Physics 3&4 Notes

BASICS OF BASICS

BASIC DEFINITIONS

1. Speed (scalar) • At instant Ø Equal to velocity at instant • Average over period of time Ø �� = Ø Distance rigorously speaking requires curve length integral calculation 2. Velocity (vector) • At instant Ø Magnitude is equal to speed at instant • Average over period of time Ø �� = Ø Displacement = change in position vector. 3. Mechanical energy • Kinetic energy + potential energy

� = � + �

GRAVITY

NEWTON’S LAW OF UNIVERSAL GRAVITATION

1. Any two bodies in universe attract each other with force that follows: � � � = � � �� : � = = � � + ℎ

• �, � measured in kg • � is distance between � �� (Usually point approximation is used, i.e. r is distance between centres of � & �, this is reasonably valid when distance between mass is large compared to size of objects) • G is gravitational constant: � ≈ 6.67 ∗ 10�� (this is quite small, i.e. only significantly heavy objects will exert significant forces)

• �⃗ measured in N. When acted on one of object in pair, it points towards other object

2. Alternative Forms: Universal Law of Gravitation can be rewritten in below forms relating to exact mass one is referring to (these are useful when calculating integrals, i.e. for potential): • Below are all quite confusing, best method is to just remember gravity is attractive, & consider specific cases accordingly. • �� : �⃗ = � �̂, �ℎ�� ̂ �� • ��: �� : �⃗ = −� �̂, �ℎ�� ̂ �� (this is just reversal of �̂ from former) • �� can be derived from above by adding negative sign in front of equation, since by Third Law force is equal & opposite in direction. 3. Shell Theorem: • spherical empty shell of mass with uniform distribution exerts gravitational force to objects outside it as if all mass in shell is concentrated at its centre • Solid sphere case: If solid sphere can be separated into layers of empty uniform shells as mentioned above, then clearly repetitive application of shell theorem show that sphere exerts gravity to objects outside it as if all its mass is concentrated in its centre. • Shell Theorem as Approximation for celestial bodies: planets, stars, etc. generally have layered mass structure that is similar to ideal shell theorem shell layer scenario, consequently, their gravitational force can usually be approximated using solid sphere case aforementioned. 4. Inverse square law (for gravity) 1 � ∝ (�� ℎ� �� ) � • Applies also for other forces, i.e. Coulomb’s force (electrostatic force) • visualization of why such effect occurs (of course, this is theory, & not universal clearly for all forces): FREEVCENOTES . COM

SEVERAL GRAVITATIONAL SYSTEMS COM 1. Solar system . • Overwhelming mass of sun maintains stability (other planets, due to mutual attraction, causes sun to ‘wobble’, but their effects are minimal due to its large mass & thus less acceleration relatively speaking wrt planets) 2. Earth-moon system • Earth less acceleration, same reason as why sun relatively less acceleration in solar system wrt to planets. • Nevertheless, moon’s gravitational force has effects on Earth, i.e. tides (hence term tidal forces)

MEASURING GRAVITATIONAL CONSTANT, G VIA TORSION BALANCE

1. Torsion balances measure very small twisting forces 2. Henry Cavendish’s experiment in 1798 had design of torsion balance that could measure forces smaller than 1 ��. He used to it measure force exerted by lead balls held small distance apart, which in turn may be used to calculate value of G.

WEIGHT

1. Refers to gravitational force object experiences, not mass like people treat it in daily life, i.e.:

��ℎ� = � • ��ℎ� = �� ( formula used in Unit 2) is just special case of general Universal Gravitation Law where object is on surface of earth. • � �� ℎ (acceleration due to gravity) is generally taken as 9.8 (�� 9.81) �� = = � � 2. Apparent weightlessness: (though in text, this term is not part of stud. design., but likely examined via alternate terminology) • Sample response from 2018 Exam rep. Ø ‘zero gravity experience’ is due to the lack of a contact or normal force () o More detailed (personal addition): due to the lack of a contact/normal force arising from the approximate equivalence in acceleration between the object and its surroundings • Refers to gravitational force ‘felt’ Ø normal reaction force acting against gravity to an object to cancel unequal acceleration between objects (usually what Phys 34 seeks for) Ø Tidal forces may used to detect gravity, but they really aren’t what you typically associate with ‘weight’ Ø May well be no real acceleration due to gravity, but weight may be perceived (e.g. centrifuge, artificial gravity) • Cases where apparent weight is different from weight: Ø Orbit Ø Freefall FREEVCENOTESØ Accelerating lift • Note that true weightlessness is when gravitational force is 0, whereas apparent weightlessness is just what one feels is 0.

OTHER

1. Gravity ‘drives universe’ • force that first caused particles to coalesce into atoms, & atoms to congregate into nebulas/planets/stars

GRAVITATIONAL FIELDS

1. Proposed in 18th century, to simplify effects of forces 2. Mental constructs initially (though now though to be legitimate entities that may say carry energy) 3. Refer to region where gravitational force exerted on all matter within 4. Every physical object of mass has accompanying gravitational field 5. Special in that field strength = acceleration (equivalence principle)

INVERSE SQUARE LAW 1. Applies to many forces 1 � ∝ � 2. A intuitive visualization

COM .

NEPTUNE DICOVERY

1. Discovered through its gravitational effect on other planets. 2. Uranus orbit appeared different from those calculated from Sun’s gravity & other known planets’ gravity → possibly unknown planet caused this, which led to discovery of Neptune

REPRESENTING GRAVITATIONAL FIELDS

1. Field lines: shows direction of field intensity, i.e. gravitational acceleration in region of space 2. Closely spaced arrows: strong field 3. Widely spaced arrows: weaker field 4. Parallel arrows: same direction (if spaced out equally, then would also imply constant field strength, so uniform field) 5. Field lines can never cross 6. Infinite number of field lines can be drawn, only few are chosen to represent to rest

GRAVITATIONAL FIELD STRENGTH

1. Gravitational fields theoretically extend infinitely out into space 2. By inverse square law, gravitational field strength is decrease by distance 3. Formula for gravitational field strength:

� �� GM g = = = (M often denotes source mass) FREEVCENOTES� � r 4. Gravitational field strength is equivalent to gravitational acceleration by equivalence principle (gravitational mass is equal to inertial mass) 5. Unit of field strength: N kg(note acceleration unit is m s, which is equivalent)

VARIATIONS IN GRAVITATIONAL FIELD STRENGTH OF EARTH

1. g often assigned 9.81 N kg or 9.8 N kg 2. It may vary depending on location, i.e. distance from centre of earth, nonuniformity of earth mass (i.e. rocks have above-average density) also causes differences, landform differences also has effects • Mineral ore rocks (higher density): stronger g • Sedimentary rocks (lower density): weaker g • Earth has equatorial bulge → g stronger slightly at poles (if we consider g as the empirical accleeraiton, not the theoretical g, then pole would also be larger due to lack of centripetal force effects) 3. Gravitational field line at surf. Earth approximately uniform from human scale (direction + magnitude) 4. Gravimeter used to find small local variations 5. Different planets have different accelerations, i.e. moon surface g (~1.6 N ��) less than on earth • Hence why moonwalk shuffle jumping used instead of walking on moon (fall too slow, so each step takes longer) 6. See variations by height below:

COM .

FREEVCENOTES COM .

FREEVCENOTES COM WORK IN GRAVITATIONAL FIELD .

1. Gravitational potential energy where gravitational field is essentially constant: i.e. near surface of earth: E = ��∆ℎ 2. Conservative force, so potential energy change due to altitude change is same regardless of path taken 3. Work non-constant gravitational field: • Sets distance at infinity as point of 0 potential energy (& may regress work from there) • Change is energy is equal to distance under force-distance graph (if it is field strength-distance graph, then the area is energy per unit mass), as shown below:

FREEVCENOTES

4. Gravitational potential energy for general cases: (derived through setting 0 point at infinity & deriving though integration of gravitational force) �� E = − � 5. For uniform gravitational field

∆� = ��∆ℎ, �� ∆� = −� = −�∆ℎ = −��∆ℎ, �� ℎ�� − 9.8, �� − �� (ℎ �� )

GRAVITATIONAL FIELD STRENGTH IN FIELD STRENGTH-DISTANCE GRAPH

1. Same as for force-distance graph, but energy calculated is now related to potential, & not potential energy

ELECTRIC & MAGNETIC FIELDS ELECTRIC FIELDS COM

ELECTRIC FIELDS . 1. Similar to gravitational field, also follow inverse square law 2. Likes charges repel, unlike charges attract 3. Field lines are arrows with direction force on small positive test charge would point towards, i.e. they go from positively charged objects to negatively charged objects 4. Electric field lines start & end at right angle to surface of objects, no gap between lines & surface 5. Electric field lines can never cross, else they would indicate that acceleration, i.e. field intensity, is in two directions at that point • For tangent cross case, mainly problematic as that would imply infinite field line density so infinite field strength 6. Around small charged spheres (point charges), field lines radiate like spokes on wheel, should at least draw 8 field lines for point charges, i.e. top, bottom, left, right & one in between each 7. Between two point charges, direction of field at any point is superposition field vector of individual fields 8. Between two oppositely charge parallel plates, field lines between plates are evenly spaced & drawn straight from positive to negative plate, i.e. uniform field (this is for infinite sized plates, realistically it would be approximation, especially at edges of plates) 9. These drawing are only 2D representations of 3D field 10. Examples of such electric field line drawings below:

FREEVCENOTES COM .

STRENGTH OF ELECTRIC FIELD

1. Negatively charged particles will experience force opposite to field lines 2. Positively charged particles will experience force in direction of field lines

BEES

1. Thought to use electric fields to communicate, find food & avoid flowers visited by another bee recently 2. Antennae are bent by electric fields & can sense amount of deflection, which flowers that has being recently visited by bees would have (their fields would have been altered) 3. Charge on their bodies help them collect pollen grains & transport them to other flowers

ELECTRIC FORCES ON FREE CHAGRES IN ELECTRIC FIELD

1. F = qE 2. Technically is vector, so: �⃗ = ��⃗ Where �FREEVCENOTES⃗ is electric field, q is charge 3. Electric field: ⃗ • �⃗ = , unit is N C • �� ℎ�� , �. �. �� ℎ�� : E = , unit is V m 4. Work in uniform electric field: W = Fd = qV 5. By conservation of energy, work done on charged object, also means work is done on electric field (in sense of negative work done on potential energy) 6. If object moves parallel to field lines, i.e. potential is not changed, then no work has been done either by or on field.

COULOMB’S LAW

1. F = = k • = � (Coulomb’s constant), value is approx 8.9875 ∗ 10 � � � (often rounded to 9.0 for 2 signif. figures) • q, � �� ℎ� �� ℎ�� (C) • r is distance between charged point (m) • � is permittivity of free space (i.e. for in space or air), �~8.8542 ∗ 10 � � � (note that this may be altered via dielectric constant, as electric fields depend on medium as well (but that is beyond course content)) • ��ℎ��, �� ℎ�� ℎ� �ℎ�� ℎ�� ? 2. For direction, one may have conventions of vectors, but it is generally better to just remember like charges repel, unlike charges attract.

ONE COULOMB IN PERSPECTIVE

1. For two point charges of 1 coulomb 1 metre apart, force would be approx.: 9.0 ∗ 10 � • Approx. equal to gravitational weight of 918000 tonnes, which is almost twice weight of Sydney Harbour Bridge, hence showing how much 1C is (it is also due to this, that in nature large accumulations of charge don’t occur often, attract each other & their electric attraction/repulsion effects more or less neutralise) • I.E. even highly charged Van de Graaff generator will only have several microcoulombs (1µC = 10�) 2. All matter is held together by electrical forces between atoms, i.e. strength of hardest steel is due to electrical attraction between ions & delocalized electrons between them 3. Electrical forces on atoms are about 10 times stronger than gravitational effects on atoms (i.e. only in cases of significant mass accumulation, i.e. collapse of giant star due to gravity can gravity overwhelm electrical forces & cause super-dense neutron star to form)

ELECTRIC FIELD AT DISTANCE FROM CHARGE

1. E = = MAGNETIC FIELD COM GENERAL NOTES . 1. Not part of course: magnetic field is relativistic effect of electric field 2. Voltaic pile: simple early battery

MAGNETISM

1. Magnetic poles: each end of magnetic dipole 2. Earth’s magnetic field: (Differentiate geographic north, magnetic north, north of the Earth magnet (the last is reverse of the former 2)

FREEVCENOTES

‘FLIPPING’ POLES 1. Earth magnetic field is not static, i.e. magnetic poles are not static like geographic poles, for many years magnetic north pole was moving at approx. 9 km per annum, accelerated in recent year to average 52 km per annum 2. Once in every few hundred thousand years, ‘geomagnetic reversal’ occurs, i.e. magnetic poles flip, i.e. compass would point south instead of north. 3. Currently, earth is well ‘overdue’ for next flip, & measurements recently show Earth magnetic field is starting to weaken faster than in past, so maybe getting close to ‘flip’ 4. Past studies suggest such reversal take hundreds or even thousands of year, but recent studies suggest reversal could occur over significantly shorter time period. 5. Below shows movement of magnetic north overtime:

COM .

MAGNETIC FIELD

1. Similar as gravitational & electrical fields, can be revealed via using iron fillings, as shown below:

FREEVCENOTES COM 2. Field lines usually drawn from north to south, as shown below, it points to direction compass would point if placed at that . position in field:

3. Near either poles of bar magnets, magnetic field lines are almost vertical, as shown below FREEVCENOTES

4. When magnet are produced close to each other, if poles are unlike, then attraction will extend magnetic field, if poles are like, then repulsion will create neutral point where there is no magnetic field 5. Different magnet shapes produce different fields, i.e. below is horseshoe magnet field: COM .

CURRENT CARRYING WIRES & MAGNETIC FIELD

1. Direction of current & consequent magnetic field (right-hand grip rule)

FREEVCENOTES COM .

MAGNETIC FIELD BETWEEN PARALLEL LINES

1. Magnetic field interaction causes repulsion/attraction of wires 2. Opposite direction fields attract (as if opposite poles, i.e. if wires have current running in same direction so field opposite) • Alternatively you may consider right hand rules of magnetic field & magnetic force, same goes for following property 3. Same direction fields repel (as if same poles, i.e. if wires have currents running in opposite directions, so field same direction)

FREEVCENOTES

3D FIELDS

1. Example of some 3D structures & field around them:

2. 3D representation of magnetic field around loop of wire

COM .

3. 2D representation around that same wire loop

FREEVCENOTES

Ø Crosses show running into page Ø Dots show running out of page Ø As how intensity of field depend on density of field lines, it also depend on density of such dots/crosses drawn MAGNETIC FIELD AROUND SOLENOID

1. Field as shown below:

COM .

2. Solenoids, i.e. wire coils, resemble bar magnets 3. Direction of field can be found by applying right hand grip rule around wires (note, current direction is important), this is then used to determine its north & south poles 4. These may be used to find electromagnets 5. Factors on strength of solenoid: (under long solenoid approximation)

� = �� = �� , � = ��ℎ ��

FORCES ON CHARGED OBJECTS DUE TO MAGNETIC FIELDS

1. Lorentz force: (electromagnetic force)

�⃗ = ��⃗ + �⃗ × �⃗ �� + �� ℎ�� 2. Magnetic force on charged object

�⃗ = ��⃗ × �⃗

• q is in coulombs (C) • B is magnetic field vector, magnitude in teslas (T) • v is velocity vector, with units m s • cross product has magnitude: vBsin(θ), direction is perpendicular to two vectors, pointing out of plane in which sweeping of v into B is anti-clockwise, as shown below:

FREEVCENOTES

3. If we only want magnitude of force, one may have: F = qvB • But v here is not velocity, instead, it is component of instantaneous velocity of particle that is perpendicular to magnetic field 4. Clearly, force is maximum when charged particle is moving at right angles to field, also no force when charged particle is travelling parallel to magnetic field 5. Example: magnetic deflection of electron beam in cathode ray tube:

DETERMINING DIRECTION OF FORCE:

1. Fundamentals of physics method: (negative particle just means opposite direction, beware) COM .

2. Physics 3,4 text method: (again, beware of negative particles)

FREEVCENOTES

FORCE ON CURRENT-CARRYING CONDUCTOR

1. Conducting wire is essentially stream of electrons, i.e. can find magnetic force on wire by calculating force on electrons 2. formula is for specific length of straight wire & uniform field (for complex scenarios, split into parts & do separately): �⃗ = ��⃗ × �⃗ = �� ℎ�� ∗ ��⃗ × �⃗ = �⃗I × �⃗ • Direction of I is direction of v • B in tesla (B) • I is current in amperes () • l is length in m • For multiple coils, i.e. stacked upon each other, simply multiply by number of such coils

COMPARING FIELDS – SUMMARY CONTACT FORCES

1. Many forces, i.e. contact on opening door can be described as contact forces 2. Forces at distance include magnetism, electricity, gravity 3. Forces at distance are helped by fields concept

DIPOLES & MONOPOLES

1. Gravity essentially is just monopoles 2. Magnetic fields practically exist solely as dipoles, there is theoretical monopole though 3. Electric fields have both monopoles (positive/negative) & dipoles 4. Gravitational fields are said to have quadrupole, which is based on how mass of object is stretched out along particular rotational axis, i.e.:

DIRECTION & SHAPE OF FIELDS

1. Static field: one that does not change with time 2. example of how fields may change depending on shape:

COM .

