Advanced Mechanics Notes · 2020. 11. 10. · Advanced Mechanics Notes . Page 2 of 127 Contents ......

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Advanced Mechanics

Notes

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Contents • Projectile Motion .................................................................................................................. 4

➢ Galileo’s Analysis of Projectile Motion ............................................................................... 4

➢ Projectile Motion Equations .............................................................................................. 4

➢ Projectile Motion Variables ............................................................................................... 4

➢ Tips & Tricks ..................................................................................................................... 5

• Circular Motion .................................................................................................................... 5

➢ Introduction ..................................................................................................................... 5

➢ Uniform Circular Motion ................................................................................................... 5

➢ Average speed .................................................................................................................. 5

➢ Angular Velocity ............................................................................................................... 6

➢ Radians & Degrees ............................................................................................................ 6

➢ Centripetal Acceleration .................................................................................................... 6

➢ Centripetal Force .............................................................................................................. 7

➢ Centripetal vs Centrifugal .................................................................................................. 7

➢ Complex Circular Motion ................................................................................................... 7

o Conical Pendulum ......................................................................................................... 7

o Banked Curves .............................................................................................................. 7

o Vertical Circular Motion ................................................................................................ 8

➢ Energy & Work.................................................................................................................. 9

o Work ............................................................................................................................ 9

➢ Torque ............................................................................................................................ 10

• Motion in Gravitational Fields ............................................................................................. 11

➢ Newton’s Law of Universal Gravitation ............................................................................ 11

➢ Altitude vs Orbital Radius ................................................................................................ 11

➢ Gravitational Field .......................................................................................................... 11

➢ Gravitational Field Strength ............................................................................................ 12

➢ Variations in the Gravitational Field Strength .................................................................. 12

➢ Satellites ......................................................................................................................... 12

➢ Newton’s Thought Experiment ........................................................................................ 13

➢ Requirements for Orbit ................................................................................................... 13

➢ Properties of Orbit .......................................................................................................... 14

o Centripetal acceleration .................................................................................................. 14

o Orbital Period (T) ............................................................................................................ 14

➢ Orbital Velocity ............................................................................................................... 14

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o Method 1 ........................................................................................................................ 14

o Method 2 ........................................................................................................................ 14

➢ Types of Orbit ................................................................................................................. 15

➢ Keplar’s Third Law ........................................................................................................... 16

➢ Gravitational Potential Energy ......................................................................................... 17

➢ Mechanical Energy in a Gravitational Field ....................................................................... 18

➢ Newton’s Thought Experiment Revisited ......................................................................... 18

➢ Escape Velocity ............................................................................................................... 18

Definitions ................................................................................................................................. 19

Formulas ................................................................................................................................... 20

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• Projectile Motion o A projectile is a macroscopic

slow-moving object.

o Diameter > 1mm

o Speed << c

o Any projectile in inertial free-fall

motion is modelled by projectile.

o Ignore friction and assume constant gravitational acceleration.

o Assume that all projectiles have no movement in the 3rd dimension.

➢ Galileo’s Analysis of Projectile Motion o “A projectile motion consists of two independent motions, the horizontal

and vertical motion. The horizontal

motion is under constant velocity and the

vertical motion is under constant

acceleration.”

➢ Projectile Motion Equations o Vertical Equations of Motion:

▪ vy = uy + at

▪ vy2 = uy

2 + 2asy

▪ sy = uyt + ½ at2

o Horizontal Equations of Motion: ▪ sx = uxt

▪ sx = 𝒖𝟐 𝒔𝒊𝒏 𝟐𝜽

𝒈

➢ Projectile Motion Variables o Definitions:

▪ Initial velocity (u)

▪ Angle of projection (𝜃)

▪ Vertical and horizontal velocity (ux, uy)

▪ Maximum height (h)

▪ Change in vertical displacement (sy)

▪ Time of flight

▪ Range (sx)

▪ Final velocity (v)

vy = 0 at

max height

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➢ Tips & Tricks o For symmetrical motion, the total time is 2 x the time to max height.

o Max range occurs at 45o.

o Collisions occur when objects are

at the same place and the same

time.

o ONLY FOR MULTIPLE

CHOICE!!!

▪ Time of flight:

❖ 𝒕 =𝟐𝒖 𝒔𝒊𝒏 𝜽

𝒈

▪ Maximum height

reached:

❖ 𝑯 =𝒖𝟐 𝒔𝒊𝒏𝟐 𝜽

𝟐𝒈

▪ Horizontal Range:

❖ 𝑹 =𝒖𝟐 𝒔𝒊𝒏 𝟐𝜽

𝒈

• Circular Motion

➢ Introduction o It is the motion of an object along the circumference of a circle.

o The object may travel at constant speed (uniform circular motion) or a

changing speed (non-uniform circular motion).

o Circular motion is used in small (car driving around a round-about) and large

scale (planets revolving around stars).

➢ Uniform Circular Motion o Objects move with constant speed not constant velocity.

o Velocity changes since the direction changes.

o Instantaneous velocity is tangential to its path.

o Time for one full cycle is called a period. (T, sec)

o Number of rotations in one second is called

frequency. (f, Hz)

➢ Average speed

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➢ Angular Velocity o Velocity can be determined from angular velocity.

o Angular velocity measures the degree (radian) at which the object left

circular motion.

o 𝝎 =𝜽

𝒕

▪ Where,

❖ ω – angular velocity (os-1 / 𝜋𝑐𝑠-1)

❖ 𝜃 – angular displacement (o or 𝜋𝑐)

❖ t – time (seconds)

➢ Radians & Degrees o If angular displacement is not given, it can be determined by:

▪ 𝒍 = 𝒓𝜽 – if in radians

▪ 𝒍 = 𝟐𝝅𝒓 (𝜃

360) – if in degrees

❖ Where,

l – arc length (m)

r – radius (m)

o Radians are a way of expressing angles.

o Converting degrees to radians (& vice-versa):

➢ Centripetal Acceleration o A change in velocity implies there is acceleration. Therefore, there must be a

net force experienced by the object.

o The object deviates inwards due to an acceleration towards the centre of

the circle. This is called centripetal acceleration.

o Centripetal acceleration always acts towards the centre. The object would

fly off along the tangential path without acceleration.

o ac always acts towards the centre of motion.

o 𝒂𝒄 =𝒗𝟐

𝒓=

𝟒𝝅𝟐𝒓

𝑻

▪ Where,

❖ ac – centripetal acceleration (ms-2)

❖ v – magnitude of the velocity (ms-1)

❖ r – radius of the circle (m)

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➢ Centripetal Force o Responsible for circular motion.

o Fnet = mac = 𝒎𝒗𝟐

𝒓

o The centripetal force is equivalent to some real force, present in the force.

o Fc always acts towards the centre of motion.

➢ Centripetal vs Centrifugal o Centrifugal force opposes centrifugal force. (Newton’s 1st Law).

o Centripetal force causes an object moving in a circular path to move out and

away from its path.

o When observed from a stationary frame

(above the motion) we see that the driver’s

inertia keeps his body moving in a straight

line (which looks like he is moving towards

the outside of the car). This explains that

centrifugal force is only an apparent force

and shouldn’t be used for calculations.

➢ Complex Circular Motion

o Conical Pendulum

o Banked Curves ▪ Used to increase the speed of the vehicle.

▪ Used at car tracks & cycling velodromes.

▪ When a vehicle turns on a flat surface, it relies

on the friction to provide the centripetal force.

This may not always be present if the surface is

smooth (icy surface or worn types).

▪ Horizontal component of the normal force

provides the centripetal force on banked curves.

▪ Design speed is the max speed on a banked curved without the

vehicle drifting higher or lower.

Fnet = Fc

= Ftension

Fnet = Fc

= Fgravity

Fnet = Fc

= Fnormal

Fnet = Fc

= Ffriction

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o Vertical Circular Motion ▪ Rollercoasters are examples; loops, dips, & hills.

▪ Travelling Along a Straight Section

❖ It undergoes uniform horizontal motion.

❖ Normal & Weight forces are the only

acting force on the person.

❖ Since there are no external forces, the

weight force balances the normal force.

❖ The man will feel his normal weight.

▪ Travelling Through the Dips

❖ The actual weight force stays

constant, but the normal force

fluctuates.

❖ The apparent weight will change

during the motion.

❖ The passenger will feel getting

pushed into the seat more as he is

in the dips & heavier than he

actually is.

❖ Wapparent > Wactual

▪ Travelling Over a Hump

❖ The weight force remains the same.

❖ The man has a smaller normal force.

❖ The seat has a smaller force on the person, making him feel

lighter.

❖ Wapparent < Wactual

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➢ Energy & Work o There are many forms of energy that can be broken into main categories:

▪ KE is energy associated with motion.

▪ PE is energy associated with the position of the object in a field.

o Energy & work analysis can be performed on circular motion.

o “For a given system of bodies where there are only kinetic and potential

energy involved then the total mechanical energy of the system is

conserved”

o Work ▪ It is the transfer of energy from one object to another.

▪ Work is a scalar quantity.

▪ W = F∥S = Fscos𝜽

❖ W – Work (J)

❖ F – force (N)

❖ s – displacement (m)

❖ 𝜃 – angle between the force vector & the displacement vectors

▪ Work can only be done when there is a change in displacement.

▪ It is possible to produce a force & result to no work and no energy

transformation (holding a book up).

▪ This is also applicable to centripetal motion.

▪ The centripetal force is

perpendicular to the

velocity/displacement at any given

instant of time. Hence there is no

work being done but a force is still

applied.

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➢ Torque o Torque is a twisting/turning effect responsible for causing something to

undergo rotational motion.

o Torque is the turning moment/point of a force.

o It is dependent on the magnitude of the applied force and the distance away

from the pivot point.

o If the force is applied through the pivot

point, the torque is zero.

o If the force is applied perpendicular to the

pivot point, the torque is at a maximum.

o T = Fd⊥

▪ T – Torque (Nm)

▪ F – Force (N)

▪ d⊥ - Perpendicular distance (m)

o In terms of the perpendicular component of the force:

o In terms of the perpendicular distance:

o It is a vector quantity.

o Represented by positive or negative.

T = Fd⊥

= Fdsin𝜃

F⊥ = Fsin𝜃

d⊥ = dsin𝜃

T = Fd⊥ = Fdsin𝜃

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• Motion in Gravitational Fields

➢ Newton’s Law of Universal Gravitation o Gravitational force is an attractive force. This exists between any two

objects.

o F = 𝑮𝑴𝒎

𝒓𝟐

▪ F – gravitational force (N)

▪ M, m – mass of objects (kg)

▪ r – centre to centre distance of separation (m)

▪ G – universal constant (6.67 x 10-11 Nm2kg-2

➢ Altitude vs Orbital Radius

o A relationship between force & radius (distance of separation).

➢ Gravitational Field o It is a field within which any mass can experience a gravitational force.

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➢ Gravitational Field Strength o It is a numerical value used to indicate the strength of a gravitational at a

point around a mass.

o Also known as gravitational acceleration.

o g = 𝑮𝑴

𝑹𝟐

➢ Variations in the Gravitational Field Strength o Several factors affect the g-value:

▪ Geographical location on Earth

❖ The is not a sphere but it is flattened at the poles.

Therefore, the g-value is slightly greater at the poles due to

the smaller radius.

▪ Altitude

❖ The g-value decreases as we move away from the planet

since g is inversely proportional to the square of the

distance of the separation.

▪ Different planetary bodies

❖ g depends on the mass and the average radius of the planet.

Different planets have different masses and radii, producing

g-values.

➢ Satellites o It is an object with stable orbit around another object.

o Orbit/orbital motion is the regular and repeating path that one object takes

around a fixed point.

o E.g. Earth is a satellite of the Sun, the Earth’s orbit is fixed around the

position of the Sun.

o Satellites can be artificial (telecommunications) or natural (moon).

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➢ Newton’s Thought Experiment o Newton conducted a thought experiment which considered what will

happen if a cannonball was fired from the top of an extremely tall mountain.

o This experiment is an extension of projectile motion to objects which are in

orbit around the Earth.

o He concluded that if the cannonball was fired with greater velocity, it could

be made to travel further around the Earth’s curvature.

o He reasoned than if the cannonball was fired with sufficient velocity, it

would be in a state of continuous falling and come back to the cannon.

o The main problem is that he ignored air resistance which would slow down

the cannonball as it orbits, it would eventually hit the ground.

o Air resistance is negligible at higher altitudes due to the thinning of the

Earth’s atmosphere.

➢ Requirements for Orbit o For an object to be in orbit it needs:

▪ Centripetal Force

❖ This is the only force acting on an object in orbit (other than

air resistance).

❖ It is supplied by the force of gravitational attraction.

▪ Momentum

❖ It gives the mass sufficient inertia

to maintain its motion of orbit.

❖ Velocity will always be a tangent to

the path of orbit.

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➢ Properties of Orbit

o Centripetal acceleration ▪ Since gravitational force provides centripetal force. Centripetal

acceleration is the same as the gravitational field strength (or

acceleration due to gravity).

o Orbital Period (T) ▪ The time taken for the satellite to make one complete orbital

rotation.

➢ Orbital Velocity

o Method 1

▪ Given, speed = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒

𝑡𝑖𝑚𝑒

▪ Substituting, v = 2𝜋𝑟

𝑇

o Method 2

▪ Where,

❖ v – orbital velocity (ms-1)

❖ G – universal constant (6.67 x 10-11 Nm2kg-2)

❖ M – mass of the plant (kg)

❖ r – radius of orbit (m)

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➢ Types of Orbit

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➢ Keplar’s Third Law

o This was derived assuming that the orbit was circular.

o In terms of complete orbits, the motion of bodies in space CAN be treated as

if circular, as the period of an elliptical orbit, & the average distance.

o The elliptical nature of orbits is so small it can be seen as circular with

minimal consequences.

o Therefore, Keplar’s third law applies for circular and non-circular motion.

o Constancy in Keplar’s Third Law,

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➢ Gravitational Potential Energy o It is the stored energy in an object due to its position in a gravitational field.

o Work must be done by an external force to bring an object from B to against

the force of gravity.

o The work done is converted into GPE,

higher the work – the higher GPE.

o Therefore, GPE is a maximum of infinity.

o U = GPE = −𝑮𝑴𝒎

𝒓

▪ Where,

❖ U – GPE (J)

❖ G – universal constant (6.67 x 10-11 Nm2kg-2)

❖ M – mass of the planet (kg)

❖ m – mass of the object (kg)

❖ r – centre to centre distance (m)

o As an object moves away for Earth, we know its GPE must increase, but at a

distance of infinity its GPE = 0.

o Therefore, we take a new reference point where infinity is 0 GPE, & all GPE

values before infinity are negative.

o Change in GPE is the only quantity that is significant, NOT the GPE by itself.

o 𝛥GPE = Work

o W = 𝛥GPE = GPE2 – GPE1

= [−𝐺𝑀𝑚

𝑟2] − [−

𝐺𝑀𝑚

𝑟1]

= −𝐺𝑀𝑚

𝑟2 +

𝐺𝑀𝑚

𝑟1

= 𝐺𝑀𝑚 [1

𝑟1−

1

𝑟2]

R – Radius of Earth

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➢ Mechanical Energy in a Gravitational Field o Recall that: E = KE + U

o K = 1

2mv2

o Substituting, orbital velocity, [𝑣 = √𝐺𝑀

𝑟)]

o K = 𝐺𝑀𝑚

2𝑟

o Therefore,

o E = KE + U

= 𝐺𝑀𝑚

2𝑟−

𝐺𝑀𝑚

𝑟

= - 𝐺𝑀𝑚

2𝑟

➢ Newton’s Thought Experiment Revisited o Newton thought of what would happen if the velocity to enter orbit was

increased.

o The result was that the cannonball would leave Earth’s gravitational field &

fly to infinity.

➢ Escape Velocity o It is the lowest velocity an object must obtain to escape a gravitational field

& go to infinity.

o According to the Law of Conservation of Energy,

❖ KElost = GPEgained

KEinitial – KEfinal = GPEfinal – GPEinitial

(1

2𝑚𝑢2 – 0 = 0 – (−

𝐺𝑀𝑚

𝑟)

1

2𝑚𝑢2 =

𝐺𝑀𝑚

𝑟

u = √ 2𝐺𝑀

𝑟

Where,

▪ U – GPE (J)

▪ KE – Kinetic Energy(J)

▪ G – Universal constant (6.67 x 10-11 Nm2kg-2

▪ M – mass of the planet (kg)

▪ m – mass of the object (kg)

▪ r – centre to centre distance (m)

Where,

▪ u – escape velocity (ms-1)

▪ G – Universal constant (6.67 x 10-11

Nm2kg-2

▪ M – mass of the planet (kg)

▪ r – centre to centre distance (m)

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Definitions WORDS DEFINITIONS

Projectile a macroscopic slow-moving object.

