of an oscillation is the angle (ωt + ε) in the equationx = A sin(ωt+ ε)(ε is called the phase constant)

1.2.2 Energy interchange in SHM

Kinetic energy, Ek is normally found using Ek = 12mv2, so when velocity is

naught, so is Ek. Ek cannot be negative (it is a scalar).

∴ Ek =1

2mA2ω2 when at maximum velocity (because vmax = Aω)

The Ek − t graph has a sinusoidal shape with all values above the x-axis and afrequency of 2f .

Ek and t are related by:

Ek =1

2mA2ω2 cos2(ωt+ ε)

Potential energy, Ep for a spring is found using Ep = 12kx

2 and since ω2 = km ,

∴ Ep = 12mω2x2 and so we know that Ep = 0 when x = 0 and that Ep is never

negative. When at amplitude, Ep = 12mA2x2. Epand t are related by:

Ep =1

2mA2ω2 sin(ωt+ ε)

Here is a graph of energy against time for Ekand Ep:


1.2.3 Resonance

Free oscillations [Natural oscillations] Free oscillations occur when an oscilla-tory system (such as a mass on a spring, or a pendulum) is dis-placed and released. [The frequency of the free oscillations is calledthe system's natural frequency.]

Forced oscillations These occur when a sinusoidally varying `driving' force isapplied to an oscillatory system, causing it to oscillate with thefrequency of the applied force.

Resonance If, in forced vibrations, the frequency of the applied force is equal tothe natural frequency of the system (e.g. mass on spring), the amplitude of theresulting oscillations is large. This is resonance.

Resonance can be useful such as in circuit tuning and microwave cooking. Res-onance should be avoided in some circumstances though, such as bridge design.

1.2.4 Resonance curve


When a system resonates, the driving frequency and oscillation of the body areΠ2 rads. out of phase.

1.2.5 Damping

Damping is removing the energy from an oscillating system.

Lightly (under) damped This is achieved by increasing resistive force

Heavily (still under) damped This is achieved by heavily resisting motion. E.g.in a mass-spring system, we could increase drag by moving from airto foil

Critically damped This is provided by an external system that removes energyfrom the oscillation, critical damping is extremely important in carsuspension

Over-damped Over-damped systems actually take longer to reach no amplitudethan critically damped

1.2.6 Resonance and damping

Lightly damped Slight reduction in maximum amplitude

Heavily damped Big reduction in max amplitude and slight reduction in naturalfrequency


1.3 Mechanics

1.3.1 Momentum

Momentum is de�ned as (by Newton and our syllabus):

The product of a body's mass and velocity

Momentum = Mass×Velocity

p = mv

(side note: momentum is de�ned by quantum mechanics as an operator which�acts on� the wave function, this explains why photons have momentum)

It is an object's ability to keep going. An object with lots of momentum isdi�cult to stop. If you change the velocity of an object, it's momentum changes.

Change in momentum = Final momentum− Initial momentum

4p = mv −mu

If we want to change an object's momentum, we must make it accelerate, andthis requires a force. The force must act for a certain period of time, this iscalled impulse:

Impulse = Force× Time

I = Ft

Impulse and change in momentum are equal (but not the same!):

Ft = mv −mu

Note that this only gives us an average force value. A contact force tends tovary with time and can be seen on a Force-time graph:


Area = Impulse

1.3.2 Elastic Collisions

Elastic collision A collision in which there is no change in total kinetic energy.

Inelastic collision A collision in which (some) kinetic energy is lost.

Perfectly inelastic collision A collision in which all kinetic energy is lost.

1.3.3 Conservation of Momentum

Momentum is always conserved when no external force a�ects the interaction.

Total momentum before = Total momentum after

(m1u1) + (m2u2) = (m1v1) + (m2v2)

1.3.4 Newton's laws

Newton's 1st law An object continues moving at constant speed in a straightline, or remains at rest, unless acted upon by a resultant force.

(If there is no change in momentum, a body will continue in uniform motion)

Newton's 2nd law The rate of change of momentum of an object is propor-tional to the resultant force acting on it, and takes place in thedirection of that force.


Force is equal to the rate of change of momentum, thus

F =mv −mu

t

Newton's 3rd law For every change in momentum, there is an equal but op-posite change in momentum.

1.3.5 Momentum of a photon

A photon is the particle representation of the electromagnetic spectrum.

Momentum of photon =Planck's Constant

Wavelength

p =h

λ

and λ = cf , so:

p =hf

c

and E = hf , so:

p =E

c

(the truth of where the photon momentum equation actually comes from is amodi�ed form of E = mc2 which is correct for bodies of all masses: E2 =p2c2 +m2c4 and photons have no mass)

1.3.6 Electromagnetic Radiation Pressure

F =4p

t

P =F

A

P =4p

A× t


1.4 Thermal physics

Boyle's Law For a �xed mass of gas at constant temperature [unless its densityis very high], the pressure varies inversely as the volume. (pV = k).

Ideal gas law An ideal gas strictly obeys the equation of state pV = nRT , inwhich n is the number of moles, T is the kelvin temperature and Ris the molar gas constant. R = 8.31 Jmol−1K−1. Except at veryhigh densities a real gas approximates well to an ideal gas.