COMPARING GRAVITATIONAL & ELECTRIC FIELDS

1. Sumarised below:

FREEVCENOTES APPLICATION OF FIELDS

SATELLITE MOTION

SATELLITE

1. Satellite: object in stable orbit around another object, they are like cannon balls fieat angle, they are constantly falling 2. Uniform circular motion orbit • May use energy consideration for explaining constant velocity in uniform circular orbit

NORMAL FORCE

1. Normal forces are any force exerted by surface perpendicular (normal) to surface 2. Normal reaction force: normal force that is also reaction force of another force, i.e. one shown below exerted on person:

3. Example of normal force, shown on ball below: COM .

4. In situations where only gravitational force & normal force acts:

�⃗ = �⃗ + �⃗

FALLING AT CONSTANT SPEED

1. freefall nature of objects shown famously by Galileo 2. Air resistance however affects acceleration in environments, i.e. why there are things like terminal velocity, where resistive forces like air resistance (drag, air friction)

NATURAL SATELLITES

1. Literally satellites formed naturally, i.e. solar system’s planets are natural satellites of sun

ARTIFICIAL SATELLITES

1. Approx. 6000 artificial satellites have been launched into Earth orbit since Space Age began in 1957 2. Artificial satellites: as name suggests, satellites that are man-made 3. only major force on Earth orbiting satellites is Earth gravity (in circular orbits, this force is effectively centripetal force, i.e. gravitational acceleration is equal to Earth gravitational field strength at point)FREEVCENOTES (for other cases, i.e. elliptical orbits, gravity of Earth is still dominant force, you could still say it is acting as cen tripetal force, but of course as stated orbit is not really circular) 4. Around 4000 of launched satellites are still in orbit (though only approx. 1200 are operational) 5. Classification of Earth orbits (by altitude, so from sea surface): • Low orbit:180km − 2000km Ø Most satellites orbit in this range, including Hubble Space Telescope • Medium orbit: 2000km − 36000km (often, upper bound more accurately defined as geosynchronous orbit) Ø Satellites in this region include: GPS (Global Positioning System) (used for navigation systems) • High orbit: ≥ 36000km (often, lower bound more accurately defined as geosynchronous orbit) Ø Including satellites that have altitude of 36000 km (more accurately 35786km above sea level) & orbit with period of 24 hours are called geostationary satellites (or geosynchronous satellites) Ø Most communication satellites are geostationary (e.g. Australian used Optus satellites for communication & deep-sapce weather picture use Japanese MTSAT-1R satellites 6. Different orbital paths: • Equatorial orbit: always travel above equator • Polar/near-polar orbit: travels over/close to North & South Poles as it orbits • Inclined orbit: lie between equatorial & polar orbits Ø Many low-orbit American NOAA satellites have inclination of 99° & orbit that allows them to pass over each part of Earth at same time each day, these are also known as Sun-synchronous satellites 7. Artificial satellites have many purposes 8. Artificial satellites often equipped with propellant tanks squirted in appropriate direction when orbit needs to be adjusted 9. 60% of satellites used for communication

SUITSAT1

1. Unusual satellite: obsolete Russian spacesuit that had radio transmitter, batteries & some sensors placed into 2. Launched by pushed off by astronaut during spacewalk, from ISS on 3 February, 2006 3. Designed to transmit signals picked up by ham (amateur) radio operators on Earth for few weeks, but transmissions ceased after few hours 4. Burned up in atmosphere over Western Australia in September 2006 5. SuitSat2: launched August 2011: contained some school student created experiments, re-entered atmosphere January 2012 after 5 months in orbit

THREE NOTABLE SATELLITES

1. Geostationary Meteorological Satellite MTSAT-1R • Japanese • Launched February 2005 • Orbits at 35800 km • Equatorial orbit, longitude of 140°E, i.e. just to north of Cape York (quite sure here it means it would pass that point) • Closet orbit point to Earth (perigee), altitude is 35776 km • Furthest orbit point to Earth (apogee), altitude is 35798 km • Ideally located for use by Australian weather forecasters • Geostationary orbit • Signals transmitted every 2 hours, received by satellite dish on roof of head office at Bureau of Meteorology in Perth • Side note: Infrared images showing temperature in atmosphere are invaluable in weather forecasting COM • Is box like, measures about 2.6 m along each side • Mass: 1250 kg . • Powered by solar panels, which when deployed overall length over 30 m 2. Hubble Space Telescope (HST) • Cooperative venture between NASA & ESA • Launched by shuttle Discovery on 25 April 1990 • Permanent unoccupied space-based observatory • 2.4 m diameter reflecting telescope, spectrographs + faint-object camera • Orbits above atmosphere • Produce images of distant stars & galaxies far clearer than those from ground-based observatories • Expected lifespan was originally around 15 years, but service & repair missions have extended its life & is still in use today 3. National Oceanic & Atmospheric Administration Satellite (NOAA-19) • Many US owned & operated NOAA satellites are located in low-altitude near-polar orbits • NOAA-19 launched in February 2009 • Inclination of 99° to equator • Low altitude, hence allows capturing high-resolution images of small bands of Earth, these data used in local weather forecasting as well as to providing large amounts of information for monitoring global warming & climate change 4. Summaries of properties of these 3 satellites

SPACE JUNK

1. ~1200 satellites still in operation, ~2800 have reached end of operational life/malfunctioned but still in orbit. 2. 2007, Chinese satellite deliberately destroyed by missile (thus creating thousands of debris pieces) 3. 2009, defunct Russian Cosmos 2251 & operational US Iridium 33 created more debris 4. Debris due to such above events FREEVCENOTES& defunct satellites are classified as space junk, there are currently ~ 22000 pieces of space junk are being tracked & monitored 5. Has been occurrences of satellite maneuvers to avoid collision with space junk 6. UN passed resolution to remove defunct satellites from LEO by moving to much higher orbits/bringing back to Earth & burn in atmosphere 7. Exaggerated map showing space debris in near-Earth orbits

KEPLER’S LAWS

1. 1st law: planets move in elliptical orbits with Sun at one focus • Generally, large body being orbited is at focus COM 2. 2nd law: line connecting planet to Sun sweeps out equal areas in equal intervals of time, as shown below: • Derivable via conservation of angular momentum & calculus consideration of infitisimals .

3. 3rd Law • general law: square of period of planet is proportional to cube of semimajor axis (from centre to perihelion) of its orbit, namely (note that this is still approximation, Newton’s Laws derived version will need to account for binary motion where central body is not assumed still): � �� = �� = , �ℎ�� ℎ� �� 4� • For Phys 34, typically only basic circular motion form, as shown below � �� = �� = � 4� Ø Newton’s Laws derivation of this circular case:

�� = = � = � �

�� = ��

2� 2�� 2�� : �� : � = = → � = FREEVCENOTES� � � 4�� → = �� → = � = � 4� � 4. Perihelion: closest point to foci, also where speed is max 5. Aphelion: furthest point to foci also where speed is min 6. Centripetal acceleration of satellites in circular orbit & gravitational force

� 4�� = = = � (�� ℎ�� ) � �

�� 4�� = = = � � �

GANYMEDE 1. One of more than 60 of Jupiter’s known satellites 2. largest of Jupiter’s known satellites 3. Biggest of all moons in solar system & larger than planet Mercury

DC MOTORS

1. First build by Michael Faraday in 1821, principles & main components have since remained largely same • Below: Faraday’s Electric Motor • Magnet mounted vertically in pool of mercury (conductive, i.e. takes current from wire) • Wire carries current, hung from support above, interacts with component of magnetic force perpendicular to wire, thereby producing horizontal force that keeps wire rotating.

COM .

2. Modern DC motors • Current carrying coil of wire in magnetic field experiences force that leads to rotation • Commuter: reverses current, due to how if not, then coil will reach certain point & stop turning • Reiteration: magnetic force experienced by straight current carrying wire in uniform magnetic field (for more complex scenarios, deal by parts): �⃗ = ��⃗ × �⃗ = �� ℎ�� ∗ ��⃗ × �⃗ = �⃗I × �⃗ • Magnetic field provided via multiple magnets, or by electromagnet • Example of such motor shown below:

FREEVCENOTES

3. practical DC motor example

4. Coils often wound around soft iron core to increase magnetic field that passes through them, this entire arrangement of core & coils is called ‘armature’. 5. Electromagnets generally able to produce larger fields than permanent magnets, hence why large electric motors tend to use electromagnets

TORQUE IN ELECTRIC MOTORS 1. For simple rectangular wire in uniform field, with AD=BC=b, AB=DC=, axis of rotation set as through midpoints of BC & COMAD, i.e. setting shown below: .

FREEVCENOTES

• Torque around rotational axis is as follows: (note that � = ) �⃗ = �⃗ = 0 (�� ℎ� �� ) �� ⃗ = �⃗ × �I⃗ × �⃗ = (� × � × �) 2 �� ⃗ = −�⃗ × −�I⃗ × �⃗ = �⃗ = (� × � × �) 2 �ℎ��: �⃗ = 2�⃗ �� , �� ℎ� rotational direction from vector formula is quite tedious for simple calculations, hence better to just use right hand rule & consider specific case. What is easier to formularize is magnitude (in absolute value sense, i.e. just size) of torque:

� = ��(�)

Ø sin (�) refer to sin ratio related to � × � × �, �� , � �� ℎ� absolute value of angle between vectors � & � × � Ø For multiple looped coils, simply multiply N in front of equation Ø In fact, for all flat coils, regardless of shape, more general formula applies:

� = ��(�), �ℎ�� ℎ� �� • From above, one may see that larger area, larger magnetic field & larger current makes for larger torque.

TORQUE

1. Basic Formula:

�⃗ = �⃗ × �⃗

• If we are only concerned with magnitude:

� = � � = ��

PARTICLE ACCELERATORS

SYNCHRONTRONS

1. Melbourne has most powerful synchrotron in southern hemisphere. 2. Is particle accelerator where electrons are accelerated around evacuated ring. 3. curved path is enabled by strong magnets 4. electrons give bursts of radiation (includes all kinds of EM radiation) as they accelerate around curves (note, circular motion at constant speed is not constant velocity, i.e. they continuously accelerate) (here we are indeed referring to radial acceeleration) COM 5. Australian Synchrotron accelerates electrons through equivalent of 3000 million volts (3GV), where electrons travel at 99.99999% speed of light (where their effective mass due to relativistic effects is about 6000 of its rest mass) . • bursts of radiation from this synchrotron is used for various research projects, these range from infrared to X-ray

PARTICLE ACCELERATOR

1. Strong magnets often used to direct collisions of high energy particles, which produces subatomic particles (i.e. found Higgs boson) that can be analysed 2. Electromagnetic fields often used to accelerate particles needed to produce subatomic particles in collisions, but generally long paths are required to gain speed, to achieve this without very long facilities mean circular tunnels, hence why many particle accelerators are of circular shapes, hence why say Australian Synchrotron near Monash University in Melbourne is circular & 70m in diameter. 3. One of first particle accelerators: Van de Graaff accelerator (Similar to Van de Graaff generator): • Developed in 1930s • Is essentially Van de Graaff generator (except used for specific purpose of using its high voltage as means of particle acceleration.), see below for typical setup

FREEVCENOTES COM .

• Discharge upon putting too close to conductor is due to ionization of air. • Can accelerate charged particles between metal electrodes to ~ 15 MeV before collision with fixed target • Theoretically you could probably design it to transfer positive or negative charge carriers, though I think electrons would be simpler usually. • Side note: below is depiction of Van de Graaff Generator (not accelerator) (which generates via utilizing triboelectric effect where simple contact of dissimilar materials cause transfer of some electrons from one to another, here, electros go from sphere onto rubber belt which gets transported down, here we use rubber as it isn’t conductor & hence won’t conduct away charge) Ø These don’t cause significant harm due to little charge stored in big metal ball & extreme voltage, i.e. when you get static electric shock, current is very short in duration (though it may be quite large), hence it does not penetrate skin & hence no significant damage is done o However, if some form of capacitor (i.e. Leyden jar) was used for process so to store more charge, discharge may become quite harmful Ø Interesting effect: placing couple of metal foils of different size (such that stack of them will all be in contact with ball) onto Van de Graaff ball, electrons will flow onto these foils. net negative charge on them will thus repel, & provided right conditions, foils will fly off one by one, as shown in video https://www.youtube.com/watch?v=rNEY3Yv9kC8 FREEVCENOTES COM .

4. Current world’s most powerful particle accelerator: LHC (Large Hadron Accelerator) • Located at CERN of France-Switzerland border • Can produce energies of 13TeV • Two sets of particles can be accelerated in opposite directions around its ring, before final collision 5. Cathode ray tube: another type of particle accelerator • Electron released from negative terminal, or hot cathode, in vacuum • Accelerate towards positive terminal, or anode, potential difference between cathode & anode is ~2-3 kV • Electrons collimated (narrowed) as pass through slit • Electrons release light when hits fluorescent screen Ø Fluorescence is emission of light by substance that has absorbed EM radiation • Used by older televisions, old visual display units & cathode ray oscilloscopes

6. Source of electrons for colliders: Electron Gun • FREEVCENOTESDiagram:

• Electrons essentially ‘boiled off’ heater element, or cathode, & accelerated towards positively charged plate anode. Ø Heat → � more energy → some may escape from metal • Electron gun equation: (i.e. kinetic energy equals change in potential energy, this refers if initial velocity is 0)

1 �� = �� 2 CHARGED PARTICLE IN MAGNETIC FIELD COM 1. Helical motion: • Given charged particle in uniform B field, will undergo helical motion with axis parallel to magnetic field. • This includes of course degenerate cases, where charged particle moves parallel to field (just travels in straight line), p. article initially at rest (stays at rest), initially moving perpendicular to field (uniform circular motion & not your typical helical motion), if not, then statement will need to be reformulated • Note: magnetic force perp. magnetic field, → velocity component parallel to field will be constant • force ⊥ �⃗ of particle, given uniform field → uniform circular motion component of helical motion, note also velocity magnitude unchanged: for direction of uniform circular motion, right hand rule (this relies on the constant perp force results in uniform circular motion argument)

�� = ��⃗ × �⃗ = �� → � = � ��

Ø m: mass (kg) Ø v: speed (��) Ø q: charge on charged particle (�) Ø B: strength of magnetic field (�)

2. Other properties like T, � & � derivable from above

THOMSON’S EXPERIMENT 1. In 1897, Thomson’s experiment using cathode rays (later renamed electrons) found particles within atoms, i.e. demonstrated subatomic particles for first time. • More detailed history Ø Thomson measured mass of cathode rays Ø Found they carried charge Ø Found they were particles ~ 1800 lighter than H (lightest atom) (via Thomson’s charge mass ratio and Millikan’s elementary charge) Ø Hence were charged subatomic particles 2. Had fluorescent screen to show deflection of electrons, set up is as shown below:

FREEVCENOTES

3. Had 2 stages • Use balancing (i.e. no deflection) electric & magnetic force to find velocity of particles ��⃗ + ��⃗ × �⃗ = 0 → �⃗ + �⃗ × �⃗ = 0 �� ℎ�� ⃗ • Magnetic field switched off, only deflect via electrical field alone, thereby allowing find e/m ratio, below is sample calculations (vertical refers to velocity perpendicular to velocity (referred for convenience as horizontal) when no forces at all are exerted) ℎ�� = �� ∗ ℎ��

1 � � ��ℎ � , �� : � ∗ � = �� → = 2 � 2�

4. Thomson’s calculated value is = 1 ∗ 10� �� (accepted value today is = 1.76 ∗ 10� ��)

ELECTROMAGNETIC INDUCTION & TRANSMISSION OF ELECTRICITY

INDUCING EMF IN MAGNETIC FIELD

1. Oersted found current produces magnetic field, Faraday thought reverse, magnetic field produce electric current should also be true. 2. Faraday wound 2 wire coils to iron ring, current joined to one ring, which created magnetic field that he expected to create current in second ring, no findings, but then when he turned on initial current, there was flicker in galvanometer of second coil, i.e. what mattered for current induction was not strength of magnetic field, but change in magnetic Field 3. Faraday’s law (below all refer to closed loop (i.e. wire loop, but it may well be imaginary loop) �� − = � (��), �ℎ�� (�� ) = �⃗ ∙ �⃗ = � �(� �� , � �� ) �� • negative sign refers to Lenz’s law, i.e. current induced is such that corresponding magnetic field produced by current induced opposes magnetic field change that induced current in first place. Ø May be derived from �⃗ = ��⃗ × �⃗ (if the flux change is viewed in frame where charges in loop are moving (else this law kinda fails)) • If there are multiple loops of wire, then each such wire will have its own EMF induced, i.e. for N coils of wire: (Note, textbook does not use calculus notation, instead, it simply uses time averaged change in flux ) �� −� = � COM �� ∆� . � = −� ∆� 4. Kirchhoff law, not part of course (relies on steady state) • Voltage law: for any loop, potential change add to 0 (so all emf consumed by potential loss) • Current law: at any junction, signed current equal 0 (conservation of charge) 5. For other calculations referring to output induced current, following Ohm’s Law is useful: � = �� • Temperature resistance relations Ø For conductors, temperature increase typically increase R (explainable via more vigorous atom vibration, hence more likely for electrons to collide, of course this does not apply for all cases) Ø For insulators, temperature increase typically decrease R (explainable with how the bound electrons under vigorous vibration may be released free to some extent, of course this does not apply for all cases) • Rules of energy dissipation (or consumption) power as a result (with some forms derived from ohm’s law) �2 � = �� = �� = � � �� 6. Induced EMF in moving conductor using Lorentz force law:

• Using Lorentz force law: �⃗ = ��⃗ × �⃗ FREEVCENOTES→ � = �⃗� = ��⃗ × �⃗ � � = �� ℎ�� = = ��⃗ × �⃗ �

→ �� : � = �� = �� (�� ℎ �� )

EDDY CURRENTS

1. In setup such as one shown below, through considering loops to apply Faraday’s law within metal plate, one may see formation of such internal electric current, i.e. Eddy currents • By lenz law current induced will make pull out push in harder (also how regenerative breaking works)

2. Eddy current through conductor with resistance will cause energy lose often as heat, hence e.g. AC generator, motor, transformer iron cores often laminated by layers with insulators so to decrease conductivity & suppress eddy currents. 3. Earth magnetic field • Due to eddy currents, (convection current in earth’s core of molten iron due to e.g. heat from radioactive decay) • This self-reinforcing current generation apparently relates to turbulence and stuff, so no need deep knowledge • Iron act like spinning disks moving in Earth magnetic field & thus gets eddy currents in them, thereby in turn produces Earth’s magnetic field

INDUCTION STOVES COM 1. Uses AC power & coil (often copper) on cooktop to generate varying magnetic field inducing eddy currents in pot/pan, which then due to resistance dissipates energy as heat, hence heating food. . 2. Heating efficiency is ~ 12% better than traditional electric cooktops & twice that of gas 3. Better than traditional electric stoves as they, like gas stoves, allow instant control of cooking power 4. Why you don’t get electrocuted: • Designed in wave that eddy current & corresponding voltage is controlled to suitable level. Such that making circuit with it will not result in any significant problems.

LENZ’S LAW & ITS APPLICATION

ELECTRIC POWER GENERATORS

1. Relate to Faraday’s Law, how magnetic flux change may induce current, i.e. EMF, i.e. electric energy 2. Induced EMF in alternator • Alternator: basic electric generator • Consist of many coils of wire wound on iron core framework (this together is called armature) • Armature made to move (often to rotate) around magnetic field, hence generating due to magnetic flux change EMF (similar to reverse of typical electric motor), typical example is shown below:

FREEVCENOTES COM .

Ø Here direction of induced current is in direction DCBA from (a) to (b) & from (b) to (c), then it will be in direction DABC from (c) to (d) & FREEVCENOTESfrom (d) to (e) (which is equivalent of (a)), i.e. EMF direction reverses every 180 degrees Ø One may also think above case via Lorentz force law as opposed to equivalent Faraday’s law Ø Below is typical AC generator, flux vs angle graph & corresponding emf graph is as follows:

• turning of armature derives energy from other sources, i.e. manual turning, steam turbines, wind turbine, etc. • In many industrial generators, coils will remain static, & have electromagnet/magnet move instead, this is so to avoid issues such as wear COM& sparking that comes with split ring commutator–carbon brush or slip ring-carbon brush designs below. • Main components of simple DC generator (similar to AC generator below, except split ring commutator is used such that alternation does not occur):.

• Main components of simple AC generator is below, essentially same as motor, except here we do not leave commutator but instead just 2 slip rings & individual carbon brushes, i.e. current does indeed alternate & does not become DC generator:

FREEVCENOTES

BACK EMF IN DC MOTORS

1. As above shows, DC motor & DC generator very much similar. 2. Consequently, DC motor can either produce force, or when something is pushing its coils, hence how some electric vehicles can generate electricity when braking (not to be confused with regenerative braking) 3. DC motors may also generate power during normal operation, this is termed ‘back emf’, this is created in line with Lenz’s law, & opposes change in magnetic flux that created it (due to rotation of motor), i.e. it will be in opposite direction to emf creating it, i.e. hence net emf used by motor is always less than supplied voltage

� = � − � 4. When load applied to motor (i.e. jamming of electric drill), speed generally reduce, hence reducing back emf (increasing net emf) & increasing current in motor, which may then lead to burning of motor & motor windings (think P=��) 5. To protect important components, resistors are often placed in series, & switched out of circuit when current drops predetermined level & switched back in for protection once level is exceeded (�ℎ�� = )

ALTERNATING VOLTAGE & CURRENT

1. AC generators often (though not necessarily) produce sinusoidal alternating current as shown below: (hence why often more useful to known average power, in sense of average magnitude of power)

COM .

2. Root mean square value, RMS value: (this is value of steady voltage that would produce same power as alternating current with peak value equal to √2 times as much) � � = ↔ � = � ∗ √2 √2 • Derivation: calculate average power & then divide by equating average AC power (area between curve & time axis) with constant power value of equivalent DC current (the need for RMS arise from how P is proportional to �, �, not just V, I) � �� sin(�) � 1 � �� 2 � 1 � � � �� → = → � = 2 � � √2 Ø A more rigorous version (not in course) �� ℎ �, � �ℎ�� , �� ℎ�� = �� (�ℎ��) � = �� = �� = � � ∫ �� ∫ �� = = � �� ∫ �� ∫ �� = , � = � � � → � = � � = → � = � � (�� , �� , � �� ) � �� ℎ�� ℎ� �� (�� ℎ�� ℎ��) 2� � � � = � sin(��) , � = 2�� = → � = = sin (��) � � � � sin (2��) FREEVCENOTES∫ �� ∫ sin(��) �� − � � 2 4� 1 � = = = = = � �� 2 √2

� sin (2��) ∫ �� ∫ sin(��) �� − 2 4� 1 � � = = = � = � = � � � 2 √2 • This value should be used when trying to find power supplied by each cycle of AC supply

SUPPLYING ELECTRICITY – TRANSFORMERS & LARGE-SCALE POWER DISTRIBUTION

WORKINGS OF TRANSFORMER (FOR AC VOLTAGE STEP UP/DOWN)

1. Typical setup shown below:

2. Iron core helps ensure all � from primary coil passes through secondary coil, • core often laminated (thin iron plates electrically insulated from each other) so to disrupt and minimize eddy current losses. 3. Ideal transformer: 100% power efficiency in step up step down processes • Realistic transformers often have practical efficiencies > 99% 4. Workings • Ampere-Maxwell’s Law: AC current in primary coil induces changing magnetic field due to current change • Faraday’s Law: fluctuating magnetic field (flux) induces AC current in secondary coil

AC VS DC

1. Power distribution system works on AC, but many devices run on DC 2. Effects of frequency on AC • High frequency more capacitive loss (not part of course so no need deep investigation here) • Skin effect (at high frequency tend to flow on surface of conductor) COM • Impedence (resistance essentially for AC) 3. Reason for AC transmission: Necessity of high V transmission & ease of AC voltage change via aforementioned transformers (DC can’t use that as it does not vary current, so no . magnetic flux varying) • Proof for high V �� = �� = � � �� ℎ �� → �� → ��

TRANSFORMER EQUATION

1. Primary coil AC frequency = secondary coil AC frequency 2. Ideal transformer equation: (apply for at instant (well technically not if you go with light transmission speed), and hence of course apply for RMS and averages) (subscript 1: primary coil, 2: secondary coil) (assumes all magnetic flux passed through) � � = � � • Since we are dealing with AC, voltage usually refers to RMS for things like ohm’s law as well

3. Step up transformer: � > �, � > � 4. Step down transformer: � > �, � > �

POWER OUTPUT

1. Energy conservation: output energy can never exceed input energy 2. ideal transformer, power output is as so: � � �� = �� ⇒ = ⇒ �� = �� 3. Transformer overload: when too much current is drawn & resistive power loss becomes too great, i.e. overheat rapidly, hence why important not to exceed rated capacity of transformer

POWER FOR CITIES: LARGE SCALE AC SUPPLY

1. Power loss in circuit: (of course, for simplicity, here we are treating everything as in series) � � = �� = � � = � 2. Here voltage refers to voltage drop across load, not voltage suppliedFREEVCENOTES 3. Reducing loss: To reduce unwanted energy loss, best is to minimize current:

� = �� If we take transformer’s power to be constant, then stepping up voltage clearly decreases current, hence less energy loss, hence why long distance transmission requires generally high voltage. 4. AC power is readily stepped up by transformer from generator to somewhere between 240 kV & 500 kV before transmission 5. Voltage of AC then stepped down at electrical substations in cities to approx. 2400 V, & later via small distribution transformers to 240 V for home/business use. 6. Alternatively, if step up step down voltage not used, to achieve same low loss would require things like more copper to increase cross-sectional area of wires, which is both expensive & heavy. 7. Transmission tower purpose • High voltages close to ground may lead to earthing • Protect ground level people and environment

LARGE SCALE ELECTRICAL DISTRIBUTION SYSTEMS 1. Typical path taken by electric energy to consumers:

AC VS DC COMPEDITION, HISTORY & FUTURE

1. ‘War of Currents’: began in late 1800s, between Thomas Edison (DC) & Nikola Tesla (AC). 2. AC won upper hand for long time due to ease of stepping up & down voltage 3. AC system voltage limit issue: Above approx. 100 kV corona loss (due to high voltage ionizing air molecules), begins to occur & beyond 500 kV it becomes no longer feasible to transmit electric power due to such effects (DC less issue as its constant voltage has a smaller peak voltage issue) 4. DC is better in sense that to achieve same power output, it does not need as high of peak voltage as AC, hence it can transmit power at high voltage without worrying about effects such as corona effect 5. Now, high voltage DC is reached more easier, i.e. through small, high frequency switching converters 6. Projects such as Three Gorges Dam in China & undersea transmission lines are planning to use DC transmission now. COM 7. AC still is needed for some devices, i.e. safety switch . NEWTONIAN THEORIES OF MOTION

NEWTON’S LAW OF MOTION

NEWTON’S FIRST LAW (LAW OF INERTIA)

1. Every object continues to be at rest or continues with constant velocity (alternatively, it remains in motion (inclusive of degenerate motion, namely at rest)) unless it experiences unbalanced force. • Seems derivable from 2nd law, but not necessarily, for its significance is in setting inertial frame in which 2nd & 3rd law applies (without fictitious forces of course)

NEWTON’S SECOND LAW

1. �⃗ = ��⃗ • For Phys 34, you need to know they are vectors, but adding sign is only good practice but not necessary • Remember to add net subscript when it is needed 2. Alternative form ��⃗ �⃗ = �� Ø For VCE, more common form is this (note that this is not not net force, but average net force across time periof) ∆� �∆� � = = ∆� ∆� 3. It may be of use to consider 2nd law for different systems of bodies in a problem (and consider the com version of second law) 4. Application: airbag/crumple zone/padding Ø Given constant mass (e.g. person, car, driver in car) & constant change in velocity (from moving to becoming at rest due to collision), there is constant change in momentum Ø airbag/crumple zone/padding increase distance for deceleration, hence longer time for deceleration (one may support this more quantitatively, e.g. set constant deceleration, but it gets quite muddy due to realistic non constant deceleration) Ø Hence smaller average net force 5. Typical energy graph for SHM (equivalent to shifted equilibrium oscilation, where gravitational potential energy is containedFREEVCENOTES in potential energy)

6. Typical energy graph for spring motion where equilibrium has not being shifted & we are still counting gravitational potential energy, strain potential energy is calculated from unstretched position

NEWTON’S THIRD LAW

1. When one body exerts force to another body (action force), second body exerts equal force in opposite direction to first body (reaction force) • two forces are of same type • same magnitude • opposite direction • reaction force exerted on different object as action force. COM Ø exerted on different object part often confused: i.e. force from table on basketball cancels gravitational attraction on it, but theyare not action-reaction force pairs, for they act on same object. actual pairs are basketball’s force on table – force from table on basketball, & gravity of Earth on basketball – gravity of basketball on earth) . 2. Wheel explanation problem • Q: why does rotation of wheel on vehicles like bikes propel vehicle forwards/backwards? • A: wheels are made to spin clockwise/anti-clockwise (be sure to identify this), section of wheel in contact with ground receives frictional force forward/backward (be sure to identify this as well) opposing velocity of that section. = , hence vehicle is propelled forward/backward (be sure to identify this again) • For variants on this problem, take this rule of thumb: be as precise as possible, don’t skip logical steps.

SPECIAL CASE STUDY: FORCES ON CAR’S WHEELS

1. Forces typically included unless told to exclude (remember not to forget any of these) • Gravity: downwards • Normal force from ground: upwards • Friction on driving wheel Ø Realistically it depends on whether at instant which direction wheel is moving wrt ground (but since here we infer absolute acceleration/deceleration (braking) we disregard other scenarios) o e.g. friction from ground might accelerate so much that for moment driving wheels is moving backwards wrt to ground, then friction switches direction to help slow vehicle Ø Typical: acceleration whilst car moving forward: in direction of motion of car (against likely engine propelled spin of driving wheel) Ø Typical: braking (deceleration) whilst car moving forward (assuming brakes are on driving wheels): opposing direction of motion of car • Friction on non-driving wheel Ø Realistically it depends on whether at instant which direction wheel is moving wrt ground (but since here we infer absolute acceleration/deceleration (braking) we disregard other scenarios) Ø Typical: acceleration whilst car moving forward: opposing direction of motion of car (so to make this wheel spin like driving wheel) Ø Typical: braking (deceleration) whilst car moving forward (assuming both driving & non-driving wheels have brakes on them): opposing direction of motion of car (as if brakes are on non-driving wheels as well, they would behave like driving wheels) • Less typical forces, should still include unless told to ignore Ø Air resistance/drag Ø Force by car held up by wheels (which is result of car’s gravity, though not same as gravity) • Forces rarely included, generally not necessary unless told otherwise Ø Rotational force from some kind of force transmission system connected to engine • As for accelerate/breaking scenario for when car is moving backwards, friction forces are simply reversed by symmetry.