Circular motion the motion of an object along the circumference of a circle.

Centripetal force a force that is directed towards the centre around which the body is moving.

Centrifugal force a force directed away from the centre around which an object is moving.

Work the transfer of energy from one object to another.

Torque a twisting/turning effect responsible for causing something to undergo rotational motion.

the turning moment/point of a force.

Gravitational force an attractive force.

Gravitational field strength It is a numerical value used to indicate the strength of a gravitational at a point around a mass.

Satellite It is an object with stable orbit around another object.

Escape velocity It is the lowest velocity an object must obtain to escape a gravitational field & go to infinity.

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Formulas

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Electromagnetism

Notes

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Contents • Charged Particles: Conductors, Electric & Magnetic Fields 24

➢ Electric Fields .................................................................................................................. 24

➢ Electric Fields Between ll Plates ....................................................................................... 24

➢ Work in Electric Fields ..................................................................................................... 25

➢ Comparing Gravitational Fields & Electric Fields ............................................................... 26

➢ Trajectory of a Charged Particle in an Electric Field vs. a Projectile in a Gravitational Field 26

➢ Projectile Motion Equations ............................................................................................ 27

o Vertical ....................................................................................................................... 27

o Horizontal ................................................................................................................... 27

➢ The Effect of a Charged Particle in a Magnetic Field ......................................................... 27

➢ Right Hand Palm Rule ...................................................................................................... 27



• Motor Effect 29

➢ Introduction ................................................................................................................... 29

➢ Carrying Conductors - Force Between Parallel Current ..................................................... 30

➢ Defining the Ampere ....................................................................................................... 30

• Electromagnetic Induction 31

➢ Introduction ................................................................................................................... 31

➢ Magnetic Flux ................................................................................................................. 31

➢ Faraday’s Law ................................................................................................................. 32

➢ Lenz’s Law ...................................................................................................................... 32

o Case Studies: ............................................................................................................... 32

➢ Applying Lenz’s Law to a Coil ........................................................................................... 33

➢ Eddy Current ................................................................................................................... 33

➢ Determining the Direction of Eddy Currents ..................................................................... 34

o Bad Method ................................................................................................................ 34

o Good Method.............................................................................................................. 34

➢ Transformers .................................................................................................................. 34

➢ From Faradays’ Law: ....................................................................................................... 35

➢ Types of Transformers..................................................................................................... 36

o Step-up Transformers .................................................................................................. 36

o Step-down Transformers ............................................................................................. 36

➢ Applications of Transformers .......................................................................................... 36

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➢ Power Loss in Transmission Lines .................................................................................... 36

➢ Applications of Transformers .......................................................................................... 37

➢ Limitations of Transformers ............................................................................................ 37

o Resistive Heat Production ............................................................................................ 37

▪ Solutions: .................................................................................................................... 37

o Incomplete Flux Linkage .............................................................................................. 38

➢ Electromagnetic Braking ................................................................................................. 38

o Advantages: ................................................................................................................ 38

o Disadvantages: ............................................................................................................ 38

• Applications of Motor Effect 39

➢ DC Motors ...................................................................................................................... 39

o Components of a DC Motor: ........................................................................................ 39

➢ Operation of a DC Motor ................................................................................................. 40

➢ Torque ............................................................................................................................ 42

➢ Torque in a DC Motor ...................................................................................................... 42

➢ Back EMF ........................................................................................................................ 43

➢ Significance of Back EMF ................................................................................................. 43

o When there is no load: ................................................................................................ 43

o When a load is introduced: .......................................................................................... 44

➢ Generators ..................................................................................................................... 44

o AC Generator: ............................................................................................................. 45

o DC Generator: ............................................................................................................. 45

➢ AC Induction Motors ....................................................................................................... 46

➢ Structure of an AC induction Motor ................................................................................. 46

➢ Operation of an AC Induction Motor ................................................................................ 46

➢ AC Induction Motor ........................................................................................................ 47

Definitions 48

Formulas 49

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• Charged Particles: Conductors, Electric & Magnetic Fields

➢ Electric Fields o A force experienced by a charge due to an electric field:

o A source of charge (Q), is the charge produced by the electric field.

o A test charge (q), is a charge used to measure the source charge’s electric field.

o The test charge experiences a force (F) inside the source charge’s electric field.

o The force can either be attractive or

repulsive; it depends on the charges of Q

& q.

➢ Electric Fields Between ll Plates

o The plates produce a uniform electric field throughout the plate.

o Field lines always travel perpendicular.

o Density of field lines represent field strength

▪ High amount of lines indicates a strong

electric field.

o A charge placed anywhere in a uniform field will experience a constant electric

field (F=Eq). Therefore, acceleration is also constant (F=ma).

Where:

▪ F – force acting on charged

particles (N)

▪ E – electric field strength (Nc-1)

▪ q – charge of an object

experiencing the force (C)

Where:

▪ E – electric field strength (NC-1)

▪ V – potential difference (V)

▪ d – distance between electric

field plates (m)

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➢ Work in Electric Fields o Work is done when a charged object is moved in an electric field.

o Furthermore, F = Eq is substituted into W = Fd to yield.

o If the charge moves against the electric field, work is done onto the field by the

charge.

o If the charge moves along the electric field, work is done by the field by the

charge.

o Equipotential lines consist of points where they all have the same electric

potential energy.

To move a point

charge from point A to

B or vice-versa, work

must be done.

Where:

▪ W – work (J)

▪ q – charge of an object

experiencing the force (C)

▪ V – electrical potential (V or JC-1)

Where:

▪ W – work (J)

▪ q – magnitude of charge (C)

▪ E – electric field strength (NC-1)

▪ d – distance between points,

parallel to the electric field (m)

Only works in uniform electric

fields such as the one in ll plates.

A

B

d

𝑬𝒅 = 𝑭𝒅

𝒒=

𝒘

𝒒= 𝜟𝒗 ; for a constant electric field.

General definition

relationship

Constant field strength

case relationships

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➢ Comparing Gravitational Fields & Electric Fields

Quantity Gravitational Field Electrical Field

Field strength in relation to distance ‘r’.

𝑔 =𝐺𝑀

𝑟2 𝐸 =

𝑘𝑞

𝑟2

Force between monopoles. 𝐹 =

𝐺𝑀𝑚

𝑟2 𝐹 =

𝑘𝑞1𝑞2

𝑟2

Potential energy changes in a uniform field.

𝛥𝐺𝑃𝐸 = 𝑚𝑔𝛥ℎ W = qV

Force due to uniform field. Fg = mg F = Eq

➢ Trajectory of a Charged Particle in an Electric Field vs. a Projectile

in a Gravitational Field

Uniform Electric Field Uniform Gravitational Field

A charge experiences a constant acceleration due to an electric field.

A mass would experience a constant vertical acceleration due to gravity.

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➢ Projectile Motion Equations

o Vertical ▪ vy = uy + at

▪ vy2 = uy

2 + 2asy

▪ sy = uyt + ½ at2

▪ uy = usin𝜃

▪ 𝑣+𝑢

2=

𝑠

𝑡

o Horizontal ▪ sx = uxt

▪ ux = ucos𝜃

➢ The Effect of a Charged Particle in a Magnetic Field o It was observed that when a charge (q) was moving at a velocity (v) in the

presence of a magnetic field (B), it would experience a force (F):

➢ Right Hand Palm Rule o Use right hand:

▪ Thumb points in the direction of the conventional current.

❖ Proton’s velocity

▪ Fingers point to the direction of the

magnetic field.

▪ Palm faces the direction of force.

Where:

▪ F – force (N)

▪ q – charge (C)

▪ v – velocity of charged particle (ms-1)

▪ B – magnetic field strength (T)

▪ 𝜃 – angle the object is moving at with respect to

the magnetic field

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➢ The Effect of a Charged Particle in a Magnetic Field o If the velocity if not perpendicular to the magnetic field. The result is a helical

motion.

o Break the oblique velocity into its horizontal & vertical components.

o The horizontal component, which is

parallel to the field, does not

experience a magnetic force.

o Only the vertical component

experiences the force.

o Combining both horizontal & vertical

components result in a spiral/helical

motion.

➢ The Effect of a Charged Particle in a Magnetic Field o If a charge were to enter a magnetic field perpendicular, it will undergo circular

motion & experience centripetal motion.

o Therefore, the magnetic force on the charge is equivalent to its centripetal

force.

Where:

▪ r – radius (m)

▪ m – mass (kg)

▪ v – velocity of charged particle (ms-1)

▪ q – charge (C)

▪ B – magnetic field strength (T)

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• Motor Effect

➢ Introduction o When a current-carrying conductor is placed in a magnetic field, it experiences

a force.

o A moving charge with a constant velocity produces a magnetic field.

o Electrons move as a stream in conductor with uniform velocity & produces a

magnetic field surrounding the conductor.

o It converts EPE to KE.

o This magnetic field interacts with an external magnetic field, which causes a

constant field in the same direction acting on the conductor.

o The magnetic field produced by a current carrying conductor surrounds the

wire in a circular fashion.

Current flowing

into the page. Current flowing

out of the page.

Where:

▪ F – force acting on conductor (N)

▪ B – strength of external magnetic field (T)

▪ I – current (A)

▪ 𝑙 – length of the conductor inside the magnetic

field (m)

▪ 𝜃 – angle between magnetic field of the

conductor

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➢ Carrying Conductors - Force Between Parallel Current o There will be a force experienced by two wires if line two conductors in a

parallel fashion.

This force is a consequence of the magnetic fields, produced by the individual

wires, interacting with one another.

o Recall Newton’s Third Law of Motion

▪ “Every action has an equal & opposite force”

❖ The force experienced by wire 1 is equal in magnitude to the

force experiences by wire 2.

❖ However, they act in the opposite direction.

➢ Defining the Ampere o Similar to how the International System of Units (SI0 defines a metre as how

far light would travel in 1

299792458 seconds, other measurements are defined by

other data sets.

▪ The ampere (A) is the amount of current needed through two parallel

identical conductors of infinite length when they are 1 metre apart to

produce a force of 2 x 10-7 Nm-1.

Where:

▪ F - force between the two parallel conductors

(N)

▪ 𝑙 - shortest length of any conductor (m)

▪ I1 – current in conductor 1 (A)

▪ I2 – current in conductor 2 (A)

▪ 𝜇0 - 4π x 10-7 TmA-1

▪ k – 2 x 10-7

▪ r – distance between the two conductors (m)

F = 2 x 10-7 N

1m

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• Electromagnetic Induction

➢ Introduction o Michael Faraday discovered that an electric current can be induced in a

conductor if there was a change in the magnetic field acting on that conductor.

o This is called electromagnetic induction.

➢ Magnetic Flux o A measure of the total magnetic field that passes through a particular area.

o Measured in Webes (Wb) = kgm2s-2A-1

o Faraday demonstrated that a change in magnetic

field would result in an induced electromotive

force (EMF) that in turn produces inducing current.

o This would be done by moving a magnet near a

conductor & observe the needle of a galvanometer

deflect in a certain direction.

▪ A deflection in a galvanometer indicates a

producing current.

Where:

▪ 𝜙 – magnetic flux (Wb)

▪ B – magnetic field strength or magnetic flux

destiny (T)

▪ A – area (m2)

▪ θ – angle between magnetic field line & the

normal to area.

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o He observed that the magnitude of the current produced was dependent on

the speed at which the magnet was moving.

o A current can also be induced by moving a

conductor.

o Recall when a charge moves in a magentic

field, it experiences a force equal to 𝑞𝑣𝐵 𝑠𝑖𝑛 𝜃.

o Using the RHP, the positive charge goes out of

the page while the negative charge goes into

the page.

➢ Faraday’s Law o “The induced EMF, in a closed circuit, is directly proportional to the rate of

change of magnetic flux.”

➢ Lenz’s Law o “An induced EMF always gives rise to a current whose magnetic field will

oppose the original change in flux”

o This law is an extension of the Law of Conservation of Energy, which states that

energy cannot be created nor destroyed.

o Lenz’s Law accounts for the negative sign in Faraday’s law. The current induced

in the coil opposes any change in the magnetic flux by flowing opposite to the

current which caused such changes.

o Thus, this law is used to explain the direction of the induced EMF.

o Case Studies:

▪ Case 1:

❖ Consider Faraday’s experiment, where a magnet moves into a

conducting coil.

❖ We know that an induced current is generated in the coil.

❖ This indicates that the initial kinetic energy has been converted to

electrical energy.

Where:

▪ ε – induced EMF (V)

▪ n – number of loops

▪ Δ𝜙 – change in magnetic flux (Wb)

▪ Δt – change in time (seconds)

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❖ If this was not the case & the induced current somehow achieved

to speed the magnet up, more kinetic energy would be produced.

❖ This violates Law of Conservation of Energy.

▪ Case 2:

❖ The magnetic field that arises from the induced EMF MUST

oppose the external flux change. Otherwise, energy is created

form nowhere, violating the Law of Conservation of Energy.

❖ Therefore, the current induced flows clockwise.

➢ Applying Lenz’s Law to a Coil 1. Determine if the flux through the coil is increasing or decreasing.

2. By Lenz’s Law, the induced current’s magnetic field must oppose this change

in flux.

3. Fingers point in direction which opposes change in flux.

i. If the magnetic field directs into the page, it is decreasing.

➢ Eddy Current o It is a special type of current that is induced when a metal plate experiences a

change in magnetic flux.

o Lenz’s Law governs the direction of eddy

currents’ flow in a loop.

The direction of current induced can also

be found using the Right-Hand Coil Rule.

1. Thumb points North.

2. Fingers wrap around the coil in the

direction of conventional current.

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➢ Determining the Direction of Eddy Currents

o Bad Method 1. Draw a loop at the location/boundary where the change in magnetic

flux is occurring.

2. Identify the direction in which the conducting plate is moving

3. Use the right-hand palm rule, such that the direction of the force

(palm) faces opposite to the direction of movement of the conductor.

4. Fingers point in the direction of the magnetic field.

▪ Perform this step inside the boundary containing the magnetic field.

5. Thumb points to the direction of the induced current.

6. The current loops around the boundary.

o Good Method ▪ Direction of current can be found by using the right-hand coil rule.

1. Thumb points in the direction that opposes the initial

change in flux.

2. Finger wrap around the eddy current in the direction of

conventional current.

➢ Transformers o They allow generated AC voltage to either by increased or decreased before it

is used.

o They function by mutual induction where a

changing current in one coil causes an induced

EMF in the area of another coil.

o A transformer has two coils (primary &

secondary coils) of conducting wires wound on

a laminated iron core.

o Iron core:

▪ Material with high permeability to

concentrate & guide the magnetic field lines inside the core.

o AC current is fed into the primary coil which induces a current in the secondary

coil.

o AC current switches current direction periodically, which results in a changing

magnetic field.

o The secondary coil experiences a change in magnetic flux; therefore, AC

current is induced.

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➢ From Faradays’ Law:

o By the law of conservation of energy, energy in the primary coil is conserved

when transmitted to the second coil through electromagnetic induction.

o Combining these equations, we obtain the full transformer equation:

Assuming that all the flux

produced in the primary coil

enters the secondary coil.

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➢ Types of Transformers

o Step-up Transformers ▪ Increases secondary voltage compared to primary voltage.

▪ Thereby, decreasing current in

secondary coil.

▪ 𝜼𝑺 > 𝒏𝒑 | 𝒗𝑺 > 𝒗𝒑 | 𝑰𝑺 < 𝑰𝒑

o Step-down Transformers ▪ Decreases secondary voltage compared

to primary voltage.

▪ Thereby, increasing current in

secondary coil.

▪ 𝜼𝑺 < 𝒏𝒑 | 𝒗𝑺 < 𝒗𝒑 | 𝑰𝑺 > 𝑰𝒑

➢ Applications of Transformers

➢ Power Loss in Transmission Lines o The resistance in the transmission lines causes an inefficient transfer of energy

when the electrical energy transmitted is eventually transformed into heat

energy.

o The power lost can be calculated through;

o To reduce the power loss, current must be reduced before entering the

transmission lines.

o This is done by set-up transformers.