The kinetic theory of gases makes the following assumptions:

• Gas consists of �point� particles, i.e. their volume is negligible comparedwith distance between particles

• The particles have the same mass

• The number of molecules is large enough to apply statistical treatment

• The particles are in constant, random and rapid motion (Brownian mo-tion)

• All collisions are perfectly elastic

• All particles are spherical in shape

• Except during collisions, interactions between particles are negligible (i.e.relativistic and quantum-mechanical e�ects are negligible, as a result, dy-namics can be considered classically)

• The average kinetic energy of the gas particles only depends on the tem-perature of the system

• The time taken for collisions is negligible compared to the time takenbetween collisions

The kinetic theory of gases leads us to believe that the pressure exerted by thegas can be attributed to movement of molecules in the gas

p =1

3ρc2 =

1

3

N

Vmc2 or pV =

1

3Nmc2

where p is pressure, ρ is the density of the gas, c2 is the mean square speed ofthe particles, N is the number of particles, V is the volume of gas and m is themass of the gas

Mole The mole is a number of particles

Avogadro's constant,NA This is the number of particles per mole


Molar mass,M The mass (in kilograms) of a mole of a gas

Relative molecular mass,Mr The mass (in grams) of a mole of gas

n =Total mass

Molar mass

The average kinetic energy of a particle in the gas can be found by combiningpV = 1

3Nmc2 and pV = nRT :

1

3Nmc2 = nRT

1

2Nmc2 =

3

2nRT

1

2nNAmc2 =

3

2nRT

1

2NAmc2 =

3

2RT

1

2mc2 =

3

2× R

NAT

1

2mc2 =

3

2kT

where k is the Boltzmann constant, hence k = RNA

1.4.1 Internal enegy

Internal energy,U , of a system This is the sum of the kinetic and potential en-ergies of the particles of the system

Since the internal energy of an ideal monatomic gas is wholly kinetic, we cansay:

U =3

2nRT


1.4.2 Heat

Heat This is energy �ow from a region at higher temperature to a regionat lower temperature, due to the temperature di�erence. In thermo-dynamics we deal with heat going into or out of a system. It makesno sense to speak of heat in a system.

As a result, heat enters of leaves a system through it's boundary or containerwall.

We consider two systems in contact that have no heat �ow between them to bein thermal equilibrium and as a a result they must have the same temperature.

1.4.3 Work

Work If the system is a gas, in a cylinder �tted with a piston, the gas doeswork of amount p∆V when it exerts a pressure p and pushes thepiston out a small way, so the gas volume increases by ∆V . Work,like heat, is energy in transit from (or to) the system.

As such:

W = p∆V

And so, if p changes, work is given by the area under the p− V graph.

1.4.4 The �rst law of thermodynamics

The �rst law of thermodynamics The increase, ∆U , in internal energy of a sys-tem is ∆U = Q−W in which Q is the heat entering the system andW is the work done by the system. Any of the terms in the equationcan be positive or negative, e.g. if 100 J of heat is lost from a systemQ = �100 J.

As such:

4U = Q−W

Note: For solids and liquids, W is normally negligible, so Q = ∆U

1.4.5 Speci�c heat capacity

Speci�c heat capacity,c The heat required, per kilogram, per degree Celsius orKelvin, to raise the temperature of a substance. UNIT: Jkg−1K−1 or Jkg−1◦C−1

As such:Q = mc∆θ


1.5 Electric �elds

1.5.1 Coulomb's law

Coulomb's law The electrostatic force, F , between two small bodies is pro-portional to the product of their charges, Q1 and Q2, and inverselyproportional to their separation squared, r2. F = Q1Q2

4Πε0r2in which

ε0 is the permittivity of free space. ε0 = 8.85× 10−12Fm−1.

As such:

F ∝Q1Q2

r2

F =kQ1Q2

r2

where k = 14Πε0

1.5.2 Electric �eld strength

An electric �eld surround a charge. If another charge enters the �eld, it expe-riences a force, but the �eld is always surrounding a charge.

Electric �eld strength,E The force experienced per unit charge by a small pos-itive charge placed in the �eld. Unit: V m−1 or NC−1.

A radial �eld surrounds a point charge, here's a diagram:

(The arrows indicate the direction a positive charge would move in)


The electric �eld strength would tell us the amount of force a +1C charge wouldexperience at that point.

E =kQ

r2

1.5.3 Uniform electric �elds

You get a uniform electric �eld between oppositely charged parallel plates:

The surface of the planet Earth (or for that matter, any planet) roughly approx-imates (Y.F.W.S.) to a uniform �eld when looking at it on a human (person-sized) scale.

E =V

d

1.5.4 Uses of uniform electric �elds

The electron would experience a force towards the positive plate and this causesan acceleration. So if we consider this situation energetically, the electric �elddoes work on the electron. The amount of work or energy gained is V Q. Itgains this as Ek, so:

V Q =1

2mv2


1.5.5 Electric potential

Electric potential,VE Electric potential at a point is the work done per unitcharge in bringing a positive charge from in�nity to that point. Unit:V [= JC−1]

So we �nd (given that W = Fs) that:

VE =kQ

r

where k is 14Πε0

, Q is the charge and r is the distance from the charge

And so the change in potential energy for a point charge moving in an electric�eld is:

∆W = q∆VE

Note: Field strength can be found from a VE − r graph by �nding negative thegradient.

Note 2: The potential di�erence/change in potential can be found from a E− rgraph by �nding the area underneath it.

1.6 Gravitational �elds

1.6.1 Newton's law of gravitation

Newton's law of gravitation The gravitational force between two particles isproportional to the product of their masses, m1 and m2, and in-versely proportional to their separation squared, r2. F = Gm1m2

r2 inwhich G is the gravitational constant. G = 6.67× 10−11Nm2kg−2.

F ∝m1m2

r2

F = −Gm1m2

r2

where g is always 6.67× 10−11

1.6.2 Gravitational �elds

Surrounding every mass is a gravitational �eld, if another mass enters the �eldit experiences a force due to gravity.

Note: We consider the mass of a spherical body to be concentrated at it's centrewhen looking at the �eld outside the body.