FREEVCENOTES

NORMAL FORCE

1. Normal force: reaction force normal to & exerted by surface • Inclusive of reaction and action forces (technically you should just forget about action reaction anyway, cos ‘cause’ force and ‘effect’ force are ambiguous) 2. Calculable for inclines via componetisation (so setting component equivalence to stuff like centripetal acceleration or make it equal to gravitational component)

GENERAL CIRCULAR MOTION STUFF

UNIFORM CIRCULAR MOTION

1. Uniform circular motion • Constant speed (not velocity, as velocity changes direction) • Constant magnitude of acceleration & force (only magnitude, as direction changes for both) • Centripetal force is net force 2. Centripetal acceleration

� 4�� = = �� = , ��

3. Centripetal force

�� 4�� = = �� = � , ��

4. Relations of period (of course uniform circular motion) 2� 2�� 2�� 2� � = = → � = , � = � � � � 5. Graviton (or centrifuges in general) • Used to make artificial gravity

CIRCULAR MOTION IN HORIZONTAL PLANE

1. Special case: String swung horizontal loop • Horizontal component of tension: centripetal force purely • Vertical component of T + gravity = 0 • Gravity & centripetal force may pythaged to find tension (use for checking as kinda beyond course) 2. Vehicle turning • Friction supplies horizontal centripetal completely COM • Normal force determines and related to how much friction may be obtained 3. Leaning into corners . • So that gravity cancels out the torque effects of the ground friction and normal force • Easiest derivation is via centrifugal force in frame of bike • The angle satisfies (note this is different from those of banked surface) �� (�) = = ��/� �

CIRCULAR MOTION ON BANKED TRACKS

1. Effects of forces (here assume 0 friction) • Normal force from track Ø Will be larger than on flat track (gravity is same, but now gravity is equal to component of normal force and not equal to normal force, so the new normal force is larger than the component and hence larger than the flat track normal force) o You’ll feel heavier from banked track (compared to if you are on a flat track and not leaning) Ø Horizontal component supplies centripetal force solely Ø Hence its vertical component must cancel with gravity �� = � = = sin(�) � � • Gravity

� = �� = cos(�)� • The combined eq. � � tan(�) = = FREEVCENOTES� �� → � = ��(�) � → � = ��(�) 2. Possible question change • Inclusive of friction force • Vertical component of friction + of normal force + gravity sum to 0 • Horizontal components of them (so of friction and normal force) sum to centripetal force 3. Means of improving safety around bends for vehicles • Slow speed (reduce centripetal force required) • Bank the curve (so normal force help to supply centripetal force as well) • Better traction on wheels (increase friction for centripetal force) • Increase radius (reduce centripetal force required) • Lower centre of gravity (so less likely to topple over due to lack of centripetal force)

CIRCULAR MOTION IN VERTICAL PLANE

1. Note below generally include closed interval (if we are to consider lose contact, then should do open interval) 2. Loop de loop (below generally refer to magnitudes) • Explaining why contact is met

Ø Use argument of � is required to keep in circular motion, hence must be in contact • Apparent weight (note ‘apparent’) at bottom

� = � + � • Maintaining contact with track at top

� ≥ � (�� ℎ��ℎ �� ℎ� = � ≥ 0) 3. Hill motion • Top of hill

�� : � ≥ � (�� ℎ��ℎ �� ℎ� = � ≥ 0) • Bottom of hill � = � + � COM PROJECTILES LUNCHED HORIZONTALLY

1. Formula for range .

� = �� 2. Formula relating t and vertical distance

1 2� 2ℎ � = �� → � = = 2 � �

PROJECTILES LAUNCHED OBLIQUELY

1. Basic parameters • Position of landing (height) from initial launch • Position of landing (horizontal) from initial launch (range) • Above positions may be shifted by value due to defining initial position as something • Time of flight • Angle of launch � • Initial velocity v 2. Formula for range in same height initial & end landing (do not use squared form in working, VCAA dislike) � sin(2�) � = � • Pronumerals Ø g = gravitational acceleration (or any uniform acceleration perpendicular to horizontal motion & opposing vertical launch component of projectile)) • Derivation

� = �� FREEVCENOTES1 0 = � = � � − ��, �ℎ�� = 9.8�� ℎ�� 2 1 2� 2��(�) � ≠ 0 → � = �� → � = = 2 � � 2��(�) 2 sin(�) cos(�) � � sin(2�) � = ��(�) ∗ = = (�� ) � � � 3. Formula relating vertical maximum height to parameters � − � −� � + � � = = (� �� ) = � 2� 2� 2 � = � 2 � � �� → �� max ℎ��ℎ�, � �� → � = = �� 4. Range

� = �� 5. Finding time of flight given max height, initial height, end height • Method 1 (recommended as avoid quadratic formula) 1 1 � = � + � , �� − �� = 4.9� = � − � , �� + �� = −4.9� = � − � 2 2 �� ℎ� �� 2 �� ℎ� �� • Method 2 (perhaps for checking) � � � − � = − = → �� 2� 9.8 ∗ 2 1 �� + �� = � − � , �� , �� = −9.8 �� + 2

CONSERVATION OF ENERGY

1. Nuff said.

CONSERVATION OF MOMENTUM

1. Conservation of momentum • Momentum is conserved in all interactions COM • Direct result of Newton’s 3rd law (or you may argue that 3rd law is result of conservation of momentum). OTHER

EQUATIONS OF CONSTANT ACCELERATION

1. � = �� + �� • Derivation: trapezium area from adding triangle to rectangle 2. � = � + �� 3. � = �� − �� • Derivation: trapezium area from subtracting triangle off rectangle 4. � = � + 2�� • Brief derivation: � − � � + � � − � � − � � = → � = = → � = � + 2�� 2 � 2� 5. � = (� + �)� • Derivation: think trapezium area 6. Side note: beware that directions are important here, sign ± on velocities, acceleration & displacements may entail significant differences in calculated results.

SPECIAL RELATIVITY

EINSTEIN’S THEORY OF SPECIAL RELATIVITY

INERTIAL FRAME OF REFERENCE

1. Frame of reference: • 4D coordinate system essentially (inclusive of time coordinates) 2. Inertial frame of reference: • Can be determined through pendulum test (for ones other than gravity) • Frame at constant velocity (not just speed, but direction too) wrt to another inertial reference frame (note this latter part, wrt to another inertial FREEVCENOTESreference frame, so in the end the definition is still realistically the Newton’s law idea) Ø Caution: not equal to travelling in straightline (travelling in straight line with same direction and magnitude always is more accurate) • An non-accelerating frame of reference (try avoiding this as acceleration is defined wrt to some reference frame really) • FOR in which physical laws of Newton’s hold (principle of relativity) Ø e.g. law of inertia, hence why Newton referred them as inertial FOR 3. No observer in the frame can reasonably conclude that the frame is accelerating (or even moving really) 4. On earth closest to inertial is poles 5. Einstein (whose relativity findings of course had some contributions from others) often did thought experiments ‘Gedanken’ experiments (German name of such experiments) • Normal of theoreticians.

GALILEAN RELATIVITY

1. this means things like:

� = � + � �� 2. Frames are equally unique, i.e. there is no such thing as absolute velocity & ‘correct’ frame of reference. 3. When travelling with inertial frame (i.e. const. velo.), impossible to know whether one is moving or not, unless if one predetermines some reference as being moving or not moving. 4. However, telling whether one is accelerating is doable. 5. Aether (in sense of ethereal substance): believed to be some sort of massless rigid medium that carried electric & magnetic fields emanated space, even in places of vacuum of space in which light may propagate (this was introduced to help resolve conflict of Galilean relativity & absolute speed of light, which kinda became abandoned theory) 6. Einstein abandoned idea of aether, & approached issue with absolute speed of light & Galilean relativity conflict via his special relativity. 7. In frame S’, which is at velocity v to frame S, following transformations apply given that frame S’ origin coincide with S at t = 0: • Space & time coordinates for events, given that frame relative velocity is along direction of x axis purely

� = � − �� = � • Velocity relativity

� = � − � • Explanation: object originally moving at velocity v would be at rest in S’, same velocity change would apply to all objects, hence –v to any velocity 8.

SPECIAL RELATIVITY

1. 2 postulates of Einstein (statements assumed to be true) • Laws of physics are same in all inertial (non-accelerated) frames of reference (means no preferred frame of reference & so is sometimes called: no law of physics can identify state of absolute rest) • speed of light in a vaccuum has constant value for all observers at rest to inertial frames of reference Ø Directly explains (or rather states) result of Michelson-Morley experiment Ø Direct result anyway from maxwell’s equations (depend only upon electromagnetic properties of medium (and not relative speed of )) 2. These two in eyes of early Newtonian mechanics were contradictory, below are 2 postulates of Newtonian mechanics recorded by Newton: • Absolute, true, & mathematical time, of itself, & from its own nature, flows equably without relation to anything external • Absolute space, in its own nature, without relation to anything external, remains always similar & immovable COM 3. For purpose of 34, special relativity should not be extended to accelerating frame of reference 4. Einstein referred to these postulates, & chose to ignore them, hence in special relativity, such postulates were broken..

MICHELSON-MORLEY EXPERIMENT

1. Attempt to measure aether, so the ‘correct frame’ (as how frame of air stationary is the ‘correct’ frame of sound propagation) 2. Earth in orbit around Sun, aether wind (analogous to air blowing) should be blowing past Earth. (so should see light speed difference) 3. American physicist Albert Michelson thought such aether winds may be measured through small changes in speed of light as Earth changed its direction of travel in orbit, in line with Galilean relativity

FREEVCENOTES

4. Apparatus used measured small differences in time taken for light to travel in 2 mutually perpendicular directions, & then had it rotated with hope of detecting small difference due to how one of directions would be travelling in line with Earth direction & other perpendicular, but no difference. This suggested either no such aether, or Earth was travelling alongside aether. 5. 6 month later, Earth travelling in opposite direction, measured again with apparatus, still no difference, so Earth is either at rest in aether no matter which direction of travelling (which is quite unlikely) or no aether at all existed. 6. Null results made many argue that experiment or theory behind it flawed. Einstein chose to introduce theory of special relativity (specifically the constancy of light as a postulate and discard the special aether frame via principle of relativity)

SIMULTANEITY

1. Simultaneity that rely on light, which has absolute velocity & disobeys Galilean relativity, depend on observer (this idea accounts in observer’s calculation of time of transmission of light influence). This is illustrated in Einstein’s Gedanken train: • Consider light bulb in middle of train carriage, it is turned on & off immediately to just give pulse, & with no surprise light reaches two front & back walls at same time, hence two events are simultaneous in observer’s frame of train • Consider observer outside moving train. To him/her, light ray from bulb after leaving will move in equal absolute velocity either way front & back. But since train is moving forwards, front & back walls will meet light rays at different times. Hence, simultaneity is broken. 2. Difference in simultaneity suggested issue with time itself, namely time flow may not be uniform as in Newton’s postulates across different reference frames, & also space & time are seemingly correlated, unlike Newton’s assumptions, hence comes spacetime coordinates: • Spacetime coordinates include time as part of 3D dimensional coordinates, i.e. sets universe as 4D relationship. 3. Experimental evidence that falsifies Newton’s laws, but not new Einsteinian mechanics: • First, we note instruments that are able to measure such false: Atomic clocks: Ø one of most precise measurements of time, from 1967 onwards (instead of past definition relying on fraction of Earth rotation time), one second is defined to be 9192631770 oscillations of 6s electron of Cs-133 isotope, used in atomic clocks to record time (these may be as accurate as 1 second difference in 1.4 million years) Ø The placed on satellites show time discrepancy to earth (of course, orbit is accelerated frame, so it gets slightly more complicated) • Long lived muons (note muon are different from pion)f Ø When certain unstable particles, i.e. pions (which have precisely known decay rate), are accelerated to almost speed of light, their life span are measured to be longer than when particles are stationary. Ø Mean stationary life of position pion is measured by stationary atomic clock as 26.033 ns Ø Mean lifetime when moving at 99% speed of light for positive pion relative to stationary atomic clock is 184.54 ns (time dilation) Ø Mean lifetime of stationary muon, measured by atomic clock: 2.198 µs Ø Muons created by cosmic radiation interacting with nuclei of oxygen atoms 15 km above surface of Earth however at 99.97% speed of light are predicted by Newtonian mechanic (using stationary decay time) to travel only about 659 m (using s=vt), but in reality such muons are detected on surface of Earth, hence can’t have just travelled 659m Ø These suggest either speed has some effect on decay rates, or that Newtonian mechanic in itself, whether its assumptions of time uniformity, or space uniformity, are false. (here, special relativity solves issues without resorting to some more complex discussion of speed relation with decay rates, & it is consistent with many other observations without residing to adding new separate patches to existing phenomenon) o any theory can be made as universal as one wants through adding exceptions & special phenomenon cases or adjusting on relevant theories, but the more of those the less elegant the theory is COM RELATIVISTIC MASS . 1. Objects apparently gains mass when moving at higher velocities, derivable through viewing collision of particles in different inertial reference frames & applying conservation of momentum in each. • Note the magnitude of quantities like energy and momentum are different across different inertial frames of reference 2. Relativistic mass: (Einstein though dislike this, he prefers relativistic momentum (after all, there’s the mass-energy equivalence))

m = γm Where m �� 3. May be used on 2nd law (this is greatly simplified for 34, realistically the non time independence of gamma makes it much more complicated)

� = �� • May arrive from more general relativistic momentum (for Phys 34, the produyct rule inclusion of gamma derivative is ignored, so you just do � =

��) �� (�� ) � = = , �� (�ℎ�� ) → � = �� = ��, � = �� 4. Revisiting theory on magnetic centripetal force on charged particle moving at right angles to magnetic field, calculated radius of resulting uniform circular motion is: mv r = qB • i.e., given mass, charge, & field strength constant, radius is proportional to velocity • However, in reality, particles in cyclotrons & synchrotrons which have velocity increase leads to radius increase to far greater extent than expected • If charge & field interactions are assumed to not change due to speed of particle (again, may add exceptions, but that is just increasing complexity of theory which has far simpler alternative), simple explanation is such that mass, or at least apparent mass (from momentum) changes upon v change. • Electrons travelling at 99.99999% speed of light appears to have 6000 times mass of stationary electron (from relativistic momentum)

TIME DILATION

TIME IN DIFFERENT FRAMES OF REFERENCE

1. When moving at high velocities, relativistic time effects becomes measurable, i.e. in space missions & even in supersonic flight, particles in particle accelerators 2. To observe time dilation in moving train, need reference clock in what is regarded as rest frame, & clock in moving object. 3. Observers are all right about their times, idea is that time can only be measured relative to some particular frame, & not in any absolute sense. 4. Due to basic postulates of absoluteness of light velocity, simple measurement of time that demonstrates easily time dilation is to use light clocks, i.e. light mirror clocks as shown below: • By simple application of two such clocks, one at rest in reference frame & one moving, one may see that distance travelled by light ray depending on FREEVCENOTESvelocity of clock may change, but due to absoluteness of light velocity, this means that time between each oscillation must change, but then if we move to frame of moving clock, time taken changes again as relative velocity of clock is now 0. Hence, one may see that objects seem have different time flows depending on their velocities. COM .