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➢ Applications of Transformers o Not all appliance run on the same voltage. Therefore, transformers exist in

household appliances to increase/decrease the supplied voltage for suitable

use, allowing them to be conveniently connected to the same power supply.

o Substations near power-plants use step-up transformers to reduce the amount

of power loss.

o Substations near households/consumers use step-down transformers to

reduce voltages to safe & practical levels.

o Allows remote communities to access grid electricity.

➢ Limitations of Transformers

o Resistive Heat Production ▪ The iron core experiences changes in flux. Therefore, eddy currents are

induced.

▪ Eddy currents in the iron core generates a significant amount of heat

energy via resistance. The iron core heats up.

▪ This is an inefficient transfer of energy. It also poses a fire hazard.

▪ Solutions:

❖ Lamination:

- Iron core is laminated by insulation sheets, introducing

electrical discontinuity in the core.

- This reduces the eddy currents that are formed inside

the core. Therefore, decreasing the heat producing

and increasing the efficiency of the transformers.

❖ Other:

- A ferrite core (made from iron oxides) can substitute

the iron core.

◊ Ferrites are great magnetic flux

transmitters; but poor electrical conductors,

so the magnitude of eddy currents are

significantly reduced.

- Overheating can cause the isolation to foil, leading to

larger currents to flow & extreme heat will follow.

- To prevent overheating, heat can be dissipated by

using:

◊ Water & oil as coolants.

◊ Fans to increase air circulation through &

around transformer.

◊ Heat sinks to disperse heat elsewhere.

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o Incomplete Flux Linkage ▪ Flux Linkage is the total magnetic flux passing through the turns of a

coil (the flux ‘links’ the turns).

▪ Each turn links the flux identically.

▪ This is applicable in an ideal scenario. In reality, it is extremely rare for

the flux to link all the turns. However, there is a flux linkage.

▪ This results in incomplete flux transmission between the primary &

secondary coil, hence inefficiency is inevitable.

➢ Electromagnetic Braking o Electromagnetic braking applies the EM induction principle.

o When the metal wheels pass through a magnetic field, eddy currents are

produced in the wheel. This is due to the change in magnetic flux (Faraday’s

Law) that the wheel experiences.

o The eddy currents are induced in a specific way to counteract the motion of

the wheel. Hence, slowing down its rotation (Lenz’s Law).

o EM braking is used by modern trains and rollercoasters.

o Advantages: ▪ Since there is no direct contact between components, friction is

removed. This reduces the need for maintenance & replacement.

❖ Little to no noise

❖ Smooth brake effect

o Disadvantages: ▪ EM braking is incapable of holding the transport system after coming

to rest. In this case, mechanical braking is required.

Where:

▪ 𝛬 – Flux Linkage (Wb)

▪ N – number of turns

▪ 𝜙 – Magnetic flux (Wb)

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• Applications of Motor Effect

➢ DC Motors o A DC motor is a device which converts electrical energy into mechanical

energy.

o DC current is fed through a coil in a magnetic field, which produces a rotation

motion due to the motor effect.

o Using a simple design and wires attached to a power supply results in two

problems:

1. The direction of force acting on each side of the coil reverses every half

rotation meaning that the coil can’t completely rotate & gets stuck at

the vertical position.

2. Even if the coil completes its rotations, eventually the wires attached

would get tangled.

o We can solve both issues by using a split-ring commutator.

o Components of a DC Motor:

▪ Armature/Coil (Rotor):

❖ Armature:

◊ The frame around which the coil of wire is wound. It

has an axis on which it can freely rotate.

❖ Coil:

◊ One or more turns of wire wound around the

armature. Current-carrying coils experience forces

which act in certain directions due to the motor

effect. This force is a rotational force, known as

torque.

◊ ↑ turns = ↑ force = ↑ torque

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▪ Stator:

❖ The stationary permanent/electromagnet that provides the

external, radial magnetic field around the coils.

❖ Radial magnetic field ensures the sides of the coil are always

travelling perpendicular to the

magnetic field to produce the

possible maximum torque

throughout its rotation.

▪ Split-Ring Commutator:

❖ A device with two metal semi-circular rings that reverses the

direction of the current flowing in each coil at every 180o,

allowing the rotor to continuously rotate in the same direction.

▪ Brushes:

❖ Conducting contacts (generally graphite or carbon) that connect

the external circuit to the split-ring commutator.

◊ Carbon & graphite are preferential as they are both

good conductors of electricity & good lubricants,

thereby reducing the friction between commutator.

➢ Operation of a DC Motor

1. Side WX experiences a force upwards because it is connected to the +

terminal. Side YZ experiences a force downwards because it is connected to

the – terminal.

o The combination of these two forces initiate the clockwise rotation

of the coil.

W

Y

Z

X

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2. As the coil rotates, it will reach a perpendicular position where the terminals

are fully disconnected & there is no force.

o However, the momentum of the coil keeps the coil rotating

clockwise & help reconnect the commutators with the brushes.

o Notice that any further rotation without a commutator, will cause

the sides to produce a force that causes the coil to rotate anti-

clockwise back to its perpendicular position.

o A commutator prevents this. Just after the coils’ perpendicular

position, the split-ring commutator changes the direction of the

current through the coil.

o The forces now act to allow the coil to continoulsy rotate clockwise.

WX

YZ

WX

YZ

WX

YZ

F

F

F

F

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3. Now side WX experiences a force downwards because it is connected to the

terminal. On the other hand, side YZ experiences a force upwards because it

is now connected to the + terminal.

o This process repeats as the motor rotates.

➢ Torque o Torque is the required force to cause an object to rotate.

o The net torque is the sum of all acting torques.

➢ Torque in a DC Motor o Consider the torque produced within a DC motor:

WX YZ

F

F

Where:

▪ T – Torque

▪ F – Force (N)

▪ d – perpendicular distance from

the turning point to line of

action of force

▪ Assume a flat position.

▪ Torque is produced on both sides AB & CD

▪ Tnet = TAB + TCD

= FABd + FCDd

= BILABd + BILCDd (Substituting F=BIL)

= BI x d(LAB + LCD)

= BIA

▪ This force is proportional to the number of

turns in a coil.

d

LCD LAB

A

B C

D

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o Now, what happens if the coil is at an angle?

o The turning force that the coil experiences in an electric motor is referred to as

the torque & is caused by the forces acting on the sides.

➢ Back EMF o Back EMF is the induced EMF produced in the coil of a motor due to its rotation

in a magnetic field.

o The rotation supplies the change in flux, thus a current is induced (Faraday’s

Law).

o As a consequence of Lenz’s Law, the induced EMF opposes the change that

causes it & therefore acts in the opposite direction to the EMF creating it.

o Therefore, back EMF works against the input voltage from the power supply.

o Back EMF reduces the net EMF:

o Net current also decreases, as voltage & current share a directly proportional

relationship (V ∝ I).

➢ Significance of Back EMF o Magnitude of back EMF is directly proportional to speed of rotation.

▪ i.e. a faster rotating motor induces a larger back EMF.

o When there is no load: ▪ Back EMF is initially zero when the coil is stationary & increases to a

maximum as the coil reaches its maximum rate of rotation.

▪ A maximum rate of rotation, the coil rotates at a constant angular

speed.

TAB = Fdsinθ = BILdsinθ

TCD = Fdsinθ = BILdsinθ

Tnet = BIL x 2dsinθ

= BI (AB x BC)sinθ

= BIAsinθ

▪ If a coil has n loops of wire. Where:

▪ T – Torque (Nm)

▪ N – Number of coils

▪ B – Magnetic field strength (T)

▪ I – Current (A)

▪ A – Area of the coil (m2)

▪ θ – angle between the normal of

the coil & the magnetic field

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▪ So, the next force acting on the coil is ZERO!

o When a load is introduced: ▪ ↑ loads = ↓ speed of rotation = ↓ back EMF = ↑ current

▪ ↑ loads (slower rotations) = ↑ currents in the coil.

▪ ↑ current = ↑ torque (sacrificing heat production).

❖ Back EMF keeps dropping until a high enough current & torque is

reached to meet the load experiment.

▪ When the motor comes to a sudden halt (drill gets stuck), back EMF

will be completely removed.

▪ This leads to extremely high currents that could burn out the motor.

▪ When a DC motor starts, there is little back EMF or rotation. This

means the coil experiences the full initial current.

▪ In order to reduce this, a variable resistor is placed in series with the

armature to provide a starting resistance.

▪ When the motor speeds up, the back EMF increases (acts like

resistance itself).

▪ The resistor switches out at higher speeds because the back EMF is

sufficient to lower the current in the coil.

▪ This resistor can switch on any time thereafter to protect the coil

against high currents, thereafter, preventing a burn out (e.g. stuck

drill).

➢ Generators o A generator is a device which converts mechanical energy into electrical energy

by applying the principle of electromagnetic induction.

o Its anatomy is extremely similar to a DC motor. The key difference is the lack of

a power supply (power is now generated).

o A coil of wire is forced to rotate about an axis in a magnetic field.

o This causes the coil to experience a change in magnetic flux, inducing an EMF

(Faraday’s Law). This then transferred to an external circuit.

o There are two types of generators: AC & DC.

▪ Both need an external source to mechanically turn the coil. E.g. water

or steam turbine.

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o AC Generator: ▪ Slip Rings:

❖ Two cylindrical metal conductors that rotate freely with the

armature.

❖ They provide constant electrical contact between the rotating

armature & external circuit.

❖ The armature’s rotation naturally produced an AC voltage which

is transmitted to the slip rings, giving an AC output.

o DC Generator: ▪ Split Rings:

❖ The flux-time graph of a DC generator is identical to an AC’s.

❖ However, its EMF-time graph is always in one direction.

❖ The split-ring commutator reverses the direction of the natural

AC voltage every half-cycle.

❖ This rectifies the output voltage to become unidirectional (DC).

o AC vs DC

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➢ AC Induction Motors o AC induction motors use AC current as opposed to DC current from DC motors.

o These motors are different since they use the principle of electromagnetic

induction to rotate the rotor, instead of the motor effect in traditional motors.

o The rotor isn’t supplied current from a supply. It is induced.

➢ Structure of an AC induction Motor o AC has a stator and rotor, just like DC.

o The rotor is an assembly of parallel conductors & end rings. They produce a

similar shape to ‘squirrel cages’.

o The stator is the stationary electrical component, which is made up of pairs of

electromagnets connected

to an AC power supply.

o The coils are wound in a way

that when a current flows

through the coils, one coil

would be the north pole &

its pair a south pole.

➢ Operation of an AC Induction Motor o Initially, one pair of electromagnets is fed AC current to produce a magnetic

field, while the remaining two pairs do not.

o Due to the nature of the AC power supply, this pair is then turned off & the

adjacent pair is turned on.

o This process continues, ultimately producing rotating magnetic field.

o Phase AC Motor:

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o Due to the rotating magnetic field, an electric current is induced in the rotor

(Faraday’s Law). This induced electric current produces its own magnetic field.

o Due to Lenz’s Law, this magnetic field is induced in such a way that opposes the

change that causes it, effectively causing the rotor to spin the same direction

as the rotating magnetic field.

▪ A magnetic field rotating clockwise equivalent to the rotor rotating

anti-clockwise (same relative motion). Lenz’s Law acts to counteract

the change that causes it, resulting in a clockwise rotation of the rotor.

o The rotor continually ‘chases’ the rotating magnetic field, always slower &

never catches up.

o The difference in rotational speed between the magnetic field & rotor is known

as the slip speed. If there is slip speed, there is relative motion between the

stator and rotor.

o Relative motion is required for torque generation in AC induction motors.

➢ AC Induction Motor o Advantages:

▪ Cost effective (absence of split-ring commutators & brushes)

▪ Reduced maintenance/reduced wear & tear (absence of split-ring

commutator & brushes)

o Disadvantages: ▪ Poor starting torque

▪ Used only in fixed-speed applications

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Electric Field A force experienced by a charge due to an electric field.

Work When a charged object is moved in an electric field.

Motor Effect When a current-carrying conductor is placed in a magnetic field, it experiences a force.

Electromagnetic Induction The production of an electromotive force across an electrical conductor in a changing magnetic field.

Magnetic Flux A measure of the total magnetic field that passes through a particular area.

Eddy Current A special type of current that is induced when a metal plate experiences a change in magnetic flux.

Transformers Allow generated AC voltage to either by increased or decreased before it is used.

Flux Linkage The total magnetic flux passing through the turns of a coil (the flux ‘links’ the turns).

Electromagnetic Brakes They slow or stop motion using electromagnetic force to apply mechanical resistance.

DC Motor A device which converts electrical energy into mechanical energy.

Torque The required force to cause an object to rotate.

Back EMF The induced EMF produced in the coil of a motor due to its rotation in a magnetic field.

Generator A device which converts mechanical energy into electrical energy by applying the principle of electromagnetic induction.

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Formulas

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The Nature of Light

Notes

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Contents • Electromagnetic Spectrum .................................................................................................. 53

➢ Prediction of EM Waves .................................................................................................. 53

➢ Measuring the Speed of Light .......................................................................................... 53

➢ Electromagnetic Wave Spectrum ..................................................................................... 55

➢ Spectra ........................................................................................................................... 57

➢ Types of Spectrum .......................................................................................................... 57

➢ Spectrology ..................................................................................................................... 58

➢ Emission Spectrum .......................................................................................................... 58

➢ Exploring Electromagnetic Spectrum ............................................................................... 59

➢ Stellar Spectra ................................................................................................................ 59

o Continuous Spectrum .................................................................................................. 59

o Stellar Temperature .................................................................................................... 60

o Translational Velocity .................................................................................................. 60

o Rotational Velocity ...................................................................................................... 60

o Chemical Composition ................................................................................................. 61

• The Wave Model of Light .................................................................................................... 62

➢ Huygens’ Principle .......................................................................................................... 62

➢ Light Diffraction .............................................................................................................. 62

➢ Path Difference ............................................................................................................... 63

➢ Interference .................................................................................................................... 63

➢ Young’s Double Slit Experiment ....................................................................................... 63

➢ Bandwidth ...................................................................................................................... 65

➢ Diffraction Grating .......................................................................................................... 65

➢ Polarisation .................................................................................................................... 66

➢ Polarisers ........................................................................................................................ 67

➢ Malus’ Law ..................................................................................................................... 68

• The Quantum Model of Light .............................................................................................. 69

➢ Blackbody Radiation ....................................................................................................... 69

➢ Classical Predications of Blackbody .................................................................................. 69

➢ The Quantum Theory ...................................................................................................... 70

➢ The Quantisation of Energy ............................................................................................. 70

➢ Analogy for Classical Physics............................................................................................ 70

➢ Analogy for Quantum Physics .......................................................................................... 70

➢ Consequences of Blackbody Radiation ............................................................................. 71

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➢ Einstein’s Particle Theory of Light .................................................................................... 71

➢ The Electron-Volt ............................................................................................................ 71

➢ Hertz’s Experiments ........................................................................................................ 72

➢ The Photoelectric Effect .................................................................................................. 73

➢ Prediction Based on the Wave Properties of Light ............................................................ 73

➢ Actual Observations ........................................................................................................ 73

➢ The Photoelectric Effect (Continuation) ........................................................................... 74

• Light & Special Relativity ..................................................................................................... 75

➢ Galilean Experiment ........................................................................................................ 75

➢ Galilean Relativity ........................................................................................................... 75

➢ Relative Velocity ............................................................................................................. 75

➢ Light in Inertial Frames .................................................................................................... 75

➢ Aether ............................................................................................................................ 76

➢ Michelson & Morley Experiment ..................................................................................... 76

➢ Einstein’s Thought Experiment ........................................................................................ 77

➢ Consequences of Special Relativity .................................................................................. 78

o Relativity of Time ........................................................................................................ 78

o Relativity of Length ..................................................................................................... 79

➢ Relativistic Momentum ................................................................................................... 80

➢ Particle Accelerators ....................................................................................................... 80

➢ Mass – Energy Equivalence .............................................................................................. 81

➢ Relativistic Mass ............................................................................................................. 81

➢ Antiparticle Annihilation ................................................................................................. 81

➢ Nuclear Reactions ........................................................................................................... 82

Definitions ................................................................................................................................. 83

Formulas ................................................................................................................................... 84

Electromagnetic Spectrum ...................................................................................................... 84

The Wave Model of Light ........................................................................................................ 84

The Quantum Model of Light .................................................................................................. 84

Light & Special Relativity ........................................................................................................ 84

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• Electromagnetic Spectrum ➢ Before the 20th century, physicists made comprehensive models and forms of

understanding.