Gravitational �eld strength,g The force experienced per unit mass by a massplaced in the �eld. Unit: ms−2 or Nkg−1.

g = −Gm

r2

and since F = ma, for a body of constant mass, the gravitational force on abody can be found using:

FG = mg

Gravitational �elds theoretically stretch out to in�nity in all directions (andtheoretically, all potential dimensions...), so if you are in a gravitational �eld,you have weight due to the �eld. So there is gravity in space due to the Earthand other planets, and depending on the distance from the Earth, the strengthcan vary.

As a result, 'weightlessness' cannot truly exist, but it can appear to occur whena mass is between two large masses since their gravitational forces' can cancelout.

1.6.3 Gravitational potential

Gravitational potential is the amount of work per unit mass that a body wouldhave to have done to it to move it from in�nity to a point in the �eld. At in�nity,the gravitational potential is zero, but to move towards in�nity, you must havework done to you, so if you add energy to a body and it's �nal energy state iszero, potential must be negative.

Gravitational potential,Vg Gravitational potential at a point is the work doneper unit mass in bringing a mass from in�nity to that point. Unit:Jkg−1.

Gravitational Potential =Work done from in�nity

Mass

Vg =W

m

If you know the mass of a body, you can calculate Vg for di�erent values of rusing:

Vg = −Gm

r

And so change in potential energy for a point mass moving in a gravitational�eld is:


4W = m4Vg

Note: ∆W or ∆UP = mg∆h for distances over which the variation of g isnegligible.

Note 2: The potential di�erence/change in potential can be found from a g − rgraph by �nding the area underneath it.

1.7 Orbits

Kepler's 1st law of planetary motion Each planet moves in an ellipse with theSun at one focus.

Kepler's 2nd law of planetary motion The line joining a planet to the centreof the Sun sweeps out equal areas in equal times.

Kepler's 3rd law of planetary motion T 2, the square of the period of the planet'smotion, is proportional to r3, in which r is the semi-major axis ofits ellipse. [For orbits which are nearly circular, r may be taken asthe mean distance of the planet from the Sun].

Doppler shift The shift in the frequency or wavelength of light when the sourceand observer are moving apart,

It can be calculated using:

∆λ

λ=

v

c

The orbital speed of matter beyond the visible section of a galaxy (i.e. in thenon-visible, but still detectable section, since we use electromagnetic radiationand Doppler e�ect to measure orbital speed) spins faster than expected forthe amount of mass which we believe to be concentrated there.

Dark matter Matter which we can't see, or detect by any sort of radiation, butwhose existence we infer from its gravitational e�ects.

Radial velocity of a star [in the context of Doppler shift] This is the compo-nent of a star's velocity along the line joining it and an observer onthe Earth.


2 Physics A2: Unit 5

2.1 Capacitance

Capacitance is the ability of a system to store charge. It is de�ned as chargestored per volt.

Capacitors are made of two parallel metal plates separated by an insulator. Weuse air or a vacuum as our insulator at A-level, but in practice, dielectrics areused. Dielectrics increase capacitance because they are more insulating than airor a vacuum.

We can �nd the capacitance of a capacitor using:

C =εA

d

where C is capacitance, ε is the permittivity of free space, A is the area of theplate, d is the distance between the plates, since we only calculate capacitancewhere air or a vacuum is the insulator, we useε0 = 8.85× 10−12Fm−1

When a p.d. (potential di�erence) is applied to a capacitor, charge is transferredfrom the power supply to the plates. The plates then carry equal and oppositecharge (the net charge being zero in agreement with the conservation of chargefrom PH1)

We can use:Q = CV

to �nd the charge on a capacitor, where Q is the amount of charge, C is thecapacitance and V is the p.d.

The energy stored by a capacitor is given by:

U =1

2QV

and you can combine this withQ = CV to form further equations for calculation.

You may need to use E = Vd in calculations, and it's not in the formula booklet,

so memorise it!

Combining capacitors is the opposite to resistors, i.e. the reciprocal of totalcapacitance ( 1

C ) is equal to the reciprocal of the capacitances ( 1C1

+ 1C2

+ ...)for series and total capacitance (C) is equal to the sum of the individual capac-itances in parallel (C1 + C2 + ...)

Capacitors follow an exponential charge and discharge, this means that we canuse an exponential equation to calculate some properties of a capacitor over a

period of time:

e.g. for charge:

Q = Q0e− t

RC


for p.d. :

V = V0e− t

RC

for current:

I = I0e− t

RC

we can see that all follow the same pattern, x = xe−t

RC , so while only the chargeform is given in the formula booklet, you can easily derive the rest.

(Note: e is a natural number, much like Π it is irrational, i.e. continues randomlyforever, and it occurs in many natural processes)

We can also see that we have a constant here, written RC or τ , this is calledthe time constant. It is the time taken for the charge (or p.d. or current) toreach 1

e of it's previous value.

You're expected to be able to take logs of equations in PH5, this is a way tomanipulate the equation to remove e. E�ectively if you write ln in front of bothsides of your equation, you can take out the e and put the power of e outsidethe ln, i.e. x = x0e

− tRC goes to ln(x) = ln(x0)− t

RC

(Note: You are expected to be able to graph exponential curves, these arebasically downward curves with a concave shape, and if you take logs you willget a straight line)

2.2 B-�elds

2.2.1 Magnetic (B) Field strength

We can �nd the magnetic �eld strength of a wire electromagnet using:

B =µ0I

2Πa

and similarly, for a solenoid:

B = µ0nI

2.2.2 Force from a magnetic �eld

For wires, we use Fleming's right hand grip rule to determine the direction ofcurrent induced by the magnetic �eld and vice versa:


The force on a wire due to a magnetic �eld is given by:

F = BIl sin θ

where F is the force, B is the magnetic �eld strength, I is the current, l is thelength of the wire, and θ is the angle between the magnetic �eld and wire

or if θ is not given, it can often be assumed to be 90◦ giving F = BIl

The force on a moving charge is similarly given by:

F = Bqv sin θ

where F is the force, B is the magnetic �eld strength, q is the magnitude of thecharge, v is the velocity of the charge, θ is the angle between the magnetic �eldand charge

or if θ is not given, it can often be assumed to be 90o giving F = Bqv

You may be asked to use circular motion equations from PH4 and equate them

with this equation to �nd values, the key equation here is F = m v2

r


2.2.3 The Hall Probe

When a current �ows in a conductor (move free electrons) in a B-�eld, themoving charge experiences a force of Bqv. As electrons enter the Hall Probe,the electrons experience a force upwards (Fleming's LHR). This means thatwe have an excess of electrons at the top and a de�ciency at the bottom whichcreates a potential di�erence. If the current is kept constant, then the forceon the moving charge Bqv will eventually be canceled out by the electrostaticrepulsion due to the electrons on the surface (Eq). The larger the current, thelarger the drift velocity, and so the bigger the force Bqv, so the larger the chargeseparation, so the larger the Hall Voltage.

VH ∝ B

(for constant current)

Bqv = Eq

Bv = E

Bv =VH

d

v =VH

Bd

and if we measure the current through the Hall Probe:

I = nAve

v =I

nAe


I

nAe=

VH

Bd

B =VHndte

Id

(because A = d× t)

B =VHnte

I

where n is the number of electrons per metre cubed.

2.3 Ion beams and particle accelerators

The �rst particle accelerator was simply a vacuum tube where electrons were��red� from a cathode and a uniform electric �eld from this cathode to theanode accelerated the particles. We can do basic calculations for this kind ofbasic accelerator using F = Eq and E = V

d , so that you get:

a =V q

md

and we can use W = q∆VE to work out the energy gained by the electron.

Since this is an extremely ine�cient method to accelerate electrons, better ac-celerators were created, we look at three, the linac, the cyclotron, and thesynchrotron.


2.3.1 Linear accelerator (Linac)

---- ----

---- --------

The linac is made up of a series of positive and negatively charge tubes whichhave an alternating p.d. sent through them. Each tube is a di�erent charge tothe last and the tubes are sized such that as the charged particle enters eachtube the p.d. switches so that it is attracted to the next tube, in order for thisto work, the tubes must get longer because the particle is moving faster as it isaccelerated. This is because the particle is only accelerated when it is not in atube and it must therefore have a charge pulling it towards the next tube justas it leaves.


2.3.2 Cyclotron

The cyclotron also uses an electric �eld to accelerate charged particles. Thesource is place in the middle between to D-shaped (semi-circular) hollow plates,and the particle is accelerated across the gap (between the plates) by the electric�eld. A magnetic �eld is used to ensure that the particle is in circular motion,but since the speed of the particle increases, it spirals outwards and eventuallyleaves the cyclotron, this limits speed which the particle can be accelerated to.We can use circular motion formulae to work out the frequency of the particle:

mrω2 = Bqv

mrω2 = Bqωr

mω = Bq

ω =Bq

m

which (given that ω = 2Πf) implies that

f =Bq

2Πm


2.3.3 Synchrotron

A synchrotron is similar to a cyclotron, except that it keeps the motion circular.It still uses an alternating pd to accelerate the particles, but it uses 4 chargedtubes to accelerate it, so it is charged 4 times per 'orbit'. In order to keepthe particle performing circular motion (in contrast to a cyclotron) it must alsoincrease the B-�eld strength and (also in contrast to a cyclotron) it must havethe AC supply increase in frequency increase as the particle increases in speed.

(large accelerators like LHC use a small accelerator and then a large storage rightbefore colliding particles and they used bending magnets as depicted above; butthe same thing can be achieved on a small scales with 4 portions of chargedtube to accelerate the particles and 4 magnets going around those 4 tubes tomaintain a circular path)

2.4 Electromagnetic induction

2.4.1 Magnetic �ux

Magnetic �ux is through a surface is the component of the magnetic B �eldpassing through that surface.

It is de�ned by the equationφ = AB cos θ

where φ is magnetic �ux, A is the surface area of item being considered, B isthe magnetic �eld strength and θ is the angle to the normal to the surface.


Much like the equation F = BIl sin θ if θ is not given, it can often be assumedto be 0◦ (at least in this case, in the case of F = BIl sin θ, θ is assumed to be90◦, hence F = BIl) hence B = φ

A .

Note: Mathematically, we can consider A (the vector of A) and B rather thanAB cos θ, but we use the cos θ form in A-level physics in order to simplify thissomewhat.

Mathematical form: φ = BA

2.4.2 Flux linkage

Flux linkage is very closely associated with magnetic �ux, it is simple the totalmagnetic �ux for a coil. So we de�ne it:

λ = Nφ = BAN

where N is the number of turns

(cos θ = 1 because of the fact that we are referring to a coil and the same �uxpasses through each of the loops which allows us to simply multiply by N)

Mathematical form: λ = BAN

Note: In actuality, �ux linkage is not actually de�ned as the total magnetic �uxfor a coil, it is de�ned as the integral of voltage across a device with respect totime (λ =

´εdt that is, where ε is voltage)

2.4.3 Faraday's Law and Lenz's law

Faraday's Law states that the induced EMF is equal to the rate of change of�ux linkage:

V = −d(BAN)

dt

or as our textbooks prefer: V = −∆(BAN)∆t since we don't learn calculus and so

we simply use di�erences in our calculations.

All of the symbols are as before, V : EMF induced, B: magnetic �eld strength,A: surface area, N : number of turns, t: time taken.

Lenz's law simply states that the current induced by a change in magnetic �uxlinkage will oppose the cause of the current, i.e. it's why there is a negative inFaraday's law and why the left and right hand rule's oppose each other (seebelow).