5. Time dilation derivation based on such light clocks: ( clock on train is one of concern, we based argument on design being oscillation based on mirror reflection in direction perpendicular to movement of train (so to simplify issue of needing to consider how mirrors also move & relate to oscillation times)) (This essentially a special case for deriving general case phenomenon) • Moving train observer case:

Ø d = ct • Stationary observer case: Ø d = d + � ∗ � = � � • Substitution & association of time in two cases: Ø c t + � ∗ � = � � FREEVCENOTES � → t = � 1 − � • You see that this above can be applied not just for trains, but generally for any relative motion. Note also that in all such scenarios, there is common reference object that is constant, time measured in train (or whatever moving object), called proper time (timed measured in frame where events occur at same space coordinates (or more specifically in same space coordinate along the axis of relative frame motion, not necessarily the other space coordinates though)) 6. Time dilation formula (essentially same as above, though made to be more general) � t = ↔ � = � � � 1 Where γ (gamma) = lorentz factor = � 1 − � t = �� , � �� ℎ� ��ℎ�� , �. �. �ℎ� �� c is speed of light in vaccum, c~299792458 ms v is relative speed of stationary frame & moving frame (for proper time) • Lorentz factor increases for velocity, hence proper time is always shortest, i.e. when you look at clock in moving object, where as you feel long time has passed, over in frame of object clock has went through much shorter time, hence it seems as if clock there is slower, i.e. time flows slower in proper time frame than other frames

LORENTZ FACTOR

1. Sometimes useful to make v subject in Lorentz factor equation, derived from proper definition, not approximation for low velocities: 1 v = c1 − γ • Another rearrangement putting c as subject reveal way of finding speed of light c, through say finding Lorentz factor through measuring time dilation in two reference frames directly through clocks. 2. Approximation of Lorentz factor for low velocities 1 � For low velocities: γ ≈ 1 + 2 � �� (1 − �)~(1 − ��)�� ≪ 1 (�ℎ�� ℎ� �� , �� (�� )) � 1 1 � �� = , � = − → �~1 + ( ) � 2 2 �

TWIN PARADOX

1. Say there is twin on some stationary inertial frame (I chose not to use Earth as that isn’t really inertial is it?), & another twin on spaceship. spaceship travels out, & twin there seems to age slower to twin on earth. But then again to twin on spaceship, twin in stationary inertial frame would also seem to age slower, so why is that? • The resolution of this is merely relativity of simultaneity 2. simpler version: same setting as above, though twin come back to meet twin in stationary inertial frame. Now, both would have seen each other age slower, so who when they meet are older? • Resolution: one twin has always been in same inertial reference frame, other has changed his/her inertial reference frame Ø This may well be resolved in special relativity as well via length contraction considerations (may be easier to reformulate qCOMuestion to have 2 ships, one for outbound and one for inbound and they switch time readings at intersection to throw away the acceleration thing) • answer is that twin on spaceship aged slower than twin on Earth . 3. Special relativity of 1905 by Einstein refers to only inertial frame of reference (constant velocity), general relativity in 1915 by Einstein refers to non-inertial frames of reference as well. (special relativity relies much on works of previous scientists, though Einstein did ring about significant physical meaning to it, as for general relativity, well, it is essentially just Einstein’s work, & is genius)

INVARIABILITY OF RELATIVITY IS BREAKING POSSIBLY

1. Some research suggests, based on lights from very distant quasars, that there have been small changes in fine structure constant (made up of speed of light, elementary charge & Planck’s constant) 2. If so, than fundamentals of theories will need to be modified, & potentially open new realm of physics, as did absoluteness of light speed bring about relativity.

LENGTH CONTRACTION

LENGTH IN DIFFERENT INERTIAL FRAME

1. Depending on frame, length measurements change for object 2. Equation: L L = ↔ � = �� • L is length in moving frame, in which entity is moving

• L IS proper length, which is length of some entity measured in reference frame in which entity is at rest. • Where γ (gamma) = lorentz factor = 3. derivation of above equation • Consider train platform & train setup use for deriving time dilation:

• Platform frame:L = � = � ∗ � • Train frame:

v� L = � ∗ � = � � = �

• Combining two equations for two frames: � L = FREEVCENOTES � 4. Lorentz factor increases with velocity, hence for observers in reference frames of velocities, their observed length of object will be shorter than those in rest frame. Hence length contraction.

DEFINING PROPER TIME & PROPER LENGTH

1. Proper time: time between two events in inertial frame such that two events occur at same point in space • You may however still apply a makeshift ‘proper time’ formula if the events occur with same coordinates in direction of relative frame motion (wouldn’t be the real proper time, but the maths expressions work out for cases of such relative frame movements) 2. Proper length: distance between two points in inertial frame such that two points are at rest.

RELATIVITY OF VELOCITY (NOT PART OF COURSE) � − � � = �� 1 − �

LORENTZ TRANSFORMATION (NOT PART OF COURSE)

� = �(� − ��) � = � � = � �� = �(� − ) �

RELATIONSHIP BETWEEN FORCE ENERGY & MASS

IMPULSE

1. Change in momentum: m∆v F = ma = = �∆� → ∆� = � = �∆� ∆t • Realistically, since F varies alongside acceleration, impulse calculation usually relies on average force, average acceleration, average speed change over time, etc. • Units: N �, �� , �� (�ℎ� �� ) • N s should be used for impulse, kg m s �ℎ�� ℎ �� ive formulae used COM • By third law (& conservation of momentum) ∆� = −∆� Ø Can be used to find change of momentum of one body if parameters of only another body is known. 2. Units: • Momentum by itself: �� (due to it being considered property when object has velocity) • Impulse: � � (due to it being considered calculated via force applied over time)

3. Implication of impulse: • High impulse, short time period, means large force exerted on average. Hence why car crashes are less catastrophic for longer crash time used to stop. (hence why cars have crumple zones (parts of cars, often in front, that literally ‘crumples’ during crashes so to extend time taken for especially people in vehicle to come to rest during crash, of course, these zones end with rigid passenger compartments to avoid other gruesome consequences) & airbags), below is typical force- time graph comparing air bag & no air bag car collision: • Concertina effect/accordion effect/slinky effect/elastic band effect (not part of course) Ø Occur when fluctuation in motion of body cause disruption in flow of elements following it

FREEVCENOTES

FORCE VS TIME GRAPHS

1. For time as horizontal axis, force as vertical axis, as shown in example below:

• Impulse is equal to area under curve, or more accurately, between horizontal axis & curve • As for when time is vertical axis & force is horizontal axis, impulse is area between curve & vertical axis.

• above can also be found via considering formula of I = F∆� & referring to average value of curve principle in calculus

WORK DONE

WORK DONE CALCULATION

1. W = �⃗ ∙ �⃗ = Fs cos (ϑ) (sometimes s is written as d) • Units: J (equivalent to kg m �) • Brief Derivation (for constant acceleration, note that this can be applied to infinisimal, hence in way used for general case) 1 � = � + 2�� → �(� − �) = �� → ∆� = �� → � = �� 2 • Alternative U34 Phys form � = �� Ø Where F is component of force in direction of d, or alternatively, d is component of displacement in direction of F 2. Work is scalar quantity 3. In cases force may be applied yet no work done (implies net force is different from force applied, that the object does work COMon some other object) 4. Work is only done on object when force or component of force is applied in direction of displacement, i.e. in uniform circular motion centripetal force which is perpendicular does no work. .

FORCE DISTANCE GRAPHS

1. Example:

• Work done in such graphs = area between curve & extension axis • Area is between those axis for � = �� → �� = ��, ��

WORK & ENERGY

1. Both work & energy are scalar quantities 2. Kinetic energy formula: (scalar) 1 E = �� 2 • May be thought as energy required to bring mass m at rest to velocity v (from that derive using definition of work (which is kinda circular reasoning of course), depending on what one uses as basic axioms) 3. Satellite energy FREEVCENOTES& velocity calculations for uniform circular orbit: ( kinetic & potential energy can be used to calculate minimum energy required to launch satellite into target orbit) • Kinetic energy • Potential energy • Orbit speed (not velocity due to how velocity as vector in uniform circular orbit in changing directions)

STRAIN POTENTIAL ENERGY

HOOKE’S LAW

1. F = −�∆� • Negative accounts for restorative force, i.e. against extension • ∆� is compression or extension of spring • � �� , �� 2. In Phys 34, springs unless otherwise told are generally considered to be ideal, i.e. massless, no damping loss, follows Hooke’s Law perfectly 3. system of 1D attached ideal spring parallel/series on object, whether in front or behind, still leaves object following Hooke’s law • Follows from some mass, namely fact that linearity defining Hooke’s law holds regardless of what extension is 4. Techniques/Special cases • Influence of any uniform force: Ø equilibrium point is shifted. Ø restorative force proportionality is still maintained Ø Hence Hooke Law (& strain potential energy) still applies, only that displacement is now calculated from new equilibrium as opposed to old, & since that uniform force (e.g. gravity) is accounted in spring force already, gravitational potential energy too is now not needed when considering potential energy, it is considered as part of spring energy 5. Strain potential energy (also called elastic potential energy) • Equal to area under Force extension graph (specifically between curve and extension axis), i.e. work done on spring:

• For spring following Hooke’s law, integration shows energy to be: 1 � = �∆� COM 2 6. Elastic limit: end of elastic region of material (refers to how region where Hooke’s law apply is called elastic behavior, .& will return all of strain potential energy when applied force is removed (in comparison, plastic region will not return to initial state & release all of energy, instead it will have some deformation due to work done being used on changing material • Recline after elastic limit broken (permanent deformation)

7. Breaking point: when material fails (this & elastic limit of material is demonstrated below)

FREEVCENOTES

NEWTONMETERS

• Essentially spring (uses 3rd law + Hooke’s Law to find force exerted) • force it measures is strictly tension, since newton meter, once in equilibrium, effectively acts as string in which force across it is tension. • Implication:

1. COM 2. Spring force = string tension (if spring replaced with string) 3. In all three cases above, same tension, hence spring scale (newtonmeter) measures same force . 4. Remember, spring exerts equal force to both ends • If not, by third law, the connections on both ends of string will have unequal force (we consider spring as a system), the system will thus not be in steady state, contradiction

KINETIC & POTENTIAL ENERGY

SOME DEFINITIONS

1. Elastic collisions: Kinetic energy is conserved; no dissipation is forms such as heat. 2. Inelastic collisions: Kinetic energy is not conserved; dissipation in forms such as heat occurs 3. Mechanical energy: • sum of kinetic energy & potential energy (constant unless work done by external) • For purpose of phys, this usually contains conservative forces for potential energy 4. Potential energy: • potential of force, i.e. gravity, spring’s restorative force, to do work on object 5. Energy may be dissipated in events such as collisions as sound, heat, light, etc.

STRAIN (ELASTIC/SPRING) POTENTIAL ENERGY

1. Formula • � = �(∆�) 2. Brief derivation • Take integral, i.e. area below force (dependent variable) distance (independent variable) (compression/extension) curve (Note that F & ds will be on different directions) ∆ ∆ 1 ∆ 1 � �� = �∆� = �(∆�) = �(∆�) 2 2

CONSERVATION OF MECHANICAL ENERGY

Ø ∆� = � • If this external force is conservative in some way, then in some way it may be incorporated into system & same argument applied, but typically this is FREEVCENOTESignored in VCE. • Corollaries (note that gravitational component is only concerned if spring mass system is vertical, i.e. parallel to gravity) 5. When external force is 0, following applies (note here we take up as positive, down as negative for ∆�) 1 1 0 = ∆� = �(∆� − (∆� )) + ��(∆� − ∆� ) + �(� − �) 2 2 6. more redundant (since often two such equations will need to be equated to find unknowns) method, equivalent, but not suggested 1 1 � = �(∆�) + ��ℎ + �� 2 2 Ø Where h is taken from some reference altitude where h=0 Ø Shifting equilibrium point (best & quickest method) (try avoiding for working as not part of course, but useful for confirmation) o Takes account of uniform gravity on attached mass, hence after shift gravity can be disregarded, hence gravitational potential energy in � calculations can also be disregarded. o Relies on fact that Hooke’s Law requires only � ∝ − , & nothing to do with where equilibrium point is. ∆ o same spring constant applies, (shift of equilibrium does not affect proportionality factor) o point of equilibrium �� + � → �� − �∆� = 0 → ∆� = , �� , �. �. �� o mechanical energy conservation theorem after shifting equilibrium 1 1 � = �(∆�) + �� 2 2 1 1 ∆� = � ∆� − (∆� ) + �� − � = 0 2 2 7. Brief proof of above with form before shifting (note here we take up as positive, down as negative for ∆�) Ø It is sufficient to prove following (that change in energy of following parts of two equations are equal ( other park, KE is shared, so no need to discuss)) (algebra reveals consistency) 1 1 �(∆� − ∆� − ∆� − ∆� ) = � ∆� − (∆� ) − ��(∆� − ∆� ) 2 2 � = , �� ℎ�� ℎ� �� to make g used in k & g used in GPE consistent (usually, since W=- ∆ ∆�, mg∆h takes g to be positive (negative of downward direction), since we here require to divide both side by g, it seems

reasonable to make g negative so to make division doable without negative 1 factor, hence −��(∆� − ∆�) is not

contradictory with usual +∆�, as here, ��∆ℎ = � due to us using original g as opposed to made positive version, hence

−��∆� − ∆� = −� = ∆� as desired �ℎ� �� , ℎ�� intended proof is complete. Ø May also be proved via F – d graph, where shifted equilibrium effectively shifts up/down new d-axis. Area consideration then reveals above.

Ø two are equal, hence original ∆� = 0 theorem would also work. Ø TQ shortcut for finding KE at maximum velocity (does not work if initial position of object is not at natural unextended position or reflection of that unextended position wrt to equilibrium point) o � = � in any oscillation (& it occurs at equilibrium position) o ��, �ℎ� � must be value from natural unstretched position of string, not shifted equilibrium position o ��: (WLOG for simplicity we place reference point for � at bottom of oscillation, below relies on g=9.8, i.e. for calculation of k negative factor must be added, though not needed for + �) 1 1 1 �(∆�) + ��∆� − 2∆� + �� = � = �2∆� (�� ) 2 2 2 COM 1 1 �� ℎ�� (∆�) + ��∆� − 2∆� �� 2 2 �� (−�) . → �ℎ�� ∆� = − , �� ℎ�� ℎ�� = , �� ℎ� �� 2� ∆� → ∆� = ∆� 1 1 1 �∆� + ��∆� − 2∆� + �� = �2∆� 2 2 2 �� 1 �� → − ∆� − ��∆� + �� = − ∗ 4 ∗ ∆� = −2��∆� 2 2 2 1 → �� = −��∆� 2 1 �� ℎ�� : � = �2∆� = −2��∆� 2 1 → � = � 4 �� Ø No need to prove max velocity at equilibrium point, just sub in straight (unless if question asks for show that, or explain that, or if question has lot of marks)

EINSTEIN’S MASS-ENERGY RELATIONSHIP

TRAVEL AT SPEED OF LIGHT

1. At speed of light, distances shrink to 0 & time flow stops 2. No ordinary matter can reach c due to mass dilation into infinite momentum (similarly, infinite kinetic energy), but light always travels as c (i.e. for light there s no time) 3. All of light’s spacetime travel is through space (well, not really as distances is 0) & none through time.

RELATIVISTIC MOMENTUM

1. Consider equation F∆t = ∆p, & since time is relative quantity, it makes sense that momentum is also relative quantity. 2. Relative momentum equation: 1 p = γmv = γp, where γ (gamma) = lorentz factor = , � = �� (�) � FREEVCENOTES1 − � � � � � ( ) → � 1 − = � � → � = � � + → � = → � = �� ± �� + � + � � � → � = � �1 + ��

p �� = �� • Derivable from F∆t = ∆p equation (says text, text, though seems dodgy)

�∆� = �∆� (�� ) ∆� ∆� = (�� ) � �∆� = ��∆� (�� , �� (�ℎ��ℎ �� 34 �� )) 3. High temperature gas • Apparent mass increases for higher temp (as particles have greater relativistic momentum) 4. Explanation of speed limit • For mass object Ø Energy required due to mass dilation tend to infinity → impossible • For general Ø Lorentz factor undefined beyond speed of light (so even if exists, it is a phenomena beyond the frameworks of special relativity (which most importantly contain a causality component that mean faster than light leads to dodginess, i.e. time travel)) 5. Einstein liked speaking of relativistic momentum than mass

COM .