➢ After the 20th century, physicists opened up with quantum theory and the theory of

relativity.

➢ Prediction of EM Waves o Hans Oersted observed a current carrying wire caused a nearby compass

needle to deflect. This shows that electricity could produce magnetism. o Michael Faraday later showed that a changing magnetic field could produce

electric current.

o James Clerk Maxwell later derived four equations involving electric &

magnetic field. He found that the interaction between a changing electric &

magnetic field resulting in a wave propagating through space.

▪ 𝜵 ⋅ 𝑬 =𝝆

𝜺𝟎 : Electric field strength depends on the density of charges.

▪ 𝜵 ⋅ 𝑩 = 𝟎 : Bnet over a surface is 0 (magnets always exist as dipoles.

▪ 𝜵 × 𝑬 = −𝝏𝑩

𝝏𝒕 : Faraday’s Law

▪ 𝜵 × 𝑩 = 𝝁𝒐𝑱 + 𝝁𝒐𝑬𝟎𝝏𝑬

𝝏𝒕 : A changing E-field or current produces B.

o Maxwell called these changing electric & magnetic fields, electromagnetic

waves. These waves are predicted to travel around 3 x 108 ms-1.

o This speed was similar to the speed of light at the time. Therefore, he

proposed that visible light is a form of EM wave.

o He hypothesised that there are other forms of EM waves, but this wasn’t

proven until 1910’s. These waves are also known as electromagnetic

radiation (EMR).

➢ Measuring the Speed of Light

o Aristotle & Kepler believed that the speed of light was infinite.

o The first movement of finite speed of light came from

Romer in 1675. He noticed that the eclipses of lo, the

innermost moon of Jupiter depended on the relative

positions of Jupiter & Earth.

o As the Earth is moving away from Jupiter (From L to K), the

time at which the eclipse occurred is later & as the Earth is

moving towards Jupiter from F to G, the time is earlier.

o Romer reasoned that it was because of light from lo had

to travel a further distance to reach the Earth when the

Earth is moving away from Jupiter.

Where:

▪ c – speed of light (ms-1)

▪ f – frequency of light (Hz)

▪ λ – wavelength of light (m)

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o Using the discrepancies in time for the eclipses of lo, Romer estimated that

it took about 22 mins for light to travel across the diameter of the Earth’s

orbit around the Sun. Romer calculated the speed of light to be about

2.2 x 108 ms-1.

o Hippolyte Fizeau performed the first measurement of the speed of light in

1840’s.

o Fizeau shone a bright light through a slot in a toothed cogwheel. This light

was then reflected by a mirror placed 8km away.

o As the cogwheel was rotated to a certain speed, the reflected light was

eclipsed by the cog. Using the rotation rate (𝝎) & the angle of rotation (𝜽)

of the wheel, the time taken for the light to complete its return journey (i.e.

16km) could be determined.

o Using 𝑠 =𝑑

𝑡, Fizeau calculated the speed of light to be 3.13 x 108 ms-1.

o Leon Foucault refined Fizeau’s method by using a rotating mirror to block

light’s path & soon determined the speed of light to be 2.98 x 108 ms-1 (very

close to today’s value of 299,792.458 kms-1).

o The distance for reflected light to travel from R to M & back to R is 2h. If the

speed of light is constant (c), then the time taken is: 𝑡 =2ℎ

𝜃 → (1)

o During this time, the mirror is rotated to an angle 𝜃. If the rotational

velocity of the mirror is 𝜔, the time taken is: 𝑡 =𝜃

𝜔 → (2)

o Equate (1) & (2) & make c the subject: 𝑐 =2ℎ𝜔

𝜃

o Further experiments (Albert Einstein – 1930’s) show more accurate values.

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o By 1983, the speed of light was declared to be exactly 299,792.458 ms-1. The

speed of light is an absolute constant regardless of the light source or the

observer.

o Due to constancy of the speed of light, one metre is defined to be how far

light travels in 1

299,792.459 or

1

c of one second.

➢ Electromagnetic Wave Spectrum

o Today there are several forms, due to Hertz’s discovery.

o The EM spectrum is divided into different bands to distinguish their

different sources and uses. Radio waves & microwaves can overlap in these

terms.

o Earth’s atmosphere filters out specific EMR’s. This is beneficial for safety

reasons as high energy EMR is dangerous to living organisms. However, it

prevents many Earth-based measurements of objects in space.

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➢ Spectra o Spectrum: A range of values of a quantity or set of related quantities.

o Visible Spectrum: Range of colours emerging from dispersion of white light.

o Electromagnetic Spectrum: A range of frequencies/wavelengths of

electromagnetic radiation.

➢ Types of Spectrum o Continuous Spectrum: shows all values on spectrum (analog clock analogy).

▪ White Light

▪ Rainbow

▪ Radiation of a blackbody

o Line Spectrum: shows limited values on spectrum (digital clock analogy).

▪ Absorption or Emission Spectrum

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➢ Spectrology o Spectrology: the analysis of spectra produced by a radiant object to obtain

information such as the chemical composition, temperature & other

features of that object.

o A spectrometer used in this study. There are two types of spectrometers:

▪ Prism Spectrometer

▪ Grating Spectrometer

o How is a visible spectrum produced by a hot object?

▪ Input energy causes electrons to be excited. They ‘jump’ to a higher

orbital shell.

▪ Electrons are unable to stay in the excited state forever.

▪ As electrons jump down to their original level, they emit EMR.

➢ Emission Spectrum

o Emission Spectrum: consists of specific wavelengths observed as bright

lines against a dark background.

o This occurs when a sample of gas is energised (electricity). The electrons of

the gaseous atoms absorb some of the energy & make a quantum jump to

the outer energy shell. They soon release the absorbed energy & fall back to

their ground state. The energy released is in the form of EM waves. Under

the visible region, it is observed as colours.

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➢ Exploring Electromagnetic Spectrum

o Absorption Spectrum: consists of dark lines as seen against a rainbow of

colours as a background.

o This occurs when a white light passes through a cool gas. Some of the

component wavelengths of white light are absorbed by the electrons of the

atoms. These electrons then make a quantum jump to the outer energy

shell. They soon release the absorbed energy & fall back to their ground

state. Since the energy released by the electrons is in all directions in space,

the energy obtained in the viewing direction is less than the original energy.

The spectral lines on these particular wavelengths appear to be darker than

the other wavelengths.

➢ Stellar Spectra o Continuous spectrum is produced when an object emits all wavelengths of

the electromagnetic spectrum. Under the visible range, a continuous band

of colours is observed.

▪ An incandescent light bulb produces a continuous spectrum.

o Continuous Spectrum ▪ A heated solid, liquid or a dense gaseous object such as a star

produces a continuous spectrum including a wide range of

wavelengths. This is because in a dense object like a solid, liquid or

even gas at high pressure the atoms are close together & the

electron energy levels overlap, causing the transition of the

electrons in all wavelengths.

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o Stellar Temperature ▪ The visible spectrum of the stars shows that hot stars radiate more

EMR with higher frequency (lower wavelength) than cooler stars.

▪ Short wavelengths correspond to the blue of the visible spectrum

while longer wavelengths correspond to the red end.

❖ Surface Temperature

◊ Wein’s Law:

o Translational Velocity ▪ This is determined by analysing the Doppler Effect on the

absorption lines. If a star is approaching the observer, every

absorption line in the spectrum of the star is shifted towards the

blue end of the spectrum by the same amount. If the star is moving

away, all the lines are shifted towards the red end.

▪ Greater Red Shift = Faster the star is travelling from the Earth

o Rotational Velocity ▪ The spectral lines of a rotating star are broadened due to the red &

blue shift.

▪ Broader Line = Faster Rotation of

Star

Where:

▪ τ – Absolute Temperature

(Kelvins)

▪ λpeak – the wavelength that carries

the longest amount of energy at a

given temperature (m)

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o Chemical Composition ▪ Each chemical element has a unique emission spectrum consisting

of lines corresponding to the energy level transitions within the

element. The chemical composition of a star can be determined by

comparing the absorption spectrum of a star to the emission

spectra of the elements on Earth.

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• The Wave Model of Light ➢ There were two competing theories, by Newton & Huygens, about the nature of

light in the 17th century.

➢ Newton’s corpuscular theory prevailed & slowed the acceptance of the wave model

of light.

➢ In the early 19th century, the first convincing experiment proved the wave model of

light.

➢ Light is now known to display both properties, particles & waves, depending on the

circumstances.

➢ Huygens’ Principle o Huygens’ principle is used to explain how 2D waves propagate.

o “Every point on a wavefront can be considered the

source of circular secondary wavelets. This new

wavefront will be tangential to the wavelets”.

➢ Light Diffraction o Diffraction is the bending or spreading of waves around the edge of an

object or through an opening.

o It is a wave property.

Newton

o Proposed light was comprised of

‘corpuscles’ or particles.

o Predicted light would be slower

in glass than air.

o Theory could not explain certain

behaviours of light.

Huygens

o Proposed light was a wave, similar to

ocean waves.

o Predicted light waves would be faster

in glass than air.

o Explains how diffraction, interference

& polarization occurs.

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➢ Path Difference o It is the extra distance travelled compared to another.

o Constructive Interference:

o Destructive Interference:

➢ Interference o When waves are diffracted, when it passes through narrow slits, the

wavelets will interfere with each other producing interference/diffraction

patterns.

o There are two types of interference of light waves (Constructive &

Deconstructive).

o This can be determined by considering the path difference, a measure of

the difference in distance from different sources to the same point.

➢ Young’s Double Slit Experiment o Thomas Young first demonstrated the concept of interference of light in

1801. His experiment concluded that light was a wave.

o He allowed light from a monochromatic source to pass through a narrow slit

to obtain a narrow beam of light which was then passed through the double

narrow slits.

o He predicted that if light was a particle, light would pass through the two

slits only & create two bars of light on the background surface.

o However, he observed that light was diffracted from the two slits &

interfered to produce a constructive &

destructive interference patterns

(bright & dark bands) on a distant

screen.

Where,

▪ n - integer

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Bright – Second order bright spot – occurs when constructive

Dark – Second order dark spot – occurs when destructive

Bright – First order bright spot – occurs when constructive

Dark – First order dark spot – occurs when destructive

Bright – Central bright maximum – PD = 0 (0th order) – refers to central position

Dark – First order dark spot – occurs when destructive

Bright – First order bright spot – occurs when constructive

Dark – Second order dark spot – occurs when destructive

Bright – Second order bright spot – occurs when constructive

∵ r1 & r2 are >>>> d, r1 & r2 are almost

parallel.

The path difference can be approximated by

drawing the altitude of point S1 & with

respect to line r2.

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➢ Bandwidth o Bandwidth: The distance between successive bright (or dark) fringes.

o From the diagram & the equation of interference, we have:

➢ Diffraction Grating o This consists of large numbers of equidistant parallel lines engraved on a

glass or metal surface.

o Due to a large number of slits on a grating, it produces a sharper image on a

screen.

o Diffraction experiments usually use only

monochromatic light (light of only one

colour/wavelength). When white light, which

contains different colours/wavelengths, is used

in a diffraction grating, each different colour is

diffracted by a different amount and forms its

own set of coloured fringes.

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➢ Polarisation o This evidence for the wave nature of light.

o Polarisation: When the oscillations of electromagnetic waves are restricted

to one dimension.

o A polariser is a device that allows oscillation in one plane.

o Light is a transverse wave which consists of electric & magnetic fields that

oscillate perpendicularly to each other & to the direction of propagation.

o Light can be polarised by restricted its vibrations to one particular plane.

o Only transverse waves can be polarised.

o A polarised beam of transverse wave is one

whose vibrations occur in one direction

(perpendicular to its propagation).

o Longitudinal waves can’t be polarised, since

its vibrations occur in the same direction as

propagation.

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➢ Polarisers o A polaroid sheet is a common polariser which is made by embedding in a

sheet of plastic, crystals of certain substances & then stretching the plastic

so that the crystals or molecules all align in one direction.

o When unpolarised light falls on a polariser such as a polaroid sheet, one

component is transmitted. Thus, the intensity of light is reduced by 50%.

o If a second polaroid sheet (called the analyser) is place after the first

polaroid sheet, with the polarising axes of both vertical, the plane-polarised

light emerging from the first polaroid sheet will go through the analyser

without any change in the nature of its polarisation & intensity.

o However, if the second polaroid sheet is arranged that its polarising axis is

at a right angle to that of the first polaroid sheet, no light does through the

second polaroid.

This vector diagram represents

the random of unpolarised light.

Note: that these random

vibrations can be resolved into

either vertical or horizontal

components. Within the diagram

above, the vertical component

can pass through.

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➢ Malus’ Law o The intensity of light transmitted through the second polaroid sheet

depends on the angles of the polarising axes between the first and second

polaroid sheet.

Where:

▪ I – Intensity of transmitted planed .

. polarised light

▪ I0 – Intensity of incident planed light

▪ θ – the angle of the polarisation . .

.....between the first & second polariser

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• The Quantum Model of Light

➢ Blackbody Radiation o An object’s colour is determined by light (EMR) being reflected, light being

radiated or a combination of both of these methods.

o Blackbody: A perfect absorber & emitter of radiation

or energy. The radiation emitted from a blackbody is

entirely due to its temperature.

o A hollow furnace is an example of this.

o The intensities of the emitted radiation obtained at different temperatures

was graphed against wavelengths.

o At a given temperature, the curve will have a peak, representing the

wavelength with the highest intensity.

o As the temperature increases, the

electromagnetic radiation emitted not

only increases in total intensity but is

strongest at shorter wavelengths (higher

frequencies).

➢ Classical Predications of Blackbody o Classical physics predicted that as the wavelength of the emitted radiation

↓, the intensity of the radiation would ↑, without limit.

o This was known as the ultraviolet catastrophe, as it violated the Law of

Conservation of Energy & didn’t

match experimental observation.

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➢ The Quantum Theory o In 1900, Max Planck, offered an alternative hypothesis that explained the

experimental observations.

o His theory made a new radical assumption that the energy from that

blackbody can’t possess just any value, but rather has energy which is a

multiple of a minimum value related to the frequency of oscillation.

o Planck suggested that energy absorbed or released by an object exists in

small bundles.

➢ The Quantisation of Energy o The energy of each packet (also known as quantum of energy) is given by:

o The energy released or absorbed by an object could only be a whole

number multiple of ‘hf’.

➢ Analogy for Classical Physics o According to 20th century physics, the energy liberated or absorbed by an

object could be any arbitrary value.

o However, calculations using this idea can’t reproduce the graphs obtained

experimentally from the radiation of a blackbody.

➢ Analogy for Quantum Physics o Energy is not absorbed or emitted continuously but rather, in small packets

known as quantum (plural is ‘quanta’). Energy is quantised.

Where:

▪ E – energy of ‘packet’ (J)

▪ h – Planck’s constant (6.63 x 10-34)

▪ f – frequency of EM radiation (Hz)

Where:

▪ n – quantum number (integer)

An object can lose or gain any value of

potential energy as it goes down or up

a ramp.

An object rolling off a staircase can

only lose discrete (quantised)

values of energy as it descends

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➢ Consequences of Blackbody Radiation o Even though Planck’s quantum theory successfully explained the blackbody

radiation curve, it was based on a radical assumption that had no

experimental evidence to support it.

o In 1905, Einstein proposed a particle theory of light based on Planck’s

Quantum Theory. The particles of light (& other EM waves) would have an

energy determined by Planck’s Equation (E = hf).

o Einstein’s theory successfully explained the photoelectric effect & provided

a strong experimental support for Planck’s Quantum Theory.

➢ Einstein’s Particle Theory of Light o The energy of each photon particle in an EM wave is: E = hf.

➢ The Electron-Volt o Electron-Volt (eV): An alternative unit for measuring small amounts of

energy. 1eV is defined as the amount of energy gained by an electron when

it moves through a potential difference of 1 volt.