2.4.4 Fleming's right and left hand rules

(not to be confused with the right hand grip rule)


(for motors)

and (for genera-tors)

We use Fleming's left-hand rule to �nd the direction of the force, �eld orcurrent from the other. The left-hand rule puts your thumb, �rst �nger andsecond �nger perpendicular (at right angles) and makes the thumb the directionof the force, the �rst �nger the direction of the �eld and the second �nger thedirection of the current. This rule is used when a magnetic �eld induced by acurrent interacts with a magnetic �eld from a magnet, in a generator where acurrent is induced by a magnetic �eld, the same rule is applied but with theright-hand.


2.4.5 Alternating current and the root mean square (rms)

Alternating current (AC) is di�erent from direct current (DC) in that the voltageis not constant (V = IR), so when we look AC circuits, we must �nd the rootmean square pd.

As you can see on the diagram above, the peak is larger than our quoted voltage(i.e. power supplies are called 230V when their peak is around 325V), but weconsider what we call the Irms and Vrms instead. It comes from the relationship

P = V 2

R , because we consider the current to be equivalent when they producethe same power output, since this will produce the same brightness light bulb.

∴ P =(Vrms)

2

R

so since R, the resistance is constant, by squaring the voltage and �nding theroot mean square voltage we know that this will produce equivalent power tothe

Vrms =V0√2

where Vrms is root mean square voltage (the quoted one), and V0 is the peakvoltage and hence:

Irms =I0√2

So in one cycle of AC we �nd that P̄ = IrmsVrms = (Irms)2R = (Vrms)

2

R .

2.4.6 The oscilloscope

On an oscilloscope, the vertical axis is a volts per division (volts/div) axis andthe horizontal axis is seconds per division (sec/div) axis. The division is set bya dial, i.e. each division (square) of our oscilloscope is one of a division set ona dial, the vertical axis is set on a volt/div dial and the horizontal is set on a


sec/div dial, both dials may be some multiple of that number too (i.e. milli,micro, etc). You can easily use this to work out values, simply treat it like anyother graph, it's just that the scale is controlled by the dial.

2.5 Radioactivity and radioisotopes

There are three types of nuclear radiation (the only ionising types of radiation),alpha (α), beta (β) and gamma (γ).

α is most highly ionising type of radiation and hence is the least penetrating(since it ionises quickly and therefore doesn't penetrate far). β is less ionisingthan α and hence more penetrating and γ is even less ionising and hence themost penetrating. Both α and β are particles, but γ is the an electromagneticwave (i.e. part of the E.M. spectrum like visible light). α is a helium nucleus, i.e.42He, whereas β comes from two processes, β+ and β− decay, β− is simply anelectron, and when it is released by decay, it is accompanied by an anti-neutrino(ν̄e). Similarly, β+ is simply a positron (e+) and it is accompanied by a neutrino(νe). When we look at radiation, we tend to only talk about β−decay.

Type of radiation Alpha (α) Beta (β) Gamma (γ)

How ionising isit?

Very Less Not very

How penetratingis it?

Can usually bestopped by a sheetof paper (or a fewcentimeters of air)

Can be stopped by3mm of aluminium

Can be stopped byaround 15cm of

lead (Pb)

Particle/Radia-tion

42He e− Electromagnetic

radiation

Charge +2e −e none

Danger tohumans

Most dangerous,but only if ingested

or inhaled

Dangerous if inclose proximity

Dangerous if inlarge

doses/quantities

We can detect radiation using a Geiger�Müller tube, and we can di�erentiatewhich type it is either by checking how penetrating it is, or by using a magnetic�eld to de�ect it to see which way (if any) it de�ects.

2.5.1 Background radiation

Radiation isn't something only found in nuclear facilities, it is found every-where. We call this radiation that is not generated intentionally experimentallybackground radiation. The largest source of background radiation is from nat-urally occurring radon gas. Other natural sources (from Earth, these are thelargest contributors to background radiation) include our food and drink andbuildings and the ground, which all contribute from radioactive isotopes such aspotassium-40, carbon-14, uranium and thorium. Other sources include cosmic


rays which arise from high-energy particles arriving from space and the remain-der of background radiation is man-made. Man-made radiation is mostly fromX-ray imaging, but a tiny percentage is from nuclear power and weapon test-ing (although of course this varies by location). A normal background count isaround half a count per second.

2.5.2 Calculating radioactivity

Radioactivity can be calculated probabilistically, because while each individualdecay is entirely random, the over decay is entirely predictable.

Note: If you are wondering what di�erence between random and predictable,look up stochastic. Stochastic is something random that is not predictable.

So if we know the activity, i.e. number of decays (disintegrations) per second,we know that this is proportional to the number of nuclei (A ∝ N) and hence:

A = λN

where λ is a constant which we call the decay constant, it is di�erent for everynucleus. We also know that we can say that the number of decays per secondis negative the di�erential (rate of change of) the number of nuclei with respectto time, i.e. A = −dN

dt or as WJEC prefers:

A = −∆N

∆t

Note: This is because we de�ned activity as the number of decays per second,which is the negative change in number of nuclei per second.

So now much like the capacitance, we can then de�ne an exponential (decay inthis case) equation:

N = N0e−λt

and we can also use the same form with activity:

A = A0e−λt

In radioactivity sometimes it is more useful to know how long it takes for halfthe nuclei to decay, so we can say:

T 12=

ln 2

λ

where half-life (T 12) is de�ned as the time taken for half the nuclei to decay (this

is simply calculated from the N = N0e−λt equation above), but you should

remember this equation since it is given to you and it will drastically save time.