6. Relativistic velocity � − � � = �� , �ℎ�� ℎ�� , � �� , �. �. �� 1 − �

THAT CERTAIN EQUATION

1. Mass-Energy Equivalence

E = mc

• Energy is frame dependent (as is in classical mechanics) • Revisiting classical kinetic energy momenta relationship: E = �� = �� = • Einstein showed from relativistic momentum & usual assumptions about work, forces & energy that classical expression for kinetic energy was not correct at high speeds, instead, it should be this: E = (� − 1)�� = γmc − �� Ø For small velocities, above reduces to classic �� Ø Even for up to 0.10c, classical expression is accurate to better than ±1% • Rewriting relativistic energy yields: γmc = � + �� = �� Ø Einstein interpreted lhs as total energy, hence: E = �� Ø → � consisted on kinetic energy & some other part depending only on rest mass, hence inferred mc = rest mass energy 2. Equation means mass-energy equivalence, hence setting foundations for predictions of nuclear reactions. 3. Example of fission: Uranium atom split into 2 fission fragment, ~ 200 million eV of energy released (most chemical reactions involve just few electron volts) 4. Mass defects in nuclear reactions & released energy corresponds to predictions of Einstein’s equation 5. Nuclear fusion in Sun generates about 4 million ton of mass into energy �. • FREEVCENOTES

NUCLEAR FUSION

1. When 2 nuclei are combined to form larger nucleus (usually light nuclei):

2. Mass of reactants is slightly greater than product mass, hence releases energy (some fusions are of course endothermic, e.g. fusion of 2 nuclei iron) 3. Energy released follows relation with mass defect: ∆E = ∆mc 4. Fusion is difficult as positively charged nuclei repel each other, need to get close enough so that strong nuclear force overcomes electromagnetic repulsion & any other repulsive forces (e.g. electromagnetic force)

5. Typically, temperatures of orders of hundred of millions of degrees and extreme pressures required for fusion (high � helps overcome energy barrier set by stuff like EM force), as are conditions inside sun. 6. Fusion has already been achieved, it’s the scalability and self-sustainability the issue (I mean, you have table top fusors already) due to the plasma that would usually come as a result • Plasma (atoms ionised) pretty much consequence of energy required in fusion (when such large energy, can’t really exist as anything else other than plasma)COM 7. Fusion takes place in many stars 8. Sun is second/third generation star, formed from remnants of other stars that exploded much earlier in history of galaxy.. 9. Electronvolt (eV) • energy gained by 1 elementary charge (not equal to electron charge which is negative) across potential of 1V 1�� = 1.6 ∗ 10� �� 1� = 1.6 ∗ 10 � 10. Stars maintain fusion through strong gravitational attraction of its giant gas clouds, which establish extreme pressure & temperatures at core. 11. Fusion reaction in Sun (for past 5 billion years, & expected to continue for another 5 billion years): Ø This process releases approx. 25 MeV of energy (quite sure this is for a 4 protons to make helium 4) (of course this is just some of the fusions in sun, not all)

MAGNETISM & RELATIVITY

1. Magnetism is relativistic phenomenon 2. Magnetism is force related to movement of charge, i.e. it is velocity dependent. I.E. change inertial reference frame and it should disappear, but it doesn’t (hence why Einstein’s paper regarding special relativity was named ‘On the Electrodynamics of Moving Bodies’) 3. Consider two wires with current, two wires should move towards each other. Now consider frame of current, i.e. when electrons are at rest, such magnetic force is 0, i.e. only electrostatic force left (which doesn’t really do anything since wires aren’t really net charged due to positive ions alongside electrons), but still wires should move towards each other in same way (whether they move toward or not should not depend on frame). (below does the same direction current case) • These all rely on the idea that ther wires are not net charged in the rest frame of the wire (so positive charge), this is fine as in this frame, the free electrons can move so to balance it (in other frames, it is the positive charges, which despite moving cannot free move so they do not necessarily balance charge) • ExplanationFREEVCENOTES: due to length contractions of different frames, density of positive charges in wire appear different in different frames. • The positive charges in opposite wire seem contracted, whilst the electrons in that wire at rest • Density of positive negative charge different (to if a frame where positive charges at rest) • In this charge at rest frame there suddenly exist now electrostatic force • This is what substitutes the ‘magnetism’ effect in this frame (this effect, despite slow speeds of electron, still is considerable due to the large number of charges) • Hence a further theoretical explanation of why electromagnetism are essentially 2 aspects of same phenonmenon • For opposite direction current flow Ø Electron in other wire moves even faster, length contracts further, electron density larger, and its density increase is to a greater degree than positive charge increase cos lorentz factor change is larger for larger velocities

PROPERTIES OF MECHANICAL WAVES

LONGITUDINAL & TRANSVERSE WAVES MECHANICAL WAVES

1. Waves that need medium, i.e. sound waves require mediums such as water & air 2. Transfers energy from one place to another through medium 3. Particles vibrate back & forth or up & down about average position 4. There is net transfer of energy but no net transfer of matter (i.e. ocean waves carry much energy to shores, but most of ocean’s water doesn’t travel to shores)

PULSES VS PERIODIC WAVES

1. Pulse: single part of wave, single disturbance 2. Periodic waves: multiple pulses periodically 3. difference between pulse & periodic waves is shown below:

COM TRANSVERSE WAVES . 1. See image:

FREEVCENOTES

2. Wave displacement is perpendicular to direction of travel 3. Crest is upper peak 4. Trough is lower peak 5. Example, water waves (this is however approximation, practically, both transverse & longitudinal waves are present in water waves, i.e. cork bobbling in gentler waves will both go up & down as well as forward & backwards (very slight though, so effectively it is fine to treat water as transverse waves))

LONGITUDINAL WAVES

1. See below:

2. Rarefactions: areas of expansion 3. Compressions: areas of compression 4. Often, longitudinal waves are made into equivalent transverse waves, with peaks as compression (high pressure) & troughs as rarefactions (low pressure) 5. Vibration, wave displacement is in same direction (more accurately, parallel to) travelling direction of wave 6. Example, sound wave, see below for visualization:

COM .

FREEVCENOTES COM .

FREEVCENOTES MEASURING MECHANICAL WAVES

BASICS

1. Features of waves: • Amplitude: maximum displacement of particle from average or rest position (usually, this is half distance between peak & trough, i.e. for your typical sinusoidal waves, consequently typical total distance particle will move through in complete cycle is twice amplitude) • Wavelength (λ): distance between any two successive points in phase (i.e. 2 peaks): unit: m • Frequency (f): number of complete cycles that pass through given point per second: unit: Hz • Period (T): time takes for complete cycle to pass through given point: unit: s Ø T = ↔ � = • Speed 2. Displacement-distance graphs: (often for transverse waves) • Refer to wave at particular point in time for different locations • Example:

3. Displacement-time graphs: (again, often for transverse waves) • Refer to displacements of specific point of wave across different times • Example:

COM .

4. Direction of motion of particular particle can be determined from displacement-time graph 5. Direction of motion of wave can be determined from displacement-time graph 6. Amplitude & period can be determined from displacement-time & displacement-distance graphs

WAVE EQUATION

1. v = fλ, v unit is m s

DOPPLER EFFECT

1. Phenomenon of waves whenever there is relative movement between source of waves & observer. 2. Source of wave away from observer: apparent decrease in frequency • source moving away: apparent wavelength increase, speed same, hence frequency decrease • observer moving away: apparent speed of wave decrease, wavelength same, hence frequency decrease 3. Source of wave towards from observer: apparent increase in frequency • Source moving in: shorter wavelength, same speed, hence frequency increase • Observer moving in: quicker speed, same wavelength, hence frequency increase 4. Apparent frequency affected, though not actual frequency 5. Doppler effect calculations: FREEVCENOTESv � ± � � ± � f = = = � λ �� ± �� ± � • Determining ± easier via examining specific case than set formula to avoid miscalculations 6. Example, pitch of incoming car siren sounds higher (increasing frequency), pitch of outgoing car siren sounds lower (lower frequency) 7. Approximation for when speed of source & observer are small relative to those of wave: ∆v f = 1 + f , ∆� = ±� ± � v • Derivation � + � � − � � = � = � 1 + � + � � + � � ≫ �. ��, �� ℎ �� ℎ�� ∆� ⇒ � ≈ � 1 + , �� ⇒ � > � �� , �� , �� • ± determinable by case, i.e. any thing that give relative motion towards has +, which corresponds to increase in frequency, vise versa. 8. Lord Rayleigh’s predicted effect: source moving at double speed of sound, musical piece emitted by source would be heard in correct time & frequency, but backwards

WAVE INTERACTIONS

MEDIUM CHANGE

1. When medium changes, i.e. from water to air, energy carried may undergo 3 processes: (note though as mediums are hardly ever completely uniform, such processes may well occur in the same approximate material) • Reflection: Ø One type of reflection (without phase reversal)

Ø Another type of reflection (with phase reversal, i.e. shift in phase of π, or 180°, or shift in phase of ) COM .

Ø Magnitude of reverse transverse waves are usually not same as original wave due to how some energy are absorbed by post or transmitted (i.e. continue) through new medium • Absorption by new medium: (& converted to forms such as heat) • Transmission 2. Even change in density 3. Example of where both reflection & transmission occur:

FREEVCENOTES

• Heavier rope gets transmitted smaller proportion of wave as it is heavier • Lighter rope gets reflected larger proportion of wave as it is lighter

REFLECTED WAVEFRONTS

1. Law of reflection: angle of incidence ϑ & angle of reflection ϑ are same, these are angles to normal of reflection surface:

2. Plane wave: where wave front is nearly straight, often occurs when wave has travelled long distances away from source 3. Diffuse reflections: reflections of waves on irregular, rough surfaces, thereby causing reflecting that is often spread over broad area, as shown below:COM .

4. Ray: arrowed line drawn perpendicular to wave fronts & in direction wave is moving, as shown below:

FREEVCENOTES

SONIC DEPTH FINDER

1. Uses echo-sounding device to measure depth through measuring time it takes for echo (sound wave reflection) to reach back to ship after reflecting on sea bed, see details & diagram below:

COM .

2. Uses knowledge of speed of propagation of sound waves in water

SUPERPOSITION

1. Superposition principle of waves: layering up of multiple waves intersecting at same point simply leads to displacement equal to vector sum of displacements at that 2. Example shown below: location

FREEVCENOTES COM .

3. For longitudinal waves wave superposition: • Transverse waves are often used to represent compression waves, where crests represent compressions (areas of high pressure), & troughs represent rarefactions (areas of pressure) • Example using sound waves:

FREEVCENOTES COM .

COCKTAIL PARTY EFFECT

1. In crowded room with many voices, i.e. cocktail party, one may still distinguish who one is speaking to. 2. Brains does this via innate ability to undo superposition of waves & recognize person’s voice from another & suppresses cognition of all other voices than one from person who is speaking to listener 3. Already reason why some people can hear their own names amidst bunch of voices by many different people.

RESONANCE

1. All objects that can vibrate have specific frequency (resonant frequency (also called natural frequency)) 2. Resonance: when object is exposed to vibrations at frequency equal to their resonant frequency, which leads to causing large vibration in that object even if source vibration is quite weak 3. Resonance may create vibrations so strong that object gets destroyed, i.e. how singers supposedly can break glass by singing particularly high notes

STANDING WAVES IN STRINGS

STANDING WAVES IN STRING 1. Example: violin • Standing wave vibration between fixed bridges of violin & finger of violinist 1. Standing waves: result of super positioning of 2 waves of same amplitude & frequency (only producible by such waves) travelling in opposite directions (often, one such wave is reflection of another) (they are named standing waves for waves don’t seem to be travelling along medium) 2. Standing waves are related to production of wide variety of sounds relating to speech FREEVCENOTES& music. (i.e. standing waves with different frequencies relate to pitch made by voices) 3. Nodes: where ropes is undergoing destructive interference, & rope stays still 4. Antinodes: where rope oscillates with maximum amplitude 5. Frequencies for string at which standing waves may be produced are called resonant frequencies of string (like aforementioned resonant frequencies) 6. Examples of standing waves with fixed ends: COM .

FREEVCENOTES

HARMONICS

1. Strings of musical instruments generate large variety of waves (of dif frequencies) 2. Waves travel in string in both directions & reflect from fixed ends. 3. Most such vibrations will interfere in random fashion & die away, those corresponding to resonant frequencies of string will form standing waves & remain 4. Resonant frequencies produced by such multiple standing waves are called harmonics 5. Standing wave of 1 antinode is called fundamental frequency (usually has largest amplitude, i.e. greatest influence on sound, generally amplitude decreases for each subsequent harmonic) 6. Higher-level harmonics are called overtones (by musicians) (fundamental, 1st overtone, 2nd overtone, etc.) 7. Usually, all possible harmonic are produced in string simultaneously ( strings then vibrate nearby air & these creating sounds that are heard as notes) 8. wavelength of harmonic for strings with fixed ends are calculated as follows: 2l λ = , n is order of harmonic, i. e. first (1 antinode) harmonic, second (2 antinode) ℎ��, ��. n 9. frequencies of harmonics for strings with fixed ends: v �� f = = λ 2� • v = velocity of wave in string

10. Standing waves where only 1 end of string fixed: can only form odd numbered harmonics (since only these satisfy condition of node at one end & antinode at other) (note overtones country stay same and not only odd) • Examples are shown below: COM .

• In general: FREEVCENOTES

Ø Alternatively, to stress odd harmonics property:

Ø Frequency calculation:

11. Resonant frequencies of string correspond to particular tension & mass per unit length (as this determines speed of wave), hence why tuning via adjusting string tension changes notes & different strings of varying densities change notes as well.

WIND INSTRUMENTS COM 1. Standing waves forming in the air column of instrument 2. Reed or mouthpiece give variety of frequencies, most of which cancel and diminishes in amplitude, whilst the resonant one stays and gets pronounced. 3. Length of tube in this determined by the keys, which through open and closing tube affects effective length of tube 4. Open tube ends correspond to fixed end, displacement wave reflected here change in phase (displacement node), pressure wave here not changed by phase (pressure antinode) 5. Closed tube ends correspond to open end, displacement wave reflected here are not changed by phase (displacement antinode), pressure wave here changed by (pressure node)

3D STANDING WAVES

1. Text deals predominantly with 2D standing waves. 2. Standing waves may also form in 3D, i.e. in section of Earth’s crust

NATURE OF LIGHT

LIGHT AS WAVE

WAVE MODEL VS PARTICLE MODEL

1. What wave model succeeds in: reflection, refraction, dispersion, diffraction, interference, polarization, etc. 2. Newton argued particle model of light (‘corpuscles’ of light): each different colour represents different type of particle 3. Robert Hooke & Christiaan Huygens proposed wave model of light, similar to water waves in ocean. 4. Newton’s theory predicted light travels quicker as it travelled through solid material such as glass, wave theory predicted it would be slower in glass. 5. In 19th century, theories made wave theory dominant, i.e. through experiments like Young’s experiment

HUYGEN’S PRINCIPLE

1. See below: FREEVCENOTES COM .

2. Shows how light waves propagate 3. circle radii = wavelength of light wave 4. idea is that each point on wavefront can be treated as if they all produce circular waves, & it is these waves that link up at their furthest points to establish new wavefront. FREEVCENOTES(so shortest perpendicular distance is 1 wavelength away from previous wavefront)

REFRACTION

1. Change in direction of light due to changes in its speed, i.e. occurs when light propagates through different mediums 2. Example:

3. Visualisation through wavefront illustration: (you can further see effect if you apply Huygen’s principle here) COM .

4. As for whether refraction is away or towards normal, rules are illustrated below: (toward normal when slow down, away from normal when speed up) FREEVCENOTES

COM .

5. In fraction, wavelength changes in correspondence to velocity, but frequency remains constant (waves cannot be gained or lost) 6. Below shows some speeds of light in several materials FREEVCENOTES

7. Refractive Index (often called absolute refractive index of material, as opposed to relative refractive index used to show light speed difference between some two materials) • Calculated depending on how much speed of light changes in material in comparison to its vacuum velocity • Formula: � � = , �� • Below are some common refractive indices:

• quicker light travels, smaller its refractive index • From nature of derivation of refractive index, following holds: �� = �� = �� COM SNELL’S LAW . 1. Used to predict refraction 2. formula:

• n are refractive indices • Angle refers to angle of ray to normal (of course, pls restrict these angle to within 0 to 90 degrees) • above are shown below

FREEVCENOTES

TOTAL INTERNAL REFLECTION

1. In refraction away from normal (speed increases), angle of refraction may approach 90° (i.e. refracted along surface), at which corresponding angle of incidence is called critical angle � � sin(�) = � sin(90°) = � → sin(�) = � 2. Optic fibers transmit light using total internal reflection 3. Beyond critical angle, ray does not undergo refraction, but is instead reflected back into medium (as typical reflection law dictates) 4. See below for such effect: COM .

DISPERSION

1. Result of refraction, leads to splitting of different wavelengths of light (as relative speeds of different wavelength in same medium is different) 2. Below are wavelengths of some colours:FREEVCENOTES

3. Example, triangular glass prism:

COM .

4. When entering different medium, wavelength change is proportional to velocity change of light, consequently depending on initial wavelengths, different light rays change wavelength by different amounts, which means that each colour travels at slightly different sped in new medium, hence refracted by different amounts, hence different wavelengths, i.e. different colours separate 5. Longer wavelength, i.e. red light, travel fastest in mediums (i.e. glass prism), hence refracted least for glass prism, 6. Newtons experiment below showed that coloured light was not white light that was stained, but instead fundamental component (else second prism would add more colours & not remove colours): idea of experiment is shown below FREEVCENOTES

7. Colour dispersion in lenses • Lens are of different medium than air, so it makes sense to see dispersion in lens, as shown below: COM .