Where:

▪ E – energy per photon (J)

▪ h – Planck’s constant (6.63 x 10-34)

▪ c – speed of light in a vacuum (3 x 108 ms-1)

Conversion between J & eV:

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➢ Hertz’s Experiments o In 1873, Maxwell proposed the existence of EM waves.

o In 1887, Heinrich Hertz experimentally proved that EM waves existed.

o Hertz set up this experiment:

o Using a high voltage induction coil, Hertz was able to produce an oscillating

spark in the gap between the electrodes of his transmitter.

o His receiver was a small loop of wire, with a small gap, which was placed at

some distance from the transmitter.

o Hertz observed that when sparks were jumping across the gap in the

transmitter, sparks would also jump across the gap in the receiver, even

though the receiver was not connected to a power supply. AC electricity

creates EMR.

o Since the loops were not connected, Hertz hypothesised that the oscillating

sparks in the transmitter produced an EM wave, which then induced the

sparks across the gap in the receiver.

o He also demonstrated that the invisible radiation in his experiment (which

were radio waves) had the same properties as light. Hertz proved Maxwell’s

theory.

o Hertz also noticed that the intensity

of the sparks in the detecting loop

increases when it was illuminated

with ultraviolet light.

o Light seemed to facilitate the escape

of charges from the surfaces. Hertz

failed to investigate this further.

o This is now known as the

photoelectric effect.

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➢ The Photoelectric Effect o In 1905, Einstein proposed a new theory of light.

o Einstein proposed that light consists of tiny particles called photons. Each

photon has a discrete amount of energy which is proportional to the

frequency of light (E=hf).

o When a photon collides with an electron, at or just within the surface of a

metal, it transfers energy to the electron and follows the ‘all or nothing’

principle.

o By building on the ideas proposed by Planck, Einstein’s theory was able to

explain the photoelectric effect.

o Photoelectric Effect: the liberation of electrons from the surface of a

conductor when light strikes the surface.

o The electrons absorb energy from the incident radiation and can overcome

the potential energy barrier that normally confines them to the material.

➢ Prediction Based on the Wave Properties of Light o ↑ intensity of the incident light, the more energy is transferred to the

surface, hence more electrons are liberated with ↑ energy.

o Below a certain intensity, no electron is liberated no matter what the

frequency of the incident light is.

o There is a time lag between when the light strikes the metallic surface to

the liberation of electrons.

➢ Actual Observations o No photoelectrons are emitted unless the frequency of light is greater than

the critical value. The corresponding minimum frequency is called the

threshold frequency of the surface.

o Above the threshold frequency, electrons are liberated no matter how small

the intensity of the incident light is.

o The liberation of electron is instantaneous.

o The KE of photoelectrons is proportional to the frequency of the incident

light.

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➢ The Photoelectric Effect (Continuation) o The minimum energy required to cause the emission of an electron is called

the work function. (which is dependent on the material from which the

electron is being ejected).

o If the energy of the photon exceeds the energy required to overcome the

electrostatic forces holding the electrons in place, the excess energy will

appear as KE of the now emitted electron.

o The energy of an emitted electron is given by:

o There is one photoelectron produced per photon absorbed. Unless the

photon energy is high enough it will make no difference how intense the

beam becomes; electrons will not be emitted.

o Thus, emission is independent of intensity.

Note: Intensity refers to the number of photons.

Light of higher intensity = more photons

Where:

▪ KE – Kinetic energy of the electron

▪ hf – energy of the photon

▪ 𝜙 – work function of the electron

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• Light & Special Relativity ➢ Frame of reference: a coordinate system used to measure velocity & observe.

➢ Inertial frame of reference: one which is stationary or moving at a constant

velocity. (non-accelerating)

➢ Non-inertial frame of reference: one which is accelerating.

➢ Galilean Experiment o A cannonball is dropped from the top of the mast of a stationary ship.

o An observer on the ship observed the cannonball landed next to the base of

the mast.

o The experiment was then repeated on a ship moving at a constant speed.

o The same result was obtained by observer on the ship.

➢ Galilean Relativity o All inertial frames of references are equivalent.

o The laws of motion are the same in all inertial frame of reference.

o The results of Galileo’s experiment implied that it is impossible to detect the

state of motion of an inertial frame of reference by carrying out any

mechanical experiment within that inertial frame of reference.

➢ Relative Velocity o Relative Velocity is the velocity of an object from one frame of reference.

▪ A car travelling at 60kmhr-1,

is travelling at 60kmhr-1

relative to the ground.

➢ Light in Inertial Frames o According to Galilean principle of relativity, all velocities are relative, all

inertial frames of reference are equivalent and there is no absolute frame of

reference.

o Maxwell’s equations however indicated that EM waves travelled at a

constant speed in a particular medium, irrespective frame of reference of

the observer or the source of light.

o To resolve this issue the concept of Aether/Luminferous Aether was

proposed.

o Initially physicists thought that EM waves (such as light) travelled through a

medium called Aether. Aether was thought to be fixed in space and acted as

a stationary absolute frame of reference. All objects in space are moving

relative to this absolute frame of reference.

o The speed of light is thought to be measured relative to the Aether.

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➢ Aether o Aether: A hypothetical medium that is responsible for propagation of light

waves. It acts as a stationary absolute frame of reference.

o Properties:

▪ Permeate all of space and yet was completely permeable to all

objects.

▪ Stationary, low density, and perfectly transparent.

▪ Had great elasticity to support the propagation of light waves.

▪ If c was 3 x 108 ms-1 in this aether, it should be faster or slower in a

frame of reference moving through aether such as the Earth.

▪ In 1887, Michelson & Morley performed an experiment to test this

idea.

➢ Michelson & Morley Experiment o This experiment is designed to measure the relative motion of Earth

through Aether using a device called interferometer.

o A single beam of light was allowed to split into two identical components,

moving perpendicularly to each other.

o When the two identical beams recombined,

the interference pattern was observed.

o The whole apparatus was then rotated

through 90o.

o Because the beams are perpendicular, they cannot both be parallel to

Earth’s motion through the aether. Therefore, if aether never existed, the

speed of light relative to the Aether, would be different along the two

perpendicular paths.

o As we rotated the interferometer, the relative speed of light along these

two paths would keep changing, resulting in the changing interference

patterns.

o The interference pattern was expected to appear the same again after 90o

rotation since the two paths of light now resumed the original position only

switching roles.

o Despite conducting the experiments at various times and locations, and the

use of extremely sensitive interferometer, there was no change in the

interference pattern whatsoever. This was known as the ‘null’ result.

o Implication: The ‘null’ result of M-M experiment questioned the existence

of aether and provided the experimental evidence to support Einstein’s

special theory of relativity.

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➢ Einstein’s Thought Experiment o Einstein was able to solve the discrepancy between Galileo’s relativity &

Maxwell’s EM equations.

o Einstein used thought experiments to reach conclusions, this helped him

visualise complex problems that were not experimentally possible to test

during his time.

o Thought experiments are hypothesis, theories, principles that are proposed

to consider consequences. They generally involve a situation, multiple

scenarios and a conclusion.

o The thought experiment involves a person sitting in a moving train travelling

at the speed of light. When the person holds a mirror in front of them, do

they observe their reflection?

o There are two cases:

▪ Case 1: If they did not see their reflection then the Galilean

principle of relativity was violated.

▪ Case 2: If they saw their reflection then according to Galilean

/Newtonian relativity, the stationary observer on the ground would

see the speed of light to be twice its usual speed.

o Einstein believed:

▪ The passenger must observe their reflection, so as not to violate

general relativity. Einstein believed that Galilean’s relativity applied

to not only mechanical systems by also to EMR.

▪ Both the stationary observer and the observer in the moving train

measured the speed of light to be c. (to satisfy Maxwell’s equations

and not to contradict the first statement).

o Given this:

▪ The two observers had to disagree on something else.

▪ c = d/t

o Einstein coined this as the Special theory of relativity, and it has two

postulates.

▪ The Laws of Physics apply equally in all inertial frames of reference.

▪ The speed of light is constant as measured in any inertial frame of

reference.

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➢ Consequences of Special Relativity

o Relativity of Time

▪ The time taken for an event to occur within its own rest frame is

called the proper time (to). Measurements of this time made from

any other inertial frame of reference (tv) in relative motion to the

first, are always greater.

▪ Rest Frame: a frame of reference within which an event is

occurring. An observer outside a moving FOR will observe anything

within that FOR to be occurring slower.

▪ Twin Paradox

❖ There are two twins. One is placed on a fast-moving

spaceship while the other remains on Earth. After a long

period of time, the spaceship returns and the ages of the

two twins is compared.

❖ From the perspective of the Earth twin, tv > to. The age of

the Space twin is 21 yrs. The age of the Earth twin is 50 yrs.

❖ We must recognise who is an inertial frame for the while

time. The Earthbound twin remains in a nearly inertial frame

at all times. The spaceship is not an inertial frame at all

times. The prediction in the relative ages of the twins made

by the space twin is not justified.

Where:

▪ to – proper time

▪ Lo – proper length

(measured by an observer within his/her

own rest frame)

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o Relativity of Length

▪ Forward Journey

▪ Backward Journey

▪ Total Time

▪ Length

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❖ Proper Length (Lo): the length of an object measured within

its own rest frame.

❖ Lv: length made from any other inertial frame of reference in

relative motion parallel to that length, always less.

➢ Relativistic Momentum o Consider the scenario in which a rocket ship is accelerated from rest to a

relativistic speed.

o The rockets change in momentum:

𝛥𝑝 = 𝑚𝛥𝑣 = 𝐹𝑡

o From the rocket’s perspective (po):

𝐹𝑡0 = 𝑚0𝛥𝑣

o To the stationary observer on Earth, the time is dilated:

𝑡𝑣 =𝑡0

√1 − (𝑣𝑐)

2

o Replacing to,

➢ Particle Accelerators o Measurements inside particle accelerators show that velocities of particles

with mass don’t exceed the speed of light, no matter how much energy is

provided.

Proton Accelerator Energy (eV) Vc-%

1 Cockcroft – Walton 7.5 x 105 4

2 Linac 4 x 108 71

3 Booster 8 x 109 99.4

4 Main Injector 1.2 x 1011 99.997

5 Tevatron 1 x 1012 99.99995

Note: Because of the relativity of momentum it becomes increasingly

difficult to accelerate an object as it approaches the speed of light.

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➢ Mass – Energy Equivalence o As an object approaches the speed of light, the work/energy supplied will

no longer act to accelerate the object, but instead is converted to inertial

mass. (E = mc2)

o Einstein was able to show the energy associated with an object’s mass.

o Thus, the total energy of an object consists of two parts: the kinetic energy

& the rest energy which is related to its mass.

o Analogy: suppose we supply 100kJ of energy to accelerate a particle, 80kJ is

converted into its kinetic energy & 20kJ is converted into mass then:

o We can see that when the particle is stationary (Ek = 0), its energy is mc2.

This implies that a particle at rest has energy & related to its mass.

➢ Relativistic Mass o From relativistic momentum,

o The mass of an object measured within its own rest frame is called the rest

mass (mo). Measurements of this mass made from any other inertial frame

of reference (mv) in relative motion to the first, are always greater (mass

dilation).

➢ Antiparticle Annihilation o It has been discovered that for every subatomic particle, there exists a

corresponding anti-particle.

o For example, there exists an anti-electron (positron) which has the exact

same mass but opposite charge.

o There are also anti-protons,

anti-neutrons & etc.

o When any particle & anti-

particle meet, they mutually

annihilate each other. All the

mass is converted into energy

via, E = mc2.

Where:

▪ E – energy (J or eV)

▪ m – mass (kg)

▪ c – speed of light

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➢ Nuclear Reactions o Nuclear Fusion: the process in which two or more small nuclei combine to

form a larger nucleus with the release of a large amount of energy (more

energy released compared to nuclear fission).

o Nuclear Fission: the process in which a heavy unstable nucleus splits to

form more stable, lighter nuclei. It also emits neutrons & energy.

o Both reactions, fusion & fission release vast amounts of energy which can

be calculated using Einstein’s equation & mass-defect.

o Mass Defect: the difference between the mass of the constituent nucleons

& the mass of the nucleus.

o Einstein’s equations, E = mc2, does not apply to nuclear reactions, but also

to other mass-energy reactions such as the combustion of fuel. In many

such situations, the mass defect/difference is so small that it is usually

unnoticed.

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Spectrum A range of values of a quantity or set of related quantities.

Visible Spectrum A range of colours emerging from dispersion of white light.

Electromagnetic Spectrum A range of frequencies/wavelengths of electromagnetic radiation.

Continuous Spectrum shows all values on spectrum

Line Spectrum shows limited values on spectrum (digital clock analogy).

Spectrology the analysis of spectra produced by a radiant object to obtain information such as the chemical composition, temperature & other features of that object.

Emission Spectrum consists of specific wavelengths observed as bright lines against a dark background.

Absorption Spectrum consists of dark lines as seen against a rainbow of colours as a background.

Bandwidth The distance between successive bright (or dark) fringes.

Polarisation When the oscillations of electromagnetic waves are restricted to one dimension.

Electron-Volt (eV) An alternative unit for measuring small amounts of energy.

Photoelectric Effect The liberation of electrons from the surface of a conductor when light strikes the surface.

Frame of Reference A coordinate system used to measure velocity & observe.

Inertial Frame of Reference

One which is stationary or moving at a constant velocity. (non-accelerating)

Non-Inertial Frame of Reference

One which is accelerating.

Aether A hypothetical medium that is responsible for propagation of light waves. It acts as a stationary absolute frame of reference.

Rest Frame A frame of reference within which an event is occurring.

Proper Length (Lo) The length of an object measured within its own rest frame.

Nuclear Fusion The process in which two or more small nuclei combine to form a larger nucleus with the release of a large amount of energy.

Nuclear Fission The process in which a heavy unstable nucleus splits to form more stable, lighter nuclei. It also emits neutrons & energy.

Mass Defect The difference between the mass of the constituent nucleons & the mass of the nucleus.

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Formulas

Electromagnetic Spectrum

The Wave Model of Light

The Quantum Model of Light

Light & Special Relativity

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The Universe of Atoms

Notes

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Contents • Chemistry Revision ....................................................................................................................... 89

➢ Isotopes .................................................................................................................................... 89

• Properties of the Nucleus ............................................................................................................ 89

➢ Introduction .............................................................................................................................. 89

➢ The Strong Nuclear Force ......................................................................................................... 89

➢ Zone of Nuclear Stability .......................................................................................................... 90

➢ Radioactivity ............................................................................................................................. 90

➢ Properties of Radiation ............................................................................................................ 90

➢ Radioisotopes ........................................................................................................................... 91

➢ Nuclear Transmutations (Radioactive Decay) ......................................................................... 92

➢ Half-life in Radioactive Decay .................................................................................................. 92

➢ Radioactive Decay Equation .................................................................................................... 93

➢ Nuclear Fission.......................................................................................................................... 93

o Fermi’s Observation ............................................................................................................. 94

➢ Chain Reactions ........................................................................................................................ 94

➢ Controlled Chain Reactions ...................................................................................................... 95

➢ Uncontrolled Chain Reactions ................................................................................................. 95

➢ Nuclear Fusion .......................................................................................................................... 95

➢ Mass Defect & Binding Energy ................................................................................................. 95

➢ Binding Energy Per Nucleon ..................................................................................................... 96

➢ Energy in Nuclear Reactions .................................................................................................... 96

• The Origins of the Elements ......................................................................................................... 97

➢ Introduction .............................................................................................................................. 97

➢ Theories of the Universe .......................................................................................................... 97

➢ The Expanding Universe ........................................................................................................... 98

➢ Hubble’s Law ............................................................................................................................ 98

➢ The Big Bang ............................................................................................................................. 99

➢ The Singularity .......................................................................................................................... 99

➢ Inflation & Energy Dominant Period ....................................................................................... 99

➢ Recombination ....................................................................................................................... 100

➢ Radiation Release ................................................................................................................... 100

➢ Accretion ................................................................................................................................. 100

➢ Cosmic Background Radiation ............................................................................................... 100

➢ Stars ........................................................................................................................................ 101

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➢ Lifestyle of a Star .................................................................................................................... 101

➢ Star Surface Temperature ...................................................................................................... 103

➢ Star Luminosity ....................................................................................................................... 104

➢ Hertzsprung – Russel Diagram ............................................................................................... 104

➢ Mass Energy Equivalence ....................................................................................................... 105

➢ Nuclear Fusion ........................................................................................................................ 105

➢ Other Measurements of Stars................................................................................................ 106

• The Structure of the Atom ......................................................................................................... 108

➢ Introduction ............................................................................................................................ 108

➢ Dalton’s Atomic Model .......................................................................................................... 108

➢ Cathode Rays & The Electron ................................................................................................. 108