2.5.3 Uses of radioisotopes

A radioisotope is simply an isotope (same number of protons, di�erent numberof neutrons) of an element which is radioactive. There are many uses for thembut you only need to know two di�erent applications, for example:

1. You can use a gamma emitter such as cobalt-60 to sterilise medical equip-ment or food. Gamma is useful because it isn't very ionising so it isrelatively safe for humans but it kills all germs, bacteria and viruses.

2. You can use a beta emitter such as strontium-90 to check the thickness ofpaper as it rolls o� of a paper mill. By using a source and a detector withthe paper in between you can use the count rate to measure the thicknessand change the adjust the rollers accordingly.

2.6 Nuclear energy

2.6.1 Mass-energy equivalence

Einstein established that mass was equivalent to energy and this is the keyconcept we make use of in nuclear power, we convert mass into useful energy.However, while E = mc2 may be one of the most famous physics equations, it isslightly inaccurate and confusing. So when we look at mass-energy equivalence,the key thing to keep in mind is that while a certain amount of mass is equivalentto a certain amount of energy, you cannot say that mass is energy. The otherkey thing to note is that while mass is equivalent to energy, the truth is thatobjects with momentum also have energy, so the complete equation is E2 =(mc2)2 + pc = m2c4 + pc, so this is why light has energy but no mass (i.e.E = pc) and when we convert mass into energy we can say ∆E = c2∆m. Soyou will be asked to calculate the mass defect during a �ssion or fusion reactionand they will give you the mass in u (atomic units) for the nuclei and you willbe expected to �nd the mass defect (the di�erence in the mass before and after)and by multiplying this by u and using ∆E = c2∆m you can �nd the energygenerated (before loss). You may also be given the energies involving eV , thisis simply related to the S.I. unit the Joule (J) by e, so if you multiple 1eV by e(which is ∼ 1.60×10−19) then you will have the value in Joules. This is requiredfor all calculations (where they give you eV that is).

2.6.2 Stability and binding energy

Some nuclei are more stable than others and we can only fuse (�join� together) or�ssion (�take apart�) unstable nuclei. The most stable nucleus is Iron-56 (5626Fe)and so Iron-56 cannot be fused or �ssioned. Each nucleus has a binding energy(B.E.), this is the amount of energy which has to be supplied to the nucleus inorder to break it up into it's nucleons (protons and neutrons). If we �nd the BEper nucleon, we can establish whether our nucleus can be �ssioned or fused; if


it has more B.E. per nucleon than Iron-56 then it can only be �ssioned and ifit has less than it can only be fused. The further from the B.E. per nucleon ofIron-56 it is, the easier it is to fuse or �ssion.

U235

U238

Fe56O16

C12

He4

Li6Li7

He3H3

H2

H1

Number of nucleons in nucleus

Ave

rage

bin

ding

ene

rgy

per

nucl

eon

(MeV

)

9

8

7

6

5

4

3

2

1

00 30 60 90 120 150 180 210 240 270

2.6.3 The nuclear �ssion reactor

We look at the basic parts of most nuclear �ssion reactors. The key componentsare the control rods, the moderator and the coolant.

Control rods The control rods absorb neutrons. They are used for two purposes,to control the reaction and in the event of emergency, to shut it down.They start lowered and are raised to obtain a sustainable chain reaction(that is not out of control, i.e. not a weapon). The material used must avery good neutron absorber with a high melting point, a good example ishigh boron steel.

Moderator The moderator is used to slow down neutrons in order to increasethe probability of �ssion. A moderator should be a poor absorber ofneutrons because we don't want to absorb the neutrons, we need themfor the �ssion! Hence we need the moderator to be made of a materialwith a light nucleus to increase kinetic energy transferred and minimiseabsorption. Examples of moderators include graphite and steel.


Coolant The coolant controls the temperature of the reactor and takes away thethermal energy to our steam turbine and generator in order to generateelectricity. The coolant should be a �uid (liquid/gas) with a high heatcapacity that doesn't absorb neutrons or become radioactive. Examplesof coolants include water and super-heated steam.

Another important de�nition is chain reaction, which is where more neutronscome out than are put in; a chain reaction can either be exponentially increasing(i.e. a weapon) or equilibrium (or thereabouts) as is the case in a reactor used togenerate electricity. You should also note that the waste from a �ssion reactionis extremely problematic since it is highly radioactive (and will be for a longperiod of time).

3 Physics A2: Unit 5: Option D: Medical measurement and medical imaging 41

3 Physics A2: Unit 5: Option D: Medical measurement andmedical imaging

3.1 X-rays

The X-rays section is very similar to that in PH2, so I'm not going to repeatanything, here are some quick diagrams with the key elements:


Key de�nition:

X-rays High energy electromagnetic radiation, with photon energies between∼ 100eV and ∼ 100keV .

3.1.1 Computerised axial tomography (CAT or CT) scans

In addition to simply using an x-ray tube and a Geiger detector to �nd a1-dimensional image, we can create 3D images from CAT scans. A CAT scan isproduced by using rotating beams which move around the body to get imagesfrom all direction (slices as it were) from which a 3D image can be constructed.This is only used in emergencies since the X-ray does received is high.

3.1.2 X-ray attenuation

This is a new section in medical physics. We look at the exponentially decreasingintensity of a beam over a distance. Intensity is de�ned as the power transferredper unit area (I = P

A ) and our exponential equation is then:

I = I0e−µx

where I is the intensity at x metres away from the initial intensity (I0) and µis the constant that is called the attenuation coe�cient.

Note: The attenuation constant is di�erent for each material that the X-raystravel through since X-rays have a di�erent penetration in di�erent materials.

3.2 Ultrasound

Ultrasound is a type of electromagnetic wave. In medical physics we use apiezoelectric crystal to generate ultrasound EM waves. A piezoelectric crystalis one that (deforms and as a result) emits EM radiation when a current is putthrough it and generates a current when an EM wave passes through it. We usethis crystal in conjunction with ultrasound EM radiation because EM radiationit is re�ected more from some tissues than others and as a result is great forimaging body tissues. When we use our detector, we must use a gel between thepiezoelectric sensor and the body to prevent the ultrasound entering air since itwill re�ect from the air and never even enter the body.


