HSC Quanta to Quarks

The HSC Physics topic Quanta to Quarks includes the concept of the wave nature of an electron.

1. When we draw an electron as a wave what does the height of the wave represent?
2. Is the electron mass spread out along the wavelength?
3. In quantum mechanics an electron is described by a wave function. Is this the same as the de Broglie matter wave?
4. What is the physical interpretation of the wave function?
5. According to quantum mechanics what actually is the electron in the hydrogen atom? Is it a particle moving in an orbit?

Curl of a Vector

In electromagnetism and fluid dynamics we often determine the curl of a vector quantity. This is a mathematical operation that follows set rules and is used to determine a field vector. What does curl mean physically?

In electromagnetism when rationalised SI units are used, the curl of the electric field vector (E) is equal to minus the partial time derivative of the magnetic induction vector (B) and the curl of the magnetic field vector (H) is equal to the sum of the current density (J) and the partial time derivative of the electric displacement vector (D). These relationships are known as the Faraday-Maxwell law and the Ampere-Maxwell law respectively.  In fluid mechanics the curl of the fluid velocity vector (u) is equal to the vorticity (W).

The curl operation is a mathematical way of determining whether the work done (also called the circulation) in moving around a small circle centred at the field point is zero. If the work done is zero the field is said to be conservative or irrotational, meaning that a small paddle wheel placed at that point in the field would not spin if the field was the velocity field of a fluid, hence the name of the operation curl. Some textbooks call the curl operation rot, the significance coming from the rotation of the test paddle wheel. The mathematical rule connecting curl with work done is known as Stokes' theorem, one of the very important theorems of vector analysis along with those of Gauss and Green.

The curl operation is a very important part of the language of electromagnetism. The curl operator is part of the mathematical language that we use to show that time varying electric and magnetic fields propagate through space at the speed of light.

Trial Physics Revision 2017

In most schools in NSW the trial HSC examinations occur in three weeks. Class assessment tasks have finished so we can now concentrate on studying. How can you best prepare for these examinations? First, you must have a study timetable, just like a daily school timetable where you have 60 minutes for each subject. Stick to the schedule. Have a 10 minute break between study sessions. Secondly, in your study sessions work through past examination questions by writing down answers on lined paper. Set out your work in an organised fashion so that this habit becomes automatic in examinations. Remember that perfect practice makes perfect. Here are some Physics revision questions:

1. Reliability An experiment is performed to measure the acceleration due to gravity. Outline how we can make the result reliable.
2. Ceramic Insulators Outline the function of the ceramic insulators on electricity transmission lines.
3. Silicon and Germanium Describe the advantages of silicon over germanium in semiconductor devices.
4. Zeeman Effect Describe the Zeeman effect. How is it explained?
5. Orbital Speed The orbital speed of a satellite moving in a circular path of radius r is v. Find the orbital speed of a satellite moving about the same planet in a circle of radius 2r.
6. DC Motor A simple DC motor has one rectangular coil of wire placed in a uniform magnetic field. Explain, using a diagram, why the magnetic force on each side of the coil is constant as it spins but the torque acting on the coil is not.
7. Superconductors Superconducting magnets are used to produce very strong magnetic fields in particle accelerators. Explain how this is possible if the magnetic field inside a superconductor is zero.
8. Matter Waves A student states that an electron is described as a particle moving along a standing wave. Explain why this is incorrect.
9. Relativity A spacecraft travels at a constant velocity v between two planets that are a distance d apart. Determine the time taken for the journey according to (a) the reference frame of the planets (b) the reference frame of the spacecraft.
10. Split-Ring Commutator A split-ring commutator is connected to the output terminals of a single coil generator that is spinning at a constant rate. Draw graphs showing for one rotation of the coil (i) the current in a particular side of the coil (ii) the magnetic force on this side of the coil (iii) the current in a particular brush (iv) the electrical power in the coil (v) the mechanical power suplied to the coil
11. Resistance A copper wire is maintainted at a temperature of 2K. Describe how an electric current flows through the conductor.
12. Uncertainty Principle An observation is made of the position of an electron. How does this affect the momentum of the electron?
13. Rocket Motion A rocket starts from rest and accelerates vertically upwards due to the expulsion of exhaust gases at a constant speed relative to the rocket. The mass of the rocket decreases at a constant rate. Draw graphs showing for the same time interval the rockets (i) acceleration (ii) velocity (iii) distance travelled (iv) momentum.
14. AC An alternating current of peak value I flows in a conductor of resistance R. Show that the heat energy released during one cycle of the AC is one-half of that of a constant current I.
15. Cathode Rays A glass tube contains air. A high voltage is applied between two electrodes in the tube. Describe the appearance of the discharge in the tube as the pressure in the tube is reduced.