➢ J.J. Thomson’s Experiment ..................................................................................................... 109

➢ The Oil Drop Experiment ........................................................................................................ 110

➢ Thomson Atomic Model ......................................................................................................... 111

➢ Rutherford Atomic Model ...................................................................................................... 111

➢ The Proton .............................................................................................................................. 113

➢ The Neutron ............................................................................................................................ 113

• Quantum Mechanical Nature of the Atom ................................................................................ 115

➢ Introduction ............................................................................................................................ 115

➢ Balmer Series & Rydberg’s Equation ..................................................................................... 115

➢ Bohr’s Atomic Model .............................................................................................................. 116

➢ Limitations of Bohr’s Model ................................................................................................... 117

➢ Wave-Particle Duality of Light ............................................................................................... 117

➢ De Broglie’s Matter Wave ...................................................................................................... 117

➢ Davisson & Germer’s Experiment .......................................................................................... 118

➢ Standing Waves ...................................................................................................................... 119

➢ Impact of De Broglie’s Matter Wave ..................................................................................... 119

➢ The Uncertain Nature of Matter ............................................................................................ 120

➢ Schrodinger’s Atomic Model .................................................................................................. 120

➢ Heisenberg Uncertainty Principle .......................................................................................... 121

➢ Schrodinger’s Cat Though Experiment .................................................................................. 121

• Deep Inside the Atom ................................................................................................................ 122

➢ Introduction ............................................................................................................................ 122

➢ The Particle Zoo ...................................................................................................................... 122

➢ The Standard Model ............................................................................................................... 123

➢ Quarks ..................................................................................................................................... 123

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➢ Leptons ................................................................................................................................... 124

➢ Gauge Bosons ......................................................................................................................... 124

➢ Evidence for the Standard Model .......................................................................................... 125

➢ Higgs Bosons ........................................................................................................................... 125

Definitions .......................................................................................................................................... 126

Formulas ............................................................................................................................................. 127

Properties of the Nucleus ............................................................................................................... 127

The Origins of the Elements ........................................................................................................... 127

The Structure of the Atom .............................................................................................................. 127

Quantum Mechanical Nature of the Atom .................................................................................... 127

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• Chemistry Revision

➢ Isotopes o They are atoms of the same elements which have the same number of

protons in their atomic nuclei but differing numbers of neutrons.

o A certain isotope can be expressed by its atomic symbols.

o Nucleons: anything inside the nucleus

• Properties of the Nucleus

➢ Introduction o Einstein’s discovery of energy mass equivalence led to humans harnessing

large amounts of energy with an atom.

o Combustion reactions were used before we were able to harness energy.

These were associated with fossil fuels. Nuclear reactions allow us to

harness the abundant naturally occurring radioactive substances.

❖ Industrial & medicinal applications

➢ The Strong Nuclear Force o Properties:

❖ Experiences by all Nucleons (Protons & Neutrons).

❖ Attractive over short ranges (≈ 1 femtometre – x 10-15m)

❖ Strong enough to overcome

electrostatic repulsion forces.

❖ Becomes repulsive at even

closer distances.

4.2 fm

1.7 fm

Deuteron

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➢ Zone of Nuclear Stability o This graph shows the criteria necessary to create

a stable nucleus.

o The two factors that contribute to the nucleus’s

stability is:

❖ The Neutron – Proton Ratio

❖ The total number of nucleus

➢ Radioactivity o Radioactive substances are referred to as unstable. An unstable substance

will emit radiation until they become stable.

o Therefore, their nucleus is constantly changing due to these emissions.

o Radioactivity: spontaneous release of energy (radiation or particles) from

the nucleus.

o The three main types of radioactive emissions/decays are:

❖ Alpha

❖ Beta

❖ Gamma

➢ Properties of Radiation o Alpha

❖ Helium nucleus consisting of two protons & neutrons.

❖ Often emitted to reduce mass of a radioactive substance.

o Beta

❖ Fast moving electrons.

❖ Often emitted to increase the stability of a radioactive substance.

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o Gamma

❖ Uncharged high energy & frequency electromagnetic radiation.

❖ Often emitted to reduce the energy of a radioactive substance.

Type Nature Charge Ionising Ability

Penetrating Power

Deflection from electric/magnetic field

α Helium nucleus (4

2He) 2+ High Low Small

β Electron

(0-1e)

1- Medium Medium Great

γ EM

Radiation (γ)

0 Low High None

➢ Radioisotopes o Radioisotope: An unstable isotope will emit particles &/or radiation until it

becomes stable.

o Every atom of the same element has the same number of protons. However,

they can have different amounts of neutrons (therefore a different overall

mass).

o Different isotopes have different configurations. Some isotopes (like Carbon-

12) are stable whilst others (Carbon-14) are unstable/radioactive.

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➢ Nuclear Transmutations (Radioactive Decay) o Nuclear Transmutation: the conversion of one chemical element to another

through the emission of alpha or beta particles from the nucleus.

o Nuclear equations are used to demonstrate nuclear transmutations.

o There are other methods in which a nuclear can transmutate. These are

Fission & Fusion.

➢ Half-life in Radioactive Decay o Half-life (t1/2): time taken for half the radioactive substance to decay.

o Different radioactive isotopes decay at different rates.

o Examples:

❖ Uranium-238: 4.5 x 109 yrs.

❖ Polonium-218: 1.4 x 10-4 secs

o Implications of varying half-lives:

❖ Uranium’s long half-life makes it a dangerous isotope as it remains

for a long period of time, whilst Polonium does not linger around

long enough for it to be a useful isotope (for medical & industrial

uses).

❖ Ideally, for diagnosis imaging, isotopes with half-lives between

minutes & hrs are selected to minimise exposure whilst existing for

long enough to serve its purpose.

❖ Examples: Tc-99m to investigate bone function, bone disease & 1-

123 to assess thyroid function.

Daughter

Isotope

Parent

Isotope

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➢ Radioactive Decay Equation o The rate of decay of a certain number of atoms is proportional to the

number of atoms present:

➢ Nuclear Fission o Nuclear Fission: the process in which a heavy unstable nucleus splits to form

more stable, lighter nuclei. It also emits neutrons & energy.

Where:

▪ No – initial amount

▪ N – amount after ‘t’ time

▪ λ – decay constant (s-1)

▪ t12⁄ – half-life

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o Fermi’s Observation ❖ Enrico Fermi discovered that new radioactive elements could be

produced when a target nucleus is struck by neutrons.

❖ The advantages of neutron bombardment are:

▪ Neutrons (due to zero change) are not deflected by the

nucleus nor affected (repelled/attracted) by the electron

clouds → worked better than charged particles.

▪ Slow neutrons can enter & interact with even the most

massive, highly charged nucleus.

❖ Fermi was successful between 1934-1938.

❖ However, when applied to Uranium, Fermi & his team found that

rather than creating a single heavier isotope, several different

isotopes were produced with different measurable half-lives.

❖ This was Fermi’s first observation of fission, although he couldn’t

comprehend the nature of the process.

❖ In 1939, two Australian Physicists, Lise Meitner & Otto Frisch

explained the process with bombardment. The Uranium nucleus

broke down into two nuclei of roughly equal size. They named this

process ‘nuclear fission’.

❖ Three neutrons are produced but only one neutron is required for

nucleus fission so there was a possibility of a chain reaction.

➢ Chain Reactions o Nuclear Chain Reactions: a series of nucleus fissions imitated by a neutron

produced in a preceding fission.

o Process of Nuclear Fission:

❖ A neutron strikes the nucleus of a

heavy & unstable isotope.

❖ The nucleus becomes unstable.

❖ Causing it to vibrate then split.

o Products of Nuclear Fission:

❖ Two or three neutrons.

❖ Two smaller, lighter atoms.

❖ Heat energy.

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➢ Controlled Chain Reactions o Controlled Chain Reaction: a self-propagating nuclear reaction in which only

one neutron released by the splitting of a nucleus is allowed to hit another

nucleus to cause further fission resulting in the steady release of energy.

o The reaction will continue at a constant rate.

o Occurs in nuclear reactors & powerplants.

➢ Uncontrolled Chain Reactions o Uncontrolled Chain Reaction: a self-propagating nuclear reaction in which

more than one neutron released by the splitting of a nucleus is allows to hit

another nucleus to cause further fission resulting in energy being released

exponentially.

o The reaction will continue at an increasing rate.

o Occurs in nuclear bombs or atomic bombs.

➢ Nuclear Fusion o Nuclear Fusion: the process in which two or more small nuclei combine to

form a larger nucleus with the release of a large amount of energy (more


➢ Mass Defect & Binding Energy o Mass Defect: the difference between the mass of the constituent nucleons

& the mass of the nucleus.

o In order to separate the atoms into its

constituent nucleons (i.e. break atoms apart),

work needs to be done which means energy is

required.

o Binding Energy: the energy needed is equal to

the energy that holds the atom together in

the first place.

o The mass of the nucleus is less than the mass

of the nucleons (which make up the nucleus). This mass defect is related to

binding energy.

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o This is explained by:

❖ Einstein’s relationship E = mc2. Mass & energy are equivalent.

❖ The Law of Conservation of Energy: cannot be created nor

destroyed. It can only be converted from one form into another.

o Hence, the mass defect (missing mass) accounts for the energy that is

gained (i.e. binding energy).

➢ Binding Energy Per Nucleon o Binding Energy of a Nucleus: the energy required to completely separate a

nucleus into nucleons. Therefore, the binding energy is a measure of the

stability of a nucleus.

o Iron is the most stable of all nuclei because it has the greatest binding

energy per nucleon.

o Small nuclei, to the left, can make themselves more stable by clumping

together into bigger nuclei (i.e. fusion).

o Large nuclei, to the right of iron, can make themselves more stable by

splitting into smaller ones (i.e. fission).

➢ Energy in Nuclear Reactions o Different nuclei contain different numbers of protons & neutrons which

means they possess different amounts of binding energy.

o Hence, during a nuclear reaction when an atom changes from one to

another, energy can be released.

❖ In nuclear fission, large unstable nuclei are split into smaller, more

stable nuclei & binding energy is released.

❖ In nuclear fusion, small unstable nuclei are joined together to form

larger, more stable atoms & binding energy is released.

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• The Origins of the Elements

➢ Introduction o The universe includes all of space & time & its contents (including planets,

stars, galaxies & other forms of space & matter).

o Humans have always questioned the finite or infinite nature of the universe,

what came before the universe, what will come after the universe & other

questions we still don’t understand.

➢ Theories of the Universe o Many astronomers have wondered about the origin of the universe. Many

attempts have been made throughout hundreds of years to try and explain

how our universe came to begin & what it looks like now.

o Early on, the universe was thought to consist of our solar system & the stars.

Today it is understood that our solar system is just one of many in our

galaxy, & that our galaxy (The Milky Way) is just one of many.

o The size & scale of the universe is so large that it is nearly incomprehensible.

o If the Earth was represented by a golf ball, the Sun would be length of a

sedan, & Star Arcturus would be the length of a football field.

o 16th Century BC: Mesopotamian culture believed the Earth to be a flat disc

sitting on a ‘cosmic ocean’.

o 4th Century BC: Aristotle proposed a universe with the Earth at the centre

(Geocentric model).

o 1543: Nicolaus Copernicus publishes his Sun centred model of the universe.

It would take over 100 years for evidence to be gathered proving his idea.

o 1584: Giordano Bruno proposed a model where the Copernican solar system

is not the centre of the universe, arguing it is one of many star systems.

o 1687: Sir Isaac Newton creates laws to explain gravity, describing how

objects orbit & move throughout the universe. He also proposes that the

universe is static in size.

o 1915: Albert Einstein published his theories on relativity, linking space &

time in the universe. He also believes that the universe is static in size.

o 1922: Alexander Friedmann proposes an expanding model of the universe.

o 1923: Edwin Hubble discovered an object that sits outside of our Milky Way

galaxy. This implied that the Milky Way is one of many galaxies.

o 1929: Edwin Hubble builds on his discovery, using the Doppler Effect to

show that the universe is expanding.

o 1948: George Gamow builds on Hubble’s experiment, proposing the ideas

which would lead to the Bing Bang Theory.

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➢ The Expanding Universe o In 1924, Hubble was studying a group of stars in a nebula (a gas from which

a star is formed) called Andromeda.

o He calculated the distance of Andromeda to be 800,000 light years away.

o This distance greatly exceeded the distance of any known star at the time as

well the length of the Milky Way (approx. 100,000 light years).

o Hubble’s work suggested evidence that the stars in Andromeda existed in a

separate galaxy.

o The implication if that there must exist other galaxies in the universe apart

from our Milky Way.

o Five years later, Hubble used spectrology from Andromeda Nebula (now

known as the Andromeda galaxy) as evidence for the expansion of the

universe.

o Recall that the Doppler Effect occurs due to the relative motion of the

source of a wave & the observer. If the wave is

light, the Doppler Effect is explained as a

redshift of blueshift of spectral lines.

o Hubble observed that the spectral lines were

redshifted. This showed that Andromeda was

moving away from us. When he observed

other objects outside of our galaxy, he found a

similar result.

o This gave evidence to an expanding universe.

➢ Hubble’s Law o Hubble’s law describes the rate at which distant galaxies are moving away

from us.

o A parsec is a unit of distance in Astrophysics (study of astronomy related to

Physics). It is equal to 3.2616 light years (ly), 206265 Astronomic units (AU)

& 3.086 x 1016 metres (m).

o Hubble’s law implies that the universe is expanding at an increasing rate.

o Balloon Analogy:

❖ The surface of the balloon represents all of space.

❖ As air fills the balloon the surface of the balloon expands.

❖ If stickers on the balloon are used to represent the galaxies/star

systems, it can be seen that the stickers don’t get bigger.

❖ So, whilst the space between the galaxies expand, the actual objects

in space don’t get larger.

Where:

▪ v – velocity at which galaxy is moving away (kms-1)

▪ Ho – Hubble’s Constant (67kms-1Mpc-1)

▪ D – distance to a far-away galaxy (Megaparsecs)

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➢ The Big Bang o 1930: Physicists agreed that the universe is not static & keeps expanding.

o They concluded that the universe must have started out tiny from one point.

o 1940: George Gamow proposed the Big Band Theory.

➢ The Singularity o The universe is thought to have begun with a single point tiny point of

energy. It had extremely high temperature, density & pressure. Space,

matter, & time didn’t exist & neither did the laws of physics.

o The universe expanded from a single point, releasing immense amounts of

energy that cooled.

o The singularity occurred roughly 13.8 billion years ago. However, from zero

to approximately 10-43 seconds, there is little understanding about how the

universe looked to be.

➢ Inflation & Energy Dominant Period o From 10-36 seconds to 10-32 seconds, the universe underwent extremely

rapid exponential expansion, known as inflation.

o The universe expanded from that single point (growing 1026 time larger) to

the size of around 10 cm, releasing immense amounts of energy.

o At this point the universe is still over 10 billion degrees hot, this prevents

any particles from forming. As the universe continues to expand, it cools.

o As it cooled, energy was transformed into matter (discovered by Einstein).

o At 10-9 seconds, the universe stretched to a billion km in diameter. It is now

cool enough for fundamental particles to exist in a stable state.

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➢ Recombination o Initially, some energy was transformed into fundamental particles of matter:

❖ Electrons

❖ Quarks (building blocks of protons & neutrons)

o After a few minutes, quarks combined to form protons &

neutrons. This process is called recombination.

o Hundreds of thousands of years later, the universe cooled sufficiently for

electrons, protons & neutrons to combine & form atoms.

o The energy released was not uniform. Some areas had much higher density

of energy. Places with more energy had more energy to work with to make

protons and neutrons to build, which would eventually create planets.

➢ Radiation Release o At this point, a large amount of radiation released from the Big Bang was

freed & cooled down. This radiation still exists today.

o This is called Cosmic Background Radiation & exists as

a microwave signature.

o It was predicted by Gamow in 1948 & later discovered

in 1964.

➢ Accretion o As the universe was expanding & cooling, particles lost kinetic energy &

began to attract each other through gravity. This formed regions of high

mass & density.

o This region then began to attract other nearby materials & gain mass. This

process is called accretion.

o Due to accretion, matter in the universe formed discrete gas clouds known

as protogalaxies.

o As further accretion occurred, galaxies were formed.

o Accretion continued to happen inside galaxies to form stars (i.e. our Sun is

formed in the Milky Way).

o To this day, our universe continues to expand (13.8 billion years later).