• In some instruments such as microscopes, this distortion is called chromatic aberration, means of solution include: Ø Using lenses with very long focal lengths Ø Using ‘achromatic’ lenses. Which are compound lenses made of different types of glass with different refractive properties Ø Talking separate images using colour filters & then combining images to form single multi-coloured image

DIFFRACTION

1. Diffraction: when wave bends as it passes through narrow opening or past obstacles, as seen below:

FREEVCENOTES

2. Diffraction is significant when opening or obstacle size is similar to or smaller than wavelength of wave, i.e. as observed below:

• This relationship of significant diffraction may be expressed by ratio: (i.e. slit width smaller than or equal to wavelength) λ ≥ 1 w 3. CD of DVD are store information via small structures, which mean that light may also diffract on them. 4. effect of different wavelengths on diffraction can be seen below: COM .

DIFFRACTION & IMAGING

1. Diffraction is problem as it may result in blurred images 2. Light from 2 tiny or distant objects very close together can be diffracted so much that two appears as one object (in such scenario, objects are said to be unresolved) 3. ratio determines essentially whether object can be resolved by particular instrument

DIFFRACTION GRATING 1. Constructive interference: bright bands, as peaks FREEVCENOTES& troughs amplify 2. Destructive interference: dark bands, as peaks & troughs of wave cancels 3. Wavelets passing through small gaps will differ in where they pass gap, i.e. some at centre, & some at edges of gap in which they will diffract, consequently these light waves will interfere with each other 4. ratio describing extent of refraction also helps describe spacing of dark & light bands in diffraction pattern, see below for example of what happens when wavelength is different (this diagram is been reused) (note, below 0 intensity corresponds to dark banks, local maximum intensities corresponds to bright band)

5. Diffraction pattern: pattern of dark & light bands seen when light passes through small gap. 6. Diffraction grating: piece of material that contains large number of very closely spaced parallel gaps or slits COM • Generates much clearer diffraction patterns (than ones using natural materials) 7. Diffraction experiments usually only use monochromatic light (since diffraction depend on wavelength, i.e. monochromatic ligh. t makes it easier to see pattern) 8. Diffraction (though using materials like diffraction gratings) may also be used to diffract out different colours of light, as different wavelengths diffract to different extent for same gaps/slits/edges etc., as demonstrated below:

POLARISATION

1. Explained well by wave nature of light, hence convincing piece of evidence for wave model. 2. Polarisation occurs when transverse wave is only allowed in vibrate in 1 direction, i.e. below is vertically polarized: FREEVCENOTES

3. When light waves have different polarization as filter, it will be blocked as shown below (in contrast, it is hard to define what property such polarization would be for particle model (especially considering how simple rotation of such filters may lead to different results of passing through or not), hence why this polarization supports wave model)

4. What happens when light ray is polarized at non-perpendicular angle to filter is depicted below COM .

• Resulting wave is polarized according to filter • amplitude decreases 5. Light from sources like light globe

6. Polarizers: materials that can act as polarizing filters for light • Example of application: polarizing sun glasses, reduces glare through absorbing light polarized in particular direction • FREEVCENOTESAnother example of application: polarized filters in photography to reduce glare in photographs or to achieve specific effects POLARISED SUNGLASSES

1. Light reflected from surface of water/snow is partially polarized 2. polarizing plane of polarized sunglasses is selected to absorb this reflecting light 3. idea is illustrated below:

INTERFERENCE: FURTHER EVIDENCE FOR WAVE MODEL OF LIGHT

YOUNG’S DOUBLE SLIT EXPERIMENT

1. Background: between 17th - 19th century, most scientists considered light as stream of particles, idea based on ‘corpuscular’ theory by Sir Isaac Newton. 2. In 1803, English scientist Thomas Young performed his double slit experiment, principles are as follows: COM • Monochromatic light shown onto screen containing 2 tiny slits • Another screen placed on other side of slits to have patterns projected onto . 3. below illustrates different predictions of particle model vs wave model:

4. results of Young’s experiment was consistent with wave model’s prediction, hence supporting wave model of light proposed by Huygens & Hooke year earlier. 5. Young also used data collected & wave model to calculate from diffraction wavelength of light, which turned out to be very small, i.e. typically less than 1 µm (note, visible spectrum is 390nm to 780nm) 6. explanation to effect: • Upon passing narrow slits, plane waves were diffracted into coherent (in phase) circular waves • These circular waves would interact & cause interference, hence leading to nodal lines (dark bands, destructive interference) & antinodal lines (bright bands, constructive interference) • effect is illustrated below:

FREEVCENOTES COM .

7. Central maximum: part of double slit pattern which is usually brightest, due to waves travelling in phase with each other & giving constructive interference

PATH DIFFERENCE

1. path difference between ray travelling from two slits to specific point on plane lead to phase difference, which leads to interference

2. path difference to point P from wave source S & S �� : pd = |S� − ��| • This may be measured in meters, but much more useful to be measure in wavelength, so to better reveal consequent phase difference & predict interference. • illustration for such path difference：

FREEVCENOTES

3. Summary of when constructive & destructive interference occurs (these are in sense of complete constructive & destructive interference, for other phase differences, you’ll usually have partially constructive & partially destructive interference

4. typical double slit experiment fringes represented by intensity distance graph:

COM .

CALCULATING FRINGE SEPARATION FOR YOUNG’S EXPERIMENT

1. parameters are shown below: (∆� is fringe spacing)

2. Some proportionality relationships on fringe spacing: • ∆� ∝ � • ∆� ∝ • ∆� ∝ � 3. calculation of fringe separation: �� ∆� = � 4. Given all other parameters same, & only changing wavelength, above relation shows that fringes by longer wavelengths are further apart, as demonstrated below: FREEVCENOTES COM .

RESISTANCE TO WAVE MODEL

1. It took time for Young’s wave explanation to be accepted by scientific community. 2. In 1818, French scientist Augustin-Jean Fresnel managed to give mathematical explanation fo Young’s experiment based on Huygens’ principle 3. French scientist Simeon Poisson was passionate supporter of Newton’s particle theory, argued that same mathematics of Fresnel would predict if light was shown around round disk then there would be bright spot in middle of shadow created by disk. Poisson thought this, since no one ever observed such, proved wave model’s incorrectness. • However, one of Poisson’s colleges test this with very small bright light source & round disk, & found that bright sot predicted by Poisson’s calculations, & as result for remainder of 19th century wave theory became almost universally accepted light model: • Below shows bright spot prediction (this pattern is now known as ‘Poisson bright spot’, ironic considering that was person who thought it wouldn’t exist)

FREEVCENOTES COM .

ELECTROMAGNETIC WAVES

1. Different from mechanical waves as they do not require medium, i.e. can pass through vacuum. 2. In middle of 19th century, Maxwell considered visible light as part of broader electromagnetic radiation (EMR) 3. Changing electric field induces magnetic field, & changing magnetic field induces electric field, hence mutual induction will lead to magnetic & electric fields at right angles to each other, as shown below:

4. Such waves may be produced through charged particle moving backwards & forwards, i.e. generating changing electric field. 5. Such mutually inducting field will self-propagate, & hence form electromagnetic waves 6. Maxwell theoretically calculated propagation speed of such electromagnetic radiation through empty space which corresponded to experimental value measured by French physicist Hippolyte Fizeau in 1849. 7. Current day accepFREEVCENOTESted speed of light is 299792458 � �, denoted by c, & often approximated as 3.00 ∗ 10 � � 8. I.E. for EMR wave equation may be written as: � = ��

SEARCHING FOR AETHER (ALSO CALLED ‘LUMINIFEROUS ETHER’)

1. Mechanical waves have medium, so it was hypothesized that light propagated through this medium called aether, hence why it can propagate in what is seemingly vacuum. 2. Efforts for this eventually all failed, & people came to terms with how EMR could go through vacuum.

ELECTROMAGNETIC SPECTRUM

1. Visible lights fall within wavelengths of 390nm to 780nm, which is quite small part of entire electromagnetic spectrum, as displayed below:

2. Shorter wavelength means higher energy & vise versa via wave equation, also, higher frequency means higher energy (thus usually also higher penetrative power) 3. X-ray has very short wavelength, hence why it can penetrate say human skin & reveal structures such as bones. 4. Low penetrating power waves, i.e. AM radio waves, can’t even penetrate Earth’s atmosphere, hence why radios signals may be bounced around atmosphere & transmit information 5. Below show characteristics of some of different types of waves in EM spectrum

COM .

RADIOWAVES

1. EMR may be used to transmit information over long distances. 2. Very broad classification, range from wavelength of 1mm to hundreds of km. 3. Radio transmitters work as follows: • Converts signal, i.e. music, to alternating current, which flow into transmitting antenna, thereby causing electrons there to oscillate & produce corresponding EMR to information • wave eventually hits antenna of radio receiver, electrons in receiver’s antenna oscillate in same way as transmitting antenna, alternating current produced, receiver then reverses process & converts current into original signal • Summarised in following diagram:

FREEVCENOTES

4. AM vs FM (refers to how carrier waves may be modulated so to contain information to be delivered) • AM: amplitude modulation, change amplitude of carrier wave to match signal • FM: frequency modulation, change frequency of carrier wave to represent signal 5. Radio wave patterns are produced used ‘carrier wave’ of fixed frequency, it is ‘channel’ that radio ‘tunes into’, i.e. many radio stations use their frequency as part of their name, i.e. Nova 100.3 uses 100.3 MHz to transmit. 6. FM tend to be more clearer at signal transmission, but AM systems are much simpler circuitry wise than FM systems.

MICROWAVES 1. Shorter wavelength than radio waves 2. Greater penetrating power than radio waves 3. Producible by devices with short antennas, hence useful for small communication devices, i.e. mobile phones & wireless internet transmission, also quite useful I heating & cooking food.

INFRARED

1. Night-vision googles: enhance low light visibility by amplifying visible light & also detecting small part of infrared radiation emitted due to temperature. 2. Lie between microwave & visible light, have longer wavelength than red light, hence named infrared. 3. Useful as they are emitted by objects to varying degrees depending on their temperature, they are partly reason why fires or electric bar heaters feel warm 4. Example of infrared radiation: Earths transmits heat received from Sun in form of infrared waves (this infrared heat is important part of greenhouse effect). 5. Some objects appear red when they feel warm, & heated up, this is partly because they release red light alongside infrared radiation, latter of which is experienced as heat.

ULTRAVIOLET

1. UV rays have wavelengths shorter than violet light, hence ultraviolet 2. Not visible by human eye 3. Stronger penetrative power than visible light 4. Can penetrate human skin & damage DNA of skin cells, thereby leading to skin cancers 5. UV may be used to take images of bodies, i.e. of Sun after solar flare (& used to infer areas of different temperature for such very hot objects, this can also be done with visible light but effects wouldn’t be as good), as shown below: COM .

X-RAY & GAMMA RAYS

1. Much shorter wavelengths than visible light 2. Very high penetrating powers (hence some types can pass through things like different types of human tissue, hence why used for medical imaging) 3. Comes with danger, since X-ray can damage tissues, sometimes killing cells or damaging DNA in cell nucleus (hence leading to cancers), thus X-ray exposure usually needs to be carefully monitored to limit side effects. 4. Gamma ray exposure also very dangerous to humans.FREEVCENOTES 5. Gamma rays’ main natural source is Sun & radioactive isotopes. 6. Earth’s atmosphere protects people from most of Sun’s gamma rays. 7. Radio isotopes not common, hence rare to find quantities that produce harmful radiation doses of gamma radiation.

LIGHT & MATTER

PHOTOELECTRIC EFFECT & DUEL NATURE OF LIGHT

PLANCK’S EQUATION

1. � = ℎ� • Energy of quantum of light, i.e. of photon (J) • F is frequency of EM radiation (Hz) • H is Planck’s constant 6.63 ∗ 10 � � (��. ) • equation can be extended with wave equation � = ��: ℎ� � = � 2. This quantum idea correlates to particle model (contrary to wave mode dominant at time)

ELECTRON-VOLT

1. Used since energy when studying light is general quite small 2. Denoted by eV 3. Is amount of energy & electron gains when it moves through potential difference of 1 V, charge on electron is −1.6 ∗ 10 �, so 1�� = 1� ∗ 1� = 1.6 ∗ 10 � (well technically there is the sign issue, but you can formulate it to get rid of the issue by say a charge on a elementary charge) 4. Conversion of J & eV: • From J to eV: �� 1.6 ∗ 10 � �� • From eV to J: multiply by 1.6 ∗ 10 � ��

PHOTOELECTRIC EFFECT

1. Photoelectrons: electrons ejected from surface of metal when EM radiation is projected onto metal 2. Photoelectric effect: this photoelectron ejection effect 3. Photocurrent: flow of electrons due to photoelectric effect, which can be detected in apparatus like shown below: COM .

4. This apparatus can be used to find energy, maximal current etc. of photoelectron, by varying voltage, one may either allow all photoelectrons to go over to anode, or make electric potential that resists movement from cathode to anode which can thereby be used to measure stopping potential that is equal to say energy carried by most energetic electrons 5. Positive potential applied attracts electrons to collector anode (small positive voltage sufficiently ensures every available photoelectron is collected) 6. Negative voltage applied attracts electrons back towards cathode, smallest voltage here such that no electrons reach is called ‘stopping voltage’, effects of such voltage is shown below: (but the ± kinda disregarded, usully in 34 just consider the magnitude) FREEVCENOTES

7. Stopping voltage: maybe used to calculate maximal kinetic energy of photoelectrons. 8. Same intensity, dif. frequencies (of course above threshold frequency) produce same max current, higher frequency light however has higher stopping voltage

THRESHOLD FREQUENCY COM

1. Denoted by � 2. Minimum frequency for photoelectrons releasing. (determinable by seeing whether photocurrents detected for different frequencies given voltage designation . that sets anode as positive) 3. For light frequency greater than threshold frequency, rate of photoelectron production varies proportionally with intensity of incident light 4. For light frequency greater than threshold frequency, photoelectron emission generally occurs without appreciable time delay, regardless of intensity of light 5. Some photoelectrons emitted from first layer of atoms typically have maximum kinetic energy possible, ones from deeper surfaces loses usually some energy due to collisions on their way to surface, hence range of possible kinetic energies of photoelectrons.

EXPLAINING PHOTOELECTRIC EFFECT

1. Can’t be explained via wave model: wave model predicts that frequency should be irrelevant to ejection, since waves are form of continuous energy transfer such that under right conditions even very low energy light could accumulate enough energy to eject photoelectrons. But this is not observed, instead you have threshold frequency.FREEVCENOTES 2. Thus demonstrates particle nature of light, & out comes idea that photoelectrons are ejected due to collision of photons (this quantized particle idea was drawn from Planck’s quanta idea earlier on) 3. Einstein gave that each photon had energy: E = hf, h is Plancks constant, f is frequency 4. Planck assumed light was emitted in quantized packets, but never questioned wave assumption of light like Einstein. 5. Einstein’s idea of work function • energy required to free photoelectron from its bonding with metal • single photon is though to interact with single electron • Hence following relation: ϕ = hf, � �� ℎ� �ℎ��ℎ�� , ℎ ��

KINETIC ENERGY OF PHOTOELECTRONS

1. Photoelectric circuit equations E = ℎ� − � • E = max potential energy of photoelectron 2. Maximal kinetic energy may thus be measured by electric potential energy:

qV = E = ℎ� − � • One may thus with this use experiments to find work function of different materials 3. Alternatively, work function may be found through noticing linear nature of

E = ℎ� − � • Hence equivalently, one may extrapolate through linear regression to find value of negative work function, & hence work function, as shown below: (technically though, this is just graphical way of stating previous method)

COM .

4. For such graph above, threshold frequency at horizontal axis intercept, where photoelectrons are no loger bound to metal, but their maximal kinetic energy is 0.