Electric Current

What happens when a current flows through a conductor? In HSC Physics students learn that the resistance of a conductor is due to the collisions of the conduction electrons with the atoms in the conductor. These collisions transfer energy from the electrons to an increase in the vibrational energy of the atoms causing an increase in temperature of the conductor. What would be an estimate of the drift speed of conduction electrons through a metal;  1 mm/s, 1 m/s, 1000 m/s or 1,000,000 m/s? The answer is 1 mm/s. When a light globe is switched on an electric field is sent along the connecting wire at a speed of about one-third of the speed of light and this field pushes electrons at the far end of the wire through first, so the globe appears to glow immediately. Quantum mechanics also teaches us that electrons behave as waves. The electron waves are scattered by the irregularly spaced atoms in the conductor which are displaced from a regular pattern due to their thermal vibration.

Why is Gravitational Potential Energy Negative?

Student's learn in HSC and IB Physics classes that the gravitational potential energy of two attracting masses is negative. Why is this? Let us consult the popular Physics textbooks to see what their authors say.

Resnick, Halliday and Walker Fundamentals of Physics (10th edition, page 365)...we choose a reference configuration with U equal to zero when the separation distance between the masses is infinite. The gravitational potential energy decreases when the separation decreases. Since U=0 for r=infinity, the potential energy is negative for any finite separation and becomes progressively more negative as the particles move closer together..

Serway and Jewett Physics for Scientists and Engineers (8th edition, page 386)...the potential energy is negative because the force is attractive and we have chosen the potential energy as zero when the particle separation is infinite. Because the force between the particles is attractive, an external agent must do positive work to increase the separation between the particles. The work done by the external agent produces an increase in potential energy as the two particles are separated.

Knight Physics for Scientists and Engineers (4th edition, page 366)...All a negative potential energy means is that the potential energy of the two masses at separation r is less than their potential energy at infinite separation.

Tipler and Mosca Physics for Scientists and Engineers (6th edition, page 374)....this means that U approaches zero as r approaches infinity. At first this may seem like a strange choice because for finite values of r all values of U are negative. This just means, however, that the potential energy is at a maximum when Earth and particle are at infinite separation.

Sears, Zemansky, Young, Freedman University Physics (14th edition, page 429)...in defining U we have chosen U to be zero when the body is infinitely far from the Earth. As the body moves towards the Earth, gravitational potential energy decreases and becomes negative.

Ohanian and Markert Physics for Engineers and Scientists (3rd edition, page 289)...the potential energy is always negative and its magnitude is inversely proportional to r. If the distance r is small, the potential energy is low (the potential energy is much below zero); if the distance r is large, the potential energy is higher (the potential energy is still negative but not so much below zero). Thus the potential energy increases with distance; it increases from a large negative value to a smaller negative value or to zero. Such an increase of potential energy with distance is characteristic of an attractive force. For instance, if we want to lift a communications satellite from a low initial orbit (just above the Earth's atmosphere) into a high final orbit (such as a geostationary orbit) we must do work on this satellite (by means of a rocket engine). The work we do while lifting the satellite increases the gravitational potential energy from a large negative value (much below zero) to a smaller negative value (not so much below zero).

Studying Physics and Mathematics

With the trial examination period for 2017 quickly approaching it is important to talk about the best way to study Physics and Mathematics. The answer is to do questions. Questions may be from past exam papers or summary books. At this stage in schools new work should have been completed with a path of revision left towards the HSC. Write detailed answers to your questions. In Physics use the words clearly and make sure you answer the question that is asked. In Mathematics do not jump too many steps at once in your working as this is a very common source of error. Most importantly do not attempt at this stage to summarise or write out the text-book. This just takes up time and does not engage your brain to promote thinking and understanding.

Period of Trigonometric Functions

A very common question in HSC Maths papers is to find the period of a given trig function. Some students regret putting the period of tan(x) as 2𝛑 in last year's HSC 2Unit paper! Here is a quick tutorial set on determining the period of trigonometric functions. Answers are given below.