Types of scan Description Example

A-scan 1-dimensional, send pulsesand the responses are

displayed on an oscilloscope;the time delay allows us todetermine the structure.

A tumour wouldalter time taken to

re�ect.

B-scan 2-dimensional, an array ofdetectors (crystals) can beused or a single movingdetector can be used. A

computer displays pulses onthe screen which representthe time delay for each areaallowing us to image the

body to see di�erent element.

Foetal/prenatalscans

3.2.1 Acoustical impedance

Acoustical impedance (Z) is a constant that de�nes what fraction of ultrasounda body will re�ect. We de�ned it using:

Z = cρ

where c is the speed of sound (in the medium) and ρ is the density of thatmedium.

We can then use this to �nd the fraction of ultrasound re�ected using:

R =(Z2 − Z1)

2

(Z2 + Z1)2

where Z1 is the �rst medium's impedance and Z2 is the second medium'simpedance.

The gel we apply when using ultrasound is called a coupling medium, this is amedium that almost the same acoustical impedance as skin to reduce re�ectionsfrom the entry of the ultrasound to the skin. However, because ultrasound isused to measure re�ections, this makes it not suitable for medium's with highacoustical impedance, e.g. it can't be used with the lungs since it will be re�ectedby the air inside them.

3.3 Doppler probe

We can also use ultrasound to probe the body in a di�erent way. If we want tostudy blood �ow, we can use ultrasound and the Doppler shift e�ect. Muchlike other cases of Doppler being used in re�ection, we take ∆λ

λ = vc but

multiply the vc by two, i.e.


∆λ

λ= 2

v

c

This allows us to determine the speed of the blood �ow, and combined with thestrength of the re�ected pulse (which reveals the volume of blood) we can �ndthe �ow rate.

3.4 Magnetic resonance imaging (MRI)

MRI is based on the principles of resonance and precession. Resonance is whena system oscillates with a high amplitude due to an input frequency (in thiscase magnets are used to vibrate ions). This resonance causes the nucleonswhich are being targets to change �spin�. Nucleons change from �spin-up� to�spin-down� and vice versa. In the case of MRI, we target hydrogen nuclei sincewe know there are many of these in the body. In both cases the �spin� precesseswhich means that it rotates around the magnetic �eld direction (much like aspinning toy, e.g. a dreidel). The frequency of it's precession depends on thestrength of the magnetic �eld and this is called the Larmor frequency (i.e.it's our resonant frequency for nucleon precession). When the (electro)magnetic�eld is turned o�, the nucleons ��ip� back releasing a radio wave which canbe detected allowing us to �nd the location of the atoms. The time taken forthem to ��ip� back is called the relaxation time and this will di�ers dependingon the surrounding so we can create an image of the tissues and because ofthis, it is extremely good for all tissues containing water (H2O). We use thedi�erent concentrations of hydrogen to build up a detailed image of the tissues.It doesn't work as well with bones since they don't tend to contain hydrogen.The only major disadvantages of MRI are the high cost of the machines, theclaustrophobic aspect of the machine and the fact that it cannot be used withmagnetic metals anywhere in the body since the magnet will pull these out.This makes it better than X-rays for imaging most of the body since X-rays cancause cancer.


3.5 Comparison of X-rays, ultrasound and MRI

Technique Advantages Disadvantages

X-rays X-rays are absorbed bybone and so produce

good shadow images andso can be used in places

like the lungs.

High radiation dose(particularly for CATscans). People workingwith them need to takecare with patients and

themselves.Ultrasound No known side e�ects.

Good quality images ofsoft tissue. Moving

images can be obtained.Machines are cheap.

Cannot pass through airso cannot be used with

the lungs. Low resolution.

MRI No major side e�ects.High quality images ofsoft tissue. Image can be

made for anypart/orientation of the

body.

Images of hard tissuesuch as bone are poorquality. Di�cult for

patients withclaustrophobia and

cannot be used by thosewith magnetic metalinside. Very expensive.

3.6 Electrocardiograms (ECG)

Electrocardiograms can be used to look at the operation of the heart and blood�ow. Normally the sequence of our double circulatory blood is �ow is:

1. De-oxygenated blood enters the top right-hand chamber, the right atriumof the heart.

2. The right atrium pumps the blood past a valve which prevents back-�owand into the bottom right chamber, the right ventricle, which pumps it tothe lungs to oxygenate.

3. The oxygenated blood returns to the top left-hand chamber, the leftatrium of the heart.

4. The left atrium pumps the blood past another valve into the bottom leftchamber, the left ventricle, where it is pumped to the rest of the bodybefore returning to the heart (at step 1. again)


The four muscles of the heart are triggered to contract by a signal from thesinoatrial node. This is the node at the top-right of the heart that is con-trolled by the central-nervous and hormonal systems and it is what controls theheartbeat. The nervous impulses associated with our heartbeat can be detectedoutside the body using electrocardiography. Electrocardiography simply uses12 electrodes applied to shaved skin with gel (to improve contact). Most ofthose electrodes are placed near the heart but some on the arms and legs, butnone near the right leg since this is too far from the heart. The pd can thenbe monitored to see the heartbeat. Normally the surface potential-time graphshould look this:


P

Q

R

S

TST

SegmentPRSegment

PR Interval

QT Interval

QRS Complex

where QRS is the contraction of the ventricles, and P is contraction of the atriaand T is the relaxation of the ventricles.

Surface potential The voltage measured on the skin due to the nerve signalwithin the body.

Contraction The shortening of muscles in response to a nerve stimulus.