RESISTANCE TO NEW THEORIES

1. People resisted particle quantum model of light, for particle model could not explain well established wave like properties such as diffraction & interference of light, as highlight by experiments such as Young’s experiments. 2. Eventually though accepted, & Planck got Nobel in 1918, Einstein in 1921 (he got it for his photoelectric effect, & not to his works on relativity)

PHOTOVOLTANIC CELLS

1. Utilizes photoelectric effect: sunlight leads to photoelectrons & thus current, hence electrical energy is produced. 2. Photovoltanic cells are designed to give of photocurrents for visible light 3. Cells commonly are semi-conducting materials based on silicon ‘doped’ with small amounts of other elements. 4. Most commercially available solar cells have energy efficiency of FREEVCENOTES<20% QUANTUM NATURE OF LIGHT & MATTER

WAVE PARTICLE DUALITY

1. May not refer to light simplly as wave or particle, more accurate to view as duality 2. When detector was used in slit experiments to check which slit photons passed through wave interference & diffraction patterns disappeared & photons acted like particles. 3. Even when light was so dim that particle nature, i.e. single photons passed through were reasonably assumed for slit experiment, wave-like interaction pattern effect still existed, thereby showing duality nature.

DE BROGLIE’S WAVE-PARTICLE THEORY

1. 1924, French physicist Louis de Broglie proposed that since light, which has long been seen as wave, is also particle like, then matter, which has long been considered particle like, is also wave like. 2. He calculated wavelengths of particle:

3. This wavelength is termed ‘de Broglie wavelength’ of matter. 4. Wavelengths for daily objects are generally extremely small, hence people don’t feel wave properties of such objects. I.E. in order for such objects to have noticeable diffraction measured, it would need to pass through gaps perhaps of scale of subatomic particles 5. Electrons have wavelengths smaller than visible light, but still sufficiently large to be measurable

ELECTRON DIFFRACTION PATTERNS COM 1. Example: .

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2. De Broglie’s results wee experimentally supported by Americans Davisson & Germer in 1927 through using below apparatus:

• Electron ‘gun’ used to provide electron beam • electrons bombarded into surface of piece of nickel • speed of electrons known due to acceleration through known voltage • pair found max & min intensities, like pattern displayed in previous part, when they moved detector around different scattering angles • electrons were scattered by different layers within crystal lattice, & were undergoing interference, hence why pattern & not just random scatter, scattering is shown below: COM .

• Calculated from results electron wavelength of 0.14nm, consistent with de Broglie’s hypothesis

ELECTRON MICROSCOPES

1. Optical microscope use photon waves to magnify tiny objects, so can electron waves 2. Optical microscope limitation is that size of object under examination must be similar in size to light’s wavelength, such that light diffracts around these structures 3. Hence, light microscope limited to sizes down to 390nm, lower wavelength end of visible light. 4. Electron wavelength is often smalFREEVCENOTESler than those of visible light, hence more clear images can be resolved than with opti cal microscopes 5. Examples of electron microscope images:

THOMSON’S COMPARISON OF PHOTON & ELECTRONS （SON OF ELECTRON DISCOVERER）

1. Produced scatter electrons from beam passed through tiny crystal, & then repeated experiment with X-rays of same wavelength. 2. Diffraction patterns of two were almost identical 3. Similar diffraction & same ‘gaps’ were used for diffraction, hence electron momenta & X-ray photons must also be comparable 4. Visualisation of above diffraction similarity: COM .

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PHOTON MOMENTUM 1. De Broglie hypothesis & his equation λ = has corollary that if particle has wavelength, then it should have momentum, i.e. photons should have momentum (despite being massless) 2. This allows hypothetical tech of solar sailing, i.e. spacecrafts get thrust off rebounding of photons, where since photons have momentum, momentum transfer will exert force. 3. Marine 10 & MESSENGER spacecraft, both of which passed Mercury & Venus, used solar pressure to decelerate. (in line with solar sail concept) 4. Japanese IKAROS is first spacecraft to draw primary propulsion from solar sail. Its 196 m sail produces 1.12 mN of thrust.

LIGHT & MATTER

BOHR’S MODEL

1. Convinced people that particle model was necessary for full understanding of light alongside particle model, & it eventually led to subsequent discoveries & theories that helped reformulate understanding regarding energy & matter 2. Helped explain occurrence of specific emission spectra of atoms, i.e. Ones like below:

COM .

3. Below are Bohr’s realisations: (drew on previous results of his time, i..e Planck’s equation) • Hydrogen atoms had absorption & emission of energy in quantized manner, as displayed by absorption & emission spectra (he also drew on earlier discoveries of Planck & Einstein regarding idea of quantized quantities) • Emission & absorption spectrum correspondence showed hydrogen could emit quanta of same energy as it absorbs • If frequency of photon is below certain level, then light will pass straight through & not be absorbed • Ionisation energy of Hydrogen atom is 13.6 eV, i.e. hydrogen may become positive ion when such energy is applied • Photons of light with all energies above ionization value of hydrogen are continuously absorbed, extra energy after ionization is simply added as kinetic energy for released electron. 4. Emission spectra explanation according to Bohr’s model

FREEVCENOTES COM .

5. Below are some ideas of Bohr’s model • Electrons move in circular orbits around nucleus of hydrogen atom • Centripetal force that keeps in orbit is electrostatic force between positive nucleus & negative electron • number of allowable orbits for electrons to occupy of different radii exist for each atom (labelled n=1,2,3…, these are known as principal quantum number) • electron usually occupies lowest energy state available (ground state) • Electron does not radiate energy when in stable orbit • EM radiation (in form of photons) may be absorb by atom when photon energy is exactly equal to difference in energies between occupied orbit & higher orbit • EM radiation is emitted by excited electron that returns from higher energy to lower energy orbit, photon released will have energy exactly equal to energy difference between initial & final orbits. 6. Bohr was able to find energy difference between electron orbits, & used it to calculate wavelengths of emitted photons using Planck’s equation. 7. idea is that at � = ∞, so when electron escapes from atom, it would have 0 potential energy, similarly any orbit closer to nucleus has negative potential energy that increases, just like your typical classical mechanics orbits

FREEVCENOTES 8. Below are energy levels for hydrogen atom:

COM .

BOHR’S MODEL LIMITATION

1. It only worked well with single-electron atoms, i.e. hydrogen & ionized helium. 2. Modelled inner-shell electrons well, but not those in higher energy orbits of multi-electron atoms 3. Did not explain continuous spectrum emitted by solids. 4. There were even problems with its description of hydrogen emission spectra, i.e. some observed emission lines may be resolved to 2 very close spectral lines, which Bohr’s model could not explain.

ABSORPTION SPECTRA

1. 1814, Joseph von Fraunhofer reported dark lines appearing spectrum of sun light, these were termed Fraunhofer lines & are as shown below:

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2. ~50 year later, Kirchhoff, Bunsen ( Bunsen burner person) & others recognized correspondence between Fraunhofer lines COM& emission colours of certain gases when heated to high temperatures, hence deduced that these Fraunhofer lines were due to absorption by gases in outer atmosph. ere. EMISSION SPECTRA

1. Elements heated to high temperatures/electricl current passed through, light is produced. 2. Atoms absorb energy, become ‘excited’ where electrons jump to higher energy states, & then when they fall back to unexcited ‘ground state’ energy which was absorbed will be released as single photon. 3. Atoms usually have number of different excited states, hence producing different colours, i.e. emission spectra is possible. 4. Example of emission spectra:

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SPECTRAL ANALYSIS

1. Atomic Emission Spectroscopy: • Analysing light emitted by material when burned/current passed through & matching it to known emission spectra of elements in order to detect presence of elements. • Spectroscopes may be used to separate emitted light into separate wavelengths 2. Metal vapour lamps: produce light as atoms are excited & then emit photons as they returned to ground state, light emitted is characteristic of emission spectra • Example of vapour lamp: sodium vapour lamps commonly used in streetlights: • The excitation is via sending electricity through the ionized vapour (plasma) • The process is essentially lose electron, collide with neutral atoms, gain electron, return to lower energy state, emit light, and so on • Good as typically long life and high efficiency, e.g. compared to incandescent light bulbs, though not necessarily compared to LEDs COM .

3. Planck’s equation: ℎ� ∆�(�� ℎ�� ) = ℎ� = � • ℎ �� , ℎ~6.63 ∗ 10 � �, �� 4.14 ∗ 10 �� • Note what unit is used for energy & determine what versin of Planck’s constant to use accordingly • � �� • � �� ℎ�, �~3.00 ∗ 10 � � (�� 299792458 � �) • � �� ℎ �� ℎ��, ��

CORRESPONDENCE BETWEEN EMISSION SPECTRA DN ABSORPTION SPECTRA OF HYDROGEN

1. As shown below:

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2. Empirical formula for predicting wavelength was derivable, but no theoretical explanation of these on wave model of light was producible, hence made particle model necessary 3. Support particle model for again, discrete energy levels, if really wave, then should be able to accumulate energy regardless of wavelength, and not have specific defined energy absorption as shown above

STANDING WAVES & DUAL NATURE OF MATTER

1. De Broglie proposed how matter may have wavelengths, & argued how viewing electrons as standing wave is only explanation to how it could maintain stable energy level. 2. He reasoned as follows: (in line with standing wave) � ℎ�� , �� ℎ �� , �� : �ℎ �ℎ �� = , � �� → 2�� = �� (�) = 2� �� ℎ �� = , � = �� ℎ�� ℎ�� ℎ� �� = 2�� ℎ �� (�ℎ�� ℎ��ℎ) • �. �. �� ℎ�� ℎ �� ℎ�� of electron wavelengths: • � �� ℎ� �� :

COM .

• Visualisation of when above condition is not satisfied & destructive interference occur, hence no standing wave occurs & orbit cannot represent energy level:

FREEVCENOTES COM .

HEISENBERG’S UNCERTAINTY PRINCIPLE

QUANTUM INTERPRETATION OF ELECTRON

1. Schrodinger’s wave equation

• Schrodinger’s equation: • In Schrodinger’s model, wave properties of electron are interpreted as representing probability of finding electron in certain location • Maybe used to calculate regions of space in which electron can be found in hydrogen atom (these are now known as orbitals rather than orbits because they are complex 3D shapes unlike simple circular paths imagined by Rutherford & Bohr), see below of first 5 electron orbits of hydrogen atom:

2. Quantum mechanics: name given to physics which wave properties of electrons are studied

SCHRODINGER’S CATFREEVCENOTES 1. Used to draw intuition to counterintuitive quantum world 2. Thought experiment 3. cat placed in closed box with flask of poison, quantum mechanical system set up such that 50% chance of flask being broken & cat killed. Schrodinger argued until box is opened, cat is considered simultaneously alive & dead, in manner similar to dual nature of light

LIMITS TO MODELS AT VERY SMALL SCALES

1. Small scaled world is fundamentally different to macroscopic world 2. In macroscopic world, everything appears to have clearly definable position & motion, classical mechanics are all based on this assumption. Not so in microscopic world.

HEISENBERG’S UNCERTAINTY PRINCIPLE

1. Uncertainty is generally unavoidable going to be there when taking measurement due to limitations of device used 2. Heisenberg’s Uncertainty Principle states limit as to how accurate some quantities may be measure: • Explanation 1: very act of measuring something very small, due to delicate structure of microscopic systems, will cause change that makes measurement invalid. Hence limit. 3. part of Heisenberg’s Uncertainty Principle: • more exactly position of sub-atomic particle (i.e. electrons) is known, less is known about its momentum. Similarly, more precisely momentum of particle is measured, less certain is its exact position. • mathematical statement: (where ∆� �� , �� ℎ� ��ℎ�� )

• above equation is also sometimes called ‘indeterminacy principle’ • I.E. you will have scenarios like follows due to uncertainty principle when you try to get absolute measure of 1 of 2 quantities:

COM .

4. All matter has wave-particle duality, i.e. Heisenberg’s principle applies especially at quantum level precisely when idea of position in itself is quite ambiguous due to how wave-particle duality is quite apparent for this scale. Viewing Electron 1. Suppose one uses photon, i.e. as with optical microscopes, to view electron. 2. Small wavelength photon allows one to better see things of small scale, but small wavelength means more energy, i.e. more influence on electron itself (E=hf), i.e. impart more momentum to electron objserved. 3. I.E. to reach higher resolution, one requires to alter momentum of electron & thus also position, i.e. observation itself introduces significant uncertainties. 4. This is taken by some as explanation for Heisenberg’s uncertainty principle 5. idea of this issue is illustrated as shown:

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INTERPRETATIONS OF HEISENBERG’S UNCERTAINTY PRINCIPLE

1. Uncertainty is revealed by past experiments to show that this is fundamental property of quantum mechanical universe. 2. At its early introduction, i.e. in times of Schrodinger’s work of how electron motion can only be described with probability & hence uncertainties exist, even people like Einstein argued that ‘God does not play dice with universe’

SINGLE SLIT DIFFRACTION & UNCERTIANTY PRINCIPLE

1. Shows Heisenberg’s uncertainty principle 2. This specific setup attempts to show how diffraction may be explained through particle aspect of light. 3. diffraction pattern even with absence of photon interference (i.e. wave interference) is explained by Heisenberg’s uncertainty principle, as slit narrows, i.e. ∆� decreases, corresponding ∆� must increase, hence resulting in how diffraction pattern spreads out for narrower slits. 4. Typical setup: Ø Laser shown through narrow adjustable slit to find diffraction pattern, as with double slit experiment Ø Very dim laser used so that one may reasonably assume 1 photon passes slit at once (hence ruling out wave models argument that pattern is due to interference of waves, i.e. interfering photons, since here we are just talking individual photons)

COM .

5. Comparison with no slit at all (i.e. infinite uncertainty about displacement & thus no displacement about momentum) than one illustrated above:

6. As for why some electron paths are more likely than others, this can be explained through Schrodinger’s equation (as opposed to residing with photon interference (wave interference))

QUANTUM VIEW OF WORLD

1. Schrodinger & Heisenberg’s model supports aspects of Bohr’s model, i.e. atoms exist only in discrete states of definite energy & emission/absorption of photons when electrons transition from energy states 2. Bohr’s model is mixture of classical & quantum theories, it partially recognizes wave-particle duality of light & matter 3. Bohr’s model allowed for more new radical theories to be developed by those like Schrodinger & Heisenberg 4. Quantum mechanics currently is much more complex than course goes. 5. Quantum mechanics describe that electrons don’t exist in well-defined circular paths as depicted regularly in text. (due to wave nature, their paths are not particular in space & time like particles, but more so like clouds where position is probabilistic & its accuracy in measurement according to Heisenberg will prevent one from knowing where electron would be at next moment in time)FREEVCENOTES 6. Classical Newtonian world view, ‘deterministic’ model, is where once properties like position & speed are known, then future position can be predicted. 7. Quantum mechanic proposes uncertainty, hence different from deterministic nature of Newtonian world view & replaces it with probabilistic measures that introduce inherent unpredictability to things, & led to things like how we can only find probability of electron is in particular place around atom. 8. Visualisation of electron probability cloud idea: COM

9. According to quantum mechanics, it is meaningless to ask how electron gets from state to another when atoms emit or absorb . photons – it just does.

AREA OF STUDY 3 PRACTICAL INVESTIGATION

OTHER NOTES

1. Lami’s Theorem (direct result of sine rule application & angle chasing in force triangle formed):

2. Electric shock / electrocution • Relatively small currents, i.e those of scale of milliamps may deal sufficient damage. • Current, duration of current, are main direct cause of harm (though since voltage is related to current, & heat dissipation, it also has indirect effect o harm) • Dry skin typically resistance: 100000 ohms, wet skin has less resistance, typically 1000 ohms or less • Men tend to have less body resistance than woman (resistance goes up with length, down with diameter), men tend to have thicker arms & legs, hence lower resistance. • Be careful on differentiating skin resistance, & internal resistance of bodies • FREEVCENOTESGenerally, current is what is harmful (you can carry great voltage, but as long as no current everything is fine), on other hand , you may still be harmed if you have small voltage but large current (though very unlikely considering body resistance), as though heat dissipation will be small, electron flow will have effect on neuron signals & stuff. • Harms include Ø Heat dissipation due to resistance, hence burning tissues Ø Arc burns (electric arcs generate high temp close to body): this may occur without entailing direct current through body Ø Thermal contact burns: relates to indirect burning due to contact with something heated by electricity, i.e. overheated conductors Ø Affecting electric signals of neural networks (consider fibrillation of heart)

3. Leyden jar • Stores high voltage charge between conductors insider & outside of glass jar (think spherical capacitor, except with glass) • Charged by electrostatic generator (i.e. some kind of friction device like Van de Graaff) COM .

4. Efficiency • Efficiency = ∗ 100% • Although understandable in fraction/decimal form, it is safest to give in above fraction form

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