➢ Cosmic Background Radiation o In the early state of the universe, the temperature was so high that atoms

didn’t exist.

o The universe consisted of radiation & elementary particles. The radiation

was trapped, travelling short distances before being scattered by electrons.

As a result, the universe was opaque.

o As the universe cooled to 3000K; 380,000 years later, atoms were able to be

formed. Free electrons were no longer present to scatter the radiation.

o The universe become transparent as the radiation was now able to disperse

freely in the universe.

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➢ Stars o Star: a type of astronomical object consisting of a luminous spheroid of

plasma (dense, ionised gas) held together by its own gravity.

o It wasn’t until the time of Galileo & Newton, when scientists began to

understand that stars were sun-like objects.

o Astronomers now understand how stars form, the reactions that take place

in the star & the significant role that stars play in forming the universe.

o Except for those created by the Big Bang, every element was synthesised

inside a star.

o Different methods (including spectrology) yield information into the nature

of stars (even if they are millions of light years away from Earth).

➢ Lifestyle of a Star

o All stars begin as large gas clouds (mainly hydrogen) called nebulas. A single

nebula might be the birthplace of thousands of stars.

o If the gas is sufficiently cooled, the particles in the cloud will begin to clump

due to gravitational acceleration.

o This process starts slowly but speeds up as the cloud becomes denser.

o The cloud now consists of 2 parts: A rapidly contracting core & the slower

contracting surroundings.

o As the cloud continues to contract, its

temperature increases. The GPE (which

caused the cloud to contract) changes into

thermal energy.

o This heat creates an outwards pressure that

works against gravity, but only slightly.

o As the core gets denser & hotter it

stabilises. At this point, fusions haven’t

occurred yet & the body is called a

protostar.

o This process occurs over one million years.

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o Eventually the core may reach a sufficient temperature to trigger the

nuclear fusion of hydrogen to helium which settles to the centre (helium is

denser). The hydrogen moves to the shell around the helium core.

o Once nuclear fusion of hydrogen begins, the star is officially a main

sequence star.

o Hydrostatic equilibrium helps a main

sequence star remain a stable size.

o If the star is at equilibrium the inwards

pressure (due to gravity) is balanced by the

outwards pressure (due to nuclear reactions).

o The hydrogen supply in the core will dwindle

over time & the core will begin to collapse

under gravity.

o If the mass of the star > 0.3 Mo, the gravitational collapse increases the

temperature & helium fusion is triggered at the core. This forms a red giant.

o If the mass of the star < 0.3 Mo, the temperature increase isn’t sufficient to

activate fusion of helium. The star will become a white dwarf.

o Red Giant Star: characterised by the nuclear fusion of helium at the core.

The remaining hydrogen fusion occurs in the outer shell.

o Eventually red-giants will begin to run out of helium, & the star contracts

due to gravitational forces (it’s no longer at hydrostatic equilibrium).

o If the red giant is large enough (super red giant), the core will heat up to a

sufficient temperature to create heavier elements from carbon to iron.

o Once a star is no longer able to fuse elements, it will begin to die. The path it

takes again will dependent on its size.

o If the mass of the red giant is < 0.5 Mo, it will run out of helium & collapse. A

nova will occur where the star will release energy, gently shed its outer

layers of gas.

o This is called a planetary nebula. Whatever energy isn’t shed forms a white

dwarf.

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o If the mass of a red giant > 0.5 Mo, a supernova explosion will occur,

releasing a giant amount of energy.

o The mass of the core will then determine its corpse:

❖ m < 1.4 Mo: the core becomes a white dwarf which eventually a

black dwarf.

❖ 1.4 Mo < m < 3 Mo: gravity will be sufficient to collapse electrons

into protons forming a neutron star.

❖ m > 3 Mo: the neutrons formed by the collapse of protons &

electrons will collapse further to form a black hole.

o White Dwarf: a collapsed star with no more nuclear fusion reactions as a

source of energy.

o It will gradually radiate its energy & cool down.

o As it continues to

radiate its energy

over time, its

temperature will

decrease.

o When it no longer

emits any heat or

light, it is then

known as a black

dwarf.

➢ Star Surface Temperature o A star’s surface temperature is linked to the radiation produced by the star

in a process called black body radiation.

o Recall that the peak wavelength of a heated object corresponds to the

surface temperature & the colour of the star.

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➢ Star Luminosity o Luminosity: the energy radiated by an object

per second.

o The luminosity of our sun is 3.83 x 1026 W (Lo).

o Brightness: the energy received per square

metre per second.

➢ Hertzsprung – Russel Diagram o A Hertzsprung – Russel diagram is a graph of a star’s luminosity against its

colour/surface temperature.

o 1920: Ejnar Hertzsprung & Henry Russel independently discovered that

plotting the luminosity of stars against their surface temperature resulted in

different groupings of stars with different characteristics.

o This diagram allows astronomers to classify stars & understand its evolution.

❖ Main Sequence Stars:

▪ It becomes more luminous & massive when moving from

bottom right to top left. The source of energy is the nuclear

fusion of hydrogen at the core of the stars.

❖ Red Giants:

▪ Extraordinarily large in size.

▪ Nuclear fusion of helium occurs at the core of the stars.

❖ White Dwarfs:

▪ No nuclear fusion.

▪ Collapsed star corpses.

Where:

▪ Bo – Brightness (Js-1m-2) or (Wm-2)

▪ Lo – Luminosity (Js-1) or (W)

▪ r – distance (m)

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➢ Mass Energy Equivalence o Originally astronomers hypothesised that chemical reactions inside the sun,

generating heat & light which then travelled to Earth.

o However, based on the mass & energy output of the Sun this was ruled out.

o In order for chemical reactions (between the Sun’s atoms) to be the source

of the radiation, the Sun would need a hundred million times more atoms.

o Chemical reactions could therefore not be the driving force behind a star’s

energy output.

o Einstein is the first scientists to theorise the relationship between mass &

energy, which would help explain how the Sun is able to generate massive

amounts of energy.

o Einstein’s famous equation was published in 1905 & identifies that anything

with mass has an equivalent amount of energy.

o From examples, we can see that due to the c2 factor, even a small amount of

mass has a large associated amount of energy.

o Mass can be converted to energy through chemical reactions, nuclear

reactions & other forms of energy transfer.

o It was believed that Stars (like the Sun) utilised this property to produce the

EMR it released.

➢ Nuclear Fusion o Nuclear fusion is the combining of two (or more) small nuclei to form a

larger nucleus which released a large amount of energy. This occurs within

stars many times every second.

o Nuclear reaction equations allow us to demonstrate the requirements &

products of a nuclear fusion.

o In main sequence stars helium is formed from

hydrogen in one of two ways:

❖ The proton-proton chain reaction (PP).

❖ The carbon-nitrogen-oxygen cycle reaction

(CNO).

o The PP chain reaction occurs mainly in smaller main sequence stars (stars

less than 1.3Mo). It changes 4H atoms to 1He atom.

o This process occurs in three main steps:

Where:

▪ E – energy (J)

▪ m – mass (kg)

▪ c – speed of light (3 x 108ms-1)

Note: Ev = 1.6 x 10-19

MeV = Mega electron-volt

= (electron-volt) x 106

= 1.6 x 10-13 J

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o The CNO cycle involves six steps:

➢ Other Measurements of Stars

o Parallax is a way of measuring the distances between objects in space.

o It relies on a phenomenon known as Parallax Shift.

o Parallax Shift: the difference in the position of an object (against a

background) due to different viewing angles.

Note: In general beta decay increases the atomic

number by one & doesn’t affect the mass number.

Net Nuclear Reaction:

Note: Distance away from

object = less parallax shift

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o Early on, astronomers couldn’t detect the parallax shift of stars in the sky

(this was one of the key arguments used to justify that the Earth was

stationary).

o When better telescopes were developed in the 19th century, it was seeing

that the background stars moved very slightly over the period of one year.

o Astronomers concluded that the Earth was rotating around the Sun (& this

creates the parallax).

o The parallax of an object can be used to approximate the distance to an

object using the formula.

o The chemical composition of a star is discoverable using spectroscopy.

o Recall that an absorption spectrum is created by light passing through cool

gases.

o Spectra lines are created by

Earth’s atmosphere absorbing

certain bands of light.

o Spectra lines are also created by

the elements in the cooler

regions of the stars (the outer

layers). Studying these spectral

lines will give us clues as to the

chemical composition of the

stars.

o In summary, for the elements

found in the universe today:

❖ The lightest elements (hydrogen, helium, lithium, beryllium) began

to form during the early stages of the Big Bang.

❖ Nuclear fusion inside main sequence stars created more amounts of

helium, whilst the reactions inside the larger giant stars created

medium sized elements (The elements of carbon up to iron). White

dwarves did not undergo nuclear fusion.

❖ Supernova released enough energy to generate heavy elements all

the way up to Uranium.

Star is close enough where

parallax is observable.

Star is too far to produce

any practical parallax shift.

Where:

▪ d – distance (AU)

▪ p – angle (arc seconds) Note: 1 degree = 3600 arc seconds

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• The Structure of the Atom

➢ Introduction o Whilst physicists are interested by the very large objects in

the universe, there is also a need to understand the

smallest building blocks of matter as well.

o For most purposes, we can say that an atom is the

smallest unit of ordinary matter that we can use to

explain the properties of chemical elements.

o However, the atom itself is made up of smaller

elementary particles; the proton, neutron & electron. These

elementary particles were not discovered at the same time.

➢ Dalton’s Atomic Model o In 1803, an English chemist, John Dalton, proposed an atomic theory.

o The basic postulates of Dalton’s atomic theory were as follow:

❖ Matter is composed of neutral, structure less & indivisible atoms

❖ The atoms of one element are identical

❖ Atoms of different elements have different atomic masses.

❖ Atoms are neither created nor destroyed in chemical reactions.

❖ Chemical reactions consist of combining, separating or rearranging

atoms in simple whole number ratio.

➢ Cathode Rays & The Electron o By the 1850’s much was known about electricity & the conductor/insulator

properties of materials. However, the fundamental nature of electricity was

not yet understood.

o During this period of time, physicists we studying a device known as a

cathode ray tube (or discharge tube).

o A cathode ray tube is a vacuum glass tube which has a very low pressure.

Metal plates (called an anode & a cathode)

are placed inside the tube & connected to a

high voltage supply.

o When the voltage supply is turned on,

physicist saw that a current was flowing &

that the glass glowed on the anode side of

the glass. (Diagram is a vacuum tube)

o When a fluorescent screen was placed inside the tube a green stream/ray

could be clearly observed. This was called a cathode ray.

o Scientist began experimenting with these tubes, demonstrating a number of

important facts related to the properties of cathode rays.

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➢ J.J. Thomson’s Experiment o Joseph John Thomson was the first physicist to demonstrate that an electric

field could deflect the cathode rays. To do this, he built an almost perfect

vacuum tube & applied an electric field.

o He successfully showed that the ray deflected towards the positive plate.

This proved that the particles made up the cathode rays had a negative

charge.

o By this time, the name electron was beginning to be used to describe the

particles that made up the cathode rays.

o J.J. Thomson then set out the measure the mass of these particles.

❖ The sharp shadow produced

indicated a lack of diffraction

which suggested particle-like

behaviour.

❖ Q = mv -> must be a particle

❖ Initially there was no deflection -> waves

❖ Vacuum tubes were improved & a

deflection was observed

❖ Recharged particle

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o Thomson’s experiment involved two stages:

1. By varying a magnetic & electric field, the forces would cancel out &

leave the cathode ray un-deflected. This allowed for the velocity of

the charge to be calculated.

2. The electric field was then removed such that the cathode ray was

deflected. The radius of curvature was then used to derive a charge

to mass ratio.

o Whilst Thomson wasn’t able to measure the mass of the charges, he was

able to measure the mass to charge ratio.

➢ The Oil Drop Experiment o In 1909 Robert Millikan created a device to measure the charge of an

electron. This would then allow for the mass of the electron to be

determined too.

o The apparatus built by Millikan involved an atomiser which sprayed a fine

mist of oil into Region A.

o An electric field was set up in Region B, over tie some oil drops floated into

this area, where it was then struck by an x-ray

source. This caused the drops to become

charged.

o By adjusting the electric field, the downwards

gravity can be balanced out by the upwards

electric force, suspending the oil drop in mid-air.

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o Therefore, Millikan could determine the charge of each oil droplet. After

repeating the experiment multiple times, Millikan found that the charge of

each oil drop was always a multiple of a small value.

o This value was 1.6 x 10-19 C, which was found to be the charge of an

electron. Combined with Thomson’s experiment, the mass of an electron

was calculated as 9.109 x 10-31kg.

➢ Thomson Atomic Model o Although the mass of an electron was calculated as 9.109 x 10-31 kg at the

time the mass of an atom was already calculated to be 1.673 x 10-27 kg.

o This meant that Dalton’s model of the atom was incomplete; there was a

smaller particle inside of an atom.

o Thomson believed that the atom was still neutrally

charged, so he proposed a ‘Plum Pudding Model’.

o “An atom is a positive sphere in which electrons are

embedded”.

➢ Rutherford Atomic Model o Rutherford is a New Zealand physicist who is well known for his work with

Alpha particle radiation.

o Alpha particles are a type of radiation that was known to come from

radioactive elements. Today it is known to be comprised of 2 protons & 2

neutrons, but this was not known during Rutherford’s time. Rutherford did

know that these particles are positively charged.

o Rutherford conducted an experiment now known as the ‘Gold Foil

Experiment’. The observations from this experiment, formed the basis of the

Rutherford/nuclear atomic model.

o Geiger & Marsden (Rutherford’s assistants) carried out an alpha scattering

experiment as illustrated below:

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o The experiment involved bombarding an extremely thin piece of gold film

with alpha radiation. The alpha radiation would penetrate through the gold

& hit a fluorescent screen on the other side allowing us to determine if it

had been deflected or not.

o Assuming Thomson’s ‘Plum Puddling Model’, the electrons are so small &

randomly distributed that the larger alpha particle would pass through

almost unimpeded by the atoms of the gold foil.

o However, it was observed that the alpha particles didn’t travel through

unimpeded, they were actually deflected by the gold foil.

o The deflection occurred mostly towards the middle of the beam. This

indicated that there exists something at the centre of an atom, capable of

greatly affecting the positively charged alpha particles.

o The large angles of deflection led to the alpha particle bouncing off in many

different directions (even returning back to the source). The scientists had

just observed the first evidence of an atom’s nucleus.

o In 1911, Rutherford proposed the planetary model of atoms based on the

results obtained from his assistances’ experiment

(the alpha-scattering experiment or the gold foil

experiment).

o “An atom consists of a dense, minute, central core

called the nucleus which carries positive charges.

The small, negatively charged electrons are

orbiting around the nucleus at a large distance”.

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➢ The Proton o Eight years after creating the nuclear atomic model, Rutherford would build

on his atomic model with the discovery of the proton.

o The experiment involved firing alpha particles at nitrogen gas. A particle

roughly the same mass as a hydrogen atom was observed. This particle was

deflected by an electric field with the opposing behaviour of an electron.

o Rutherford concluded that the particle was the electron’s counterpart &

named this positive charge a proton.

o There were however certain limitations related to Rutherford’s model:

❖ Firstly, if electrons (negatively charged) orbited a positive nucleus,

why didn’t they exert an attractive force on each other? Newtonian

mechanics predicted that the electron would emit EMR & lose

energy until it fell into the nucleus.

❖ Rutherford’s model doesn’t explain another emerging phenomenon

during the time; Absorption/ Emission Spectrum.

➢ The Neutron o Rutherford identified a further problem with his own atomic model in the

early 1920’s. If the nucleus was made of protons with similar charged, then

why aren’t they repelled by each other?

o Rutherford put forward two proposals to solve this problem:

1. There were electrons inside the nucleus to negate

the repelling forces.

2. There was a neutral particle that could bind the

protons together.

o James Chadwick (student of Rutherford) was the first

physicist to prove the existence of the neutron. He

interpreted the results of an experiment conducted earlier by

Irene Curie & Frederic Joliot as evidence.

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o Curie & Joliot showed that an unknown radiation (produced by Beryllium)

was capable of knocking protons out of a sample of Paraffin wax.

(Hydrocarbon)

o Chadwick used the law of conservation of momentum to conclude that the

particle had to be as heavy as a proton.

o He then went on to show that the particle was uncharged, thus identifying

the neutron as the last component of an atom.

o In summary, the atom is comprised of three major components. Currently

represented by the Rutherford model.