For a normal pulse rate of 75 beats per minute, the shape above would be repeatevery 0.8s (800ms).

3.7 Nuclear radiation in medicine

We looked at radiation earlier and as mentioned, α, β and γ particles havedi�erent potential to damage the body. In medical physics we must note howradiation a�ect the body. The radiation can damage biological molecules be-cause it interacts with the atoms which make it up. It can knock out electrons


or otherwise a�ect the atom. This can change the make up of DNA causingmutations, but the body usually repairs this damage and many genetic changesare not damaging. This can lead to two things:

1. If it changes DNA which a�ects the cell's control mechanisms, this canlead to uncontrolled cell division (i.e. cause cancer/it's carcinogenic). Thiskind of damage is caused by long-term exposure to low levels of radiation.

2. If it changes cells which are highly susceptible to radiation damage, thatis those which are dividing, then these cells can su�er constant damage.Examples of dividing cells include hair follicles and epithelial cells (whichinclude the lining of the alimentary canal and the lungs). Low levels ofradiation can be tolerated here, but large doses over a short period of tiecan cause such damage that these cells are killed, which is why patientsundergoing radiation therapy often lose their hair.

3.7.1 Measuring radiation

We measure radiation in many di�erent ways in medical physics. As a physicistyou may use activity in becquerels (Bq) which is the number of decays persecond, but medics use other measurements. The �rst is absorbed dose:

Absorbed dose = Energy absorbed per kg of body tissue

Absorbed dose is measured in grays (Gy) which are Joules per kilogram (J(kg)−1)

We also talk about dose equivalent, this is e�ectively absorbed dose but adjustedfor the relative e�ects of the radiation:

Dose equivalent = quality factor× absorbed dose

D = Q×A

where the dose equivalent is measured in sieverts (Sv) and the quality factor issimply a constant, usually 20 for α, 1 for β and 1 for γ but it also depends onthe energy of the radiation.

3.7.2 Radionuclides (Radioactive tracers)

Radioactive tracers Chemical compounds with an atom replaced by a radioactiveisotope. They are used to track the uptake of a compound into the body.

We are expected to know about iodine-123 and iodine-131. I-123 is used toinvestigate the function of the thyroid and it decays by electron capture to anexcited state of tellurium-123:


12353 I +0

−1 e →12352 Te? →123

52 Te+ γ

The iodine is injected in the form NaI which is biologically safe and the I-123'shalf-life 13.3 hours is ideal for use as a tracer as it is long enough for productionand administration but is short enough that the body is rid of it within a fewdays. The Tellurium produced is mildly toxic, but safe in the quantities used.The emitted γ rays are investigated with a gamma camera (see below). Thiskind of decay occurs in proton-rich nuclei and their electrons actually spendsome time inside the nucleus. An electron combines with a proton to producea neutron (and an electron neutrino):

0−1e+

11 p →0

0 n+00 νe

This only occurs if there is a lower energy level available for the newly createdneutron to occupy because neutrons are more massive than protons. The nucleusis formed in an excited state and the neutron drops down to a lower energy levelemitting a photon.

Iodine-131 is used in radioactive therapy to treat thyroid cancers, again usingthe fact that the thyroid absorbs iodine. I-131 decays by β− decay to an excitedstate of xenon-131, which rapidly decays by γ emission:

13153 I →131

54 Xe+0−1 β

followed by:

13154 Xe →131

54 Xe+ γ

The β particles kill surrounding tissues, including the cancerous tissue. The γis also useful because we can use it to monitor the e�ectiveness of the uptakewith a gamma camera (again see below). I-131 is not usually used as a tracerbecause of the dangerous β emission.

3.7.3 Gamma camera

A gamma camera consists of:

Collimator A piece of lead with narrow parallel channel. The lead absorbs allγ apart from those which travel along the channels (about 99% of theincoming γ photons, hence a strong source must be used with a longexposure time).

Scintillation crystal This crystal emits a �ash of (visible) light when it absorbsa photon.


Photomultipliers Electronic devices in contact with the scintillation crystal whichamplify the light and turn it into an electric signal.

Scintillation counter A counter to register the arrival of photons.

Output display A screen (typically an LCD) which builds up the image.

3.7.4 Positron emission tomography (PET) scan

The PET scan uses β+ (i.e. positron) emission to scan the body. A goodexample is �uorine-18:

189 F →18

8 O +01 e

+

The positron is slowed down signi�cantly by collisions and eventually annihilateswith an electron producing two photons with equal but opposite momenta, i.e.they have the same wavelength and travel in opposite directions:

01e

+ +0−1 e

− → γ + γ


Each photon has 0.511MeV of energy because of the electron's mass energyequivalence.

The patient lies inside the scanner after a suitable tracer has been administeredby injection, inhalation or ingestion. The γ scanner is lined with γ detectorswhich register coincidental and opposite events, i.e. they look for γ detected onopposite sides of the patient, they look for events nanoseconds apart and hencethis is very unlikely to occur by chance. The positron typically moves less than1mm before detection, so we can pinpoint the site of it's emission pretty accu-rately. Using a long time, a large number of detectors and several slices, muchlike a CAT scan we can create a 3D image of the area. Most common uses ofa PET scan are in search for metastases (secondary cancers). Fluorodeoxyglu-cose (FDG) is usually the tracer used (with F-18 being the radioactive isotope).F-18s half-life is only 110 minutes, but this is long enough for it to be preparedon site, administered and used for scanning. It is made from bombarding waterenriched with O-18 with high-speed photons:

188 O +1

1 p →10 n+18

9 F

The oversight of this production routine is one of the primary jobs of a medicalphysicist.

Annihilation When a particle meets it's antiparticle at low speeds annihilationoccurs. Their mass' turned into photons of the associated energies.

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