Electron Proton Neutron

Discovered by Thomson (1897) Rutherford (1919) Chadwick (1932)

Location Orbiting the nucleus Inside the nucleus Inside the nucleus

Symbol e or e- p or p+ n or no

Charge (C) -1.6 x 10-19 1.6 x 10-19 0

Mass (kg) 9.11 x 10-31 1.6 x 10-27 1.6 x 10-27

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• Quantum Mechanical Nature of the Atom

➢ Introduction o The most commonly used model of the atom is known as the Rutherford’s

model. In this model electrons are seen revolving around a nucleus (made

up of protons & neutrons).

o There was however one major physics principle that the model couldn’t

explain;

o As electrons orbit the nucleus, they should emit energy, causing the orbit of

the electron to shrink until they fall into the nucleus. Why does this not

occur?

o Niels Bohr (1885 – 1962) was a Danish physicist trained under Rutherford.

o Drawing on the theories on quantisation (by Planck & Einstein) & the work

of Anders Angstrom & Jacob Balmer, Bohr was able to present an atomic

model that could explain the stable orbits.

o Anders Angstrom was the first physicist to calculate the wavelengths of four

of the spectral lines for hydrogen. These were all in the visible part of the

EM spectrum.

o Balmer then went on to derive a relationship between these wavelengths.

➢ Balmer Series & Rydberg’s Equation

o Johannes Rydberg generalised the Balmer series for all spectra

line for hydrogen (not just visible light).

Where:

▪ 𝜆 – wavelength emitted light (m)

▪ RH – Rydberg’s constant (1.097 x 107 m-1)

▪ n – 3, 4, 5, or 6

n = 3, 4, ...

Where:

▪ 𝜆 – wavelength emitted light (m)

▪ RH – Rydberg’s constant (1.097 x 107 m-1)

▪ ni & nf – any integers such that ni > nf

nf -> low energy

ni -> high energy

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➢ Bohr’s Atomic Model o Bohr combined the work of Balmer/Rydberg with a quantum understanding

to propose his atomic model.

o Hy hypothesised that in order for the electrons to not

spiral into the centre, they needsed to exist in discreet

& stable orbit (these orbits correspond to the n = 2, 3,

4, ... values from Balmer’s equation).

o In order to explain how the electrons remain in their

stable orbits, Bohr that the electrons are not continuously emitting energy.

o Instead he proposes that electrons only emit energy when they transition

from an excited state back down to a lower state. This process is what

allows for the creation of an absorption spectrum.

o The amount of energy absorbed for each transition has a discrete value. (If a

photon doesn’t supply that exact amount of energy the electron cannot

make the transition).

o Using this information, the many possible electron jump can be mapped on

an energy level diagram.

o The ground state represents the closest electron orbit from the nucleus

(n=1).

o As n increases, the electron will move to an outer orbit, unit it becomes free

from the electrostatic attraction of the nucleus (n = ∞). When this happens

the atom becomes ionised.

o The different electron jumps are often grouped into series (e.g. the Balmer

Series represents electrons jumping from an excited state down to n = 2

orbit, which releases visible light).

So, for hydrogen:

o An electron at the ground state

requires 13.6eV to escape from the

atom. (ionisation energy).

o The transition of an electron from

n = 3 to n = 2 releases 1.9eV & then

releases visible light.

Note: In some graphs the ground state is given a negative energy value

whilst other graphs have it designated as 0eV. This is due to convention.

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➢ Limitations of Bohr’s Model o Whilst the model allowed for the explanation of stable orbits & emissions it

had some limitations:

❖ The model accurately described how

long atomic number atoms behaved,

but was not as accurate for elements

with many electrons, in many shells.

❖ The presence of a magnetic field

caused strange observations to the

absorption spectra. This could not be

explained (Zeeman Effect).

❖ The model couldn’t explain how solids emitted a continuous

spectrum.

o Whilst the Bohr model has imited applications, it was very important to help

advanced the quantum approach to studying atoms.

➢ Wave-Particle Duality of Light o In 1905, Einstein’s Photon Theory demonstrated that the photoelectric

effect could only be accounted for if light was assumed to have particle-like

propoerties.

o However, the Wave Model of Light remained the only appropriate

explanation for earlier observations of light (interference, diffraction, etc.).

o Scientists were left with no other choice but to accept the dual wave-

particle nature of light.

o Electromagnetic waves are characterised by their speed (c), frequency (f) &

wavelength (𝜆):

o Electromagnetic particles (photons) are instead characterised by the energy

they carry & their subsequent momentum:

➢ De Broglie’s Matter Wave o By extension of Einstein’s Photon Energy (which claimed that EM waves

could possess a particle nature), Louis de Brouglie in 1924 predicted that

particles should also possess a wave nature.

o De Broglie hypothesised:

❖ Matter has both wave & particle properties.

❖ The wavelength associated with any particle with momentum p is:

OR

OR

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o The Broglie wavelength of an electron is smaller than that of visible light.

The wavelength of everyday objects is even smaller due to their much larger

mass.

o It is due to this reason that the wave nature of everyday objects does

unnoticed.

o However, it should be possible to demonstrate the wave nature of a particle

(such as an electron) by showing that it can be diffracted. Physicist set out to

prove this wave nature of matter.

o Recall, that diffraction requires the aperture size to be comparable to the

wavelength of the wave being diffracted.

o At the time te average aperture size of a diffraction

grating ≈ 20𝜇𝑚 meaning it was extremely hard to

observe electron diffraction.

➢ Davisson & Germer’s Experiment o In 1927, Davisson & Germer confirmed De Brogli’e momentum-wavelength

postulate by observing that electrons diffracted through a crystal lattice.

o The De Broglie’s wavelength of an electron was simply too small to be

diffracted through regular diffraction gratings.

o The spacing between Nickel atoms in the crystal lattice was small enough to

produce observab;e diffraction effects.

o This clearly indicated the wave nature of electrons.

Object Wavelength (m)

Radio Wave 1 x 10-3

Visible Red Light 380 x 10-9

Electron 24 x 10-9

Football 4.59 x 10-35

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➢ Standing Waves o Standing Wave: a pattern which results from the interference of two

identical waves travelling along the same medium.

➢ Impact of De Broglie’s Matter Wave o One of the limitations of Bohr’s model was that it could not account for the

stability of the electrons in its orbits (i.e. no scientific justification).

o Using the ‘matter wave’ proposal & the concept of electron standing waves,

De Broglie provided a successful explanation to electron stability in Bohr’s

discrete orbits.

o According to De Broglie, electrons are stable in its orbit around the atom

because the electron wave forms a standing-wave pattern so the electron

waves don’t interfere destructively. (NO ENERGY LOSS!!!)

o For the orbiting electrons to set up a standing-wave pattern, it must orbit

the nucleus at allowable orbits/energy levels such that the circumference

of the orbit equals to some integer multiple of the electron wavelengths:

Where:

▪ 𝜆 – wavelength (m)

▪ rn – radius

▪ n - integer

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➢ The Uncertain Nature of Matter o After Bohr & De Broglie’s work physicist were beginning to find the dual

nature of energy & matter hard to interpret & resolve.

o This is due to the fundamental difference between waves & particles; waves

are continuous disturbance whilst particles are discrete. Particles should

always exist in a single place in space, whilst waves spread out through

space.

o The field of quantum mechanics was solidified, with an aim to study the

duual anture of matter. The work of Schrodinger & Heisenburg set the

foundations for future understanding.

➢ Schrodinger’s Atomic Model o In 1926, Erwin Schrodinger used a mathmetical model to describe the wave-

particle duality of matter. His largest challenge was reconciling how particle

(like electrons) can be continuous & discrete.

o The solution was to think of the wave nature of a particle as a probability

known as the particle’s wave function.

o Schrodinger’s equation expresses the probability that an electron will

occupy a certain region (or be in a certain state) around the nucleus of an

atom.

o This is different to Bohr’s model as it doesn’t draw a circular path that the

electron will definitely follow.

o Instead in 3D space the model shows regions where the electron has a high

proability of existing (darker area) versus areas where there is a low

probability of it existing (lighter area).

o Schrodinger’s atomic model is sometimes described as an atomic cloud. This

quantum model is the most accepted

current atomic model.

o Different electrons will have different

probable locations. These location are

called orbitals (different from orbits,

which are only 2D) & are given

specific names based on their shape.

o Each orbital can only have 2 electrons.

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➢ Heisenberg Uncertainty Principle o The quantum model of the atom is also governed by the Heinsenberg

uncertainity principle, which was propsed in 1926.

o The principle states that the position & momentum (& therefore velocity) of

an object cannot be measured exactly at the same time.

o This is not due to the accuracy of our measuring devices. It is mainly

attributed to two things:

❖ The wave-particle duality of matter (particles are discrete while

waves are continuous).

❖ The unavoidable interactions between an object being observed &

the instrument doing the observing. (Observing an object involves

imparting energy to it, that changes the nature of the observed

object).

o This rule reaffirms the fact that we can only refer to location of particles in

terms of probabilities.

➢ Schrodinger’s Cat Though Experiment o In 1935, Schrodinger proposed a though experiment as an analogy to the

quantum nature of matter.

o The setup involves a cat placed in a sealed box along with a radioactive

substance.

o The substance has a 50% chance of detonating in the next hour (killing the

cat) & a 50% chance of not detonating (not killing the cat).

o The question was, after an hour, is the cat dead of alive?

o The quantum interpretation of the situation would predict the cat is both

alive & dead.

o This is of course an absurd observation. Cats cannot be both dead & alive.

o Schrodinger’s point (which is often missed) to show that it was crazy to

apply the laws of small quantum objects (like electrons) to large complex

objects (like cats).

o In summary, the work of Bohr, De Broglie, Schrodinger, Heisenberg (&

others) have led to a complex & evolving field quantum mechanics. This is

still very new & not well understood, but it has lead to large breakthroughs

in science, computing & other areas.

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• Deep Inside the Atom

➢ Introduction o Particle physics is a branch of physics that studies the nature of the particles

that constitute matter & radiation.

o The fundamental building blocks of matter has evolved overtime from

Dalton’s billiard ball model to the quantum cloud model. Sor far, we know

that atoms are made of 3 fundamental particles: protons, neutrons &

electrons.

o However in 1912, Physicists discovered mysterious particles bombarding

Earth from outer space. The source of these new particles were called

cosmic rays.

o Cosmic rays are fast & energetic. The particles that made up cosmic rays

would strike atoms in the atmosphere creating strange new subatomic

particles.

o As cosmic rays were unreliable,

scientists started building machines

capable of firing protons together to

mimic the collision occurring in the

atmosphere. These were called particle

accelerators & they were responsible

for discovering many additional

particles.

➢ The Particle Zoo o By the 1960’s, cosmic rays & particle accelerators had led to hundreds of

different particles being discovered & named. This large number of different

particles was collectively called the ‘the particle zoo’.

o Physicists started grouping these particles based on certain properties

(charge – interaction with EF, spin – angular momentum, mass - matter &

lifetime – time until it decays).

o To simplify these many particles, a more fundamental set of particles called

quarks were hypothesised & then discovered. These particles form the basis

of the standard model.

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➢ The Standard Model o The Standard Model is a theory that classifies all know elementary particles

(the fundamental building blocks of matter) along with the fundamental

forces.

o There are three main components of the model:

❖ Quarks

❖ Leptons

❖ Bosons

o Quarks & Leptons (fermions) are also divided into generations (I, II & III).

o Besides from these standard particles the matter particles also have

corresponding anti-matter particles (same mass, opposite charge). When a

matter & anti-matter particle meet, they collide & annihilate each other

(releasing huge amounts of energy).

➢ Quarks o Quarks are one of two families of fermions that make up matter. In total, 6

quarks have been discovered (up, down, charm, top & bottom).

o Protons & neutrons are composed of up & down quarks only. This implies

that quarks can have fractional charges.

The up & down quarks, along with the electron, form most matter

we interact with.

The quarks are found briefly in the rarer particles that are created

in particle colliders (before they decay).

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➢ Leptons o Leptons are the simplest & lightest subatomic particles. The first to be

discovered is the electron.

o The muon & tau particles are larger counterparts of the electron. These

particles quickly decay into electrons.

o This decay process also is accompanied by the production of neutrinos;

incredibly small, neutral particles that are very abundant by rarely interact

with matter.

➢ Gauge Bosons o Bosons differ from fermions; in that they are ‘force carrying particles’. These

are responsible for carrying the fundamental forces:

o Exchange particles allow for an

explanation of attractive &

repelling forces we are familiar

with.

Eg.

o Photons are exchanged

between electrons, creating a

repelling force & explain how

like charges repel.

o Gluons are exchanged between

quarks, creating an attractive

force & binding the nucleus

together.

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➢ Evidence for the Standard Model o Evidence for the standard model comes primarily from modern experiments

conducted inside particle accelerators.

o The Australian Nuclear Science & Technology Organisation (ANSTO)

operates the Australian Synchrotron (the Southern Hemisphere’s most

powerful synchrotron).

o CERN is the world’s leading particle physics laboratory & houses the Large

Hadron Collider (LHC), the largest particle accelerator ever constructed

(diameter 27km).

➢ Higgs Bosons o Until the 1960’s physicists could not explain how subatomic particles gained

their mass. The Standard Model did not explain the origins of mass or why

some particles are very heavy while others have no mass at all.

o In 1964 Robert Brout, Fancois Englert & Peter Higgs proposed that all

particles interact with an invisible field to gain their mass; the more a

particle interacts with the field, the heavier it becomes.

o The Higgs Boson ≠ The Higgs Field

❖ The Higgs Boson helps us detect the field.

❖ The Higgs Field is the actual thing that gives particles mass.

o The Higgs Boson was added to the standard model, but we are still not

entirely sure of its existence. However, the maths behind the standard

model has always assumed that the Higgs Boson exists, meaning it is

fundamental to the standard model.

o If the Higgs Boson could be observed, the standard model has further

evidence supporting it.

o In 2012, scientists at CERN identified a particle in one their experiments that

could potentially be the Higgs Boson. Since then, particle physicists had

work to try & prove that this mystery particle is the Higgs Boson.

o This illustrates that the work on the standard model is still ongoing &

incomplete.

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Nucleons anything inside the nucleus

Radioactivity spontaneous release of energy (radiation or particles) from the

nucleus

Radioisotope An unstable isotope will emit particles &/or radiation until it

becomes stable.

Nuclear Transmutation the conversion of one chemical element to another through the

emission of alpha or beta particles from the nucleus.

Half-life (t1/2) time taken for half the radioactive substance to decay.

Nuclear Fission the process in which a heavy unstable nucleus splits to form more

stable, lighter nuclei. It also emits neutrons & energy.

Nuclear Chain Reactions a series of nucleus fissions imitated by a neutron produced in a

preceding fission.

Controlled Chain Reaction

a self-propagating nuclear reaction in which only one neutron released by the splitting of a nucleus is allowed to hit another

nucleus to cause further fission resulting in the steady release of energy.

Uncontrolled Chain Reaction

a self-propagating nuclear reaction in which more than one neutron released by the splitting of a nucleus is allows to hit

another nucleus to cause further fission resulting in energy being released exponentially.

Nuclear Fusion the process in which two or more small nuclei combine to form a larger nucleus with the release of a large amount of energy (more


Mass Defect the difference between the mass of the constituent nucleons &

the mass of the nucleus.

Binding Energy the energy needed is equal to the energy that holds the atom

together in the first place.

The Law of Conservation of Energy

cannot be created nor destroyed. It can only be converted from one form into another.

Binding Energy of a Nucleus

the energy required to completely separate a nucleus into nucleons. Therefore, the binding energy is a measure of the

stability of a nucleus.

Star a type of astronomical object consisting of a luminous spheroid of

plasma (dense, ionised gas) held together by its own gravity.

White Dwarf a collapsed star with no more nuclear fusion reactions as a source

of energy.

Red Giant Star characterised by the nuclear fusion of helium at the core. The

remaining hydrogen fusion occurs in the outer shell.

Luminosity the energy radiated by an object per second.

Brightness the energy received per square metre per second.

Parallax Shift the difference in the position of an object (against a background)

due to different viewing angles.

Standing Wave a pattern which results from the interference of two identical

waves travelling along the same medium.

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Formulas

Properties of the Nucleus

The Origins of the Elements

The Structure of the Atom

Quantum Mechanical Nature of the Atom

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