1. sin(x)
2. 2cos(x)
3. tan(x)
4. csc(x)
5. sec(x)
6. cot(x)
7. 3sin(2x)
8. tan(2x)
9. cos(x/2)
10. sin(𝜋x+3)
11. 2cos(x/3+π)
12. cos(x)+sin(x)
13. cos(2x)+sin(2x)
14. cos(x/2)+sin(x/2)
15. sin(x)+sin(2x)
16. sin(x)+sin(2x)+sin(3x)

[2𝛑, 2𝜋, 𝜋, 2𝜋, 2𝜋, 𝜋, 𝜋, 𝜋/2, 4𝜋, 2, 6𝜋, 2𝜋, 𝜋, 4𝜋, 2𝜋, 2𝜋]

Ideal Gases

Question 11 in IB Physics November 2016 Paper 1 was a question on determining the gradient of the V-T graph for an ideal gas. Only 11% of candidates answered this question correctly and this statistic is the lowest in SL papers in the last 4 years. Why was this question so difficult? The correct alternative, C, has the same form as the most chosen alternative, B . Alternative B contains the gas constant R whereas alternative C contains the Boltzmann constant k, which is correct; 66% of candidates sitting for the paper confused the constants by selecting alternative B. This could have been avoided by using the formulae on page 6 of the Data Booklet. Make use of the Data Booklet in Physics examinations.

Cathode Rays

Here are some points about the nature of cathode rays for HSC Physics.

1. Cathode rays (now called electrons) are small negatively charged particles leaving the cathode and attracted to the anode in a discharge tube containing air at a low pressure when a high voltage is applied between the electrodes.
2. German scientists believed that cathode rays were a wave-like disturbance in the aether like light. Heinrich Hertz found that cathode rays could pass through thin sheets of gold and were not deflected by electric fields. Hertz left too much gas in his tube causing it to be ionised and so a weak resultant electric field existed between his deflecting plates....too weak to produce a noticeable deflection of the cathode ray beam.
3. J.J. Thomson in 1897 used a lower pressure in his discharge tube and delected a beam of cathode rays towards the positively charged plate showing that cathode rays are negatively charged particles. Thomson applied a magnetic field perpendicular to his deflecting electric field using a set up known as Helmholtz coils. The magnetic field deflected the beam perpendicular to its velocity and by adjusting the strength of the field he was able to allow the cathode rays to pass through both fields in a straight line, the electric force balancing the magnetic force. In this case v = E/B, where v is the speed of the cathode ray, E is the strength of the electric field and B is the strength of the magnetic field.
4. Thomson switched off the magnetic field and allowed the cathode rays to be deflected only by the electric field. He measured the angle 𝜽 at which the beam was deflected when it left the electric field; tan𝜽 = qEL/(mv ² ), where L is the length of the electric field, v is the speed of the cathode ray and q/m is the charge to mass ratio of the cathode ray. If the sideways deflection y of the beam is measured instead of 𝜽 the equation connecting y and q/m is y = qB ² L ² /(2mE)
5. Thomson found that the charge to mass ratio of cathode rays was a large number and was independent of the type of metal in the cathode. He concluded that cathode rays are small negatively charged partices that are present in all matter. His work established the existence of the electron as a fundamental particle of matter provided the basis for further advances in Physics such as quantum theory, semiconductor technology and superconductivity.

Some facts about cathode rays.

1. The first observation of the presence of cathode rays was made by Julius Plucker in 1858 who noticed a green fluorescence coming from the wall of the glass tube near the anode. This colour is determined by the chemical composition of the glass.
2. When a small paddle-wheel balanced on a pair of horizontal rails is placed in a discharge tube the vane always turns away from the cathode. This led William Crookes to (incorrectly) conclude that cathode rays were particles with momentum. H Starke later showed that the rotation of the vane was due to the heating of only one side of the vane by the cathode rays. The gas next to the vane had an increased pressure and so the vane was pushed away from the cathode. Thomson showed that the momentum of the beam was not sufficient to produce the observed motion. The paddle wheel experiment shows that cathode rays have a heating effect rather than momentum.

Circular Motion

Question 22 (SL) and Question 14 (HL) in the May 2016 IB Physics Paper 1  involved a mass on the end of a rod moving in a vertical circle in such a manner that the speed of the mass was constant. This question received the lowest percentage of correct responses in both papers and so it is worthwhile to discuss the Physics involved in this question. The question asks about the force exerted by the rod on the mass. The forces acting on the mass are its weight due to the Earth's gravitational field and the contact force of the rod pulling on the mass. Since the mass is moving at a constant speed in uniform circular motion the resultant force acting on the mass must always be directed towards the centre of the circle and must be constant in magnitude. As the mass moves around the circle the weight force is always directed downwards and so the direction of the contact force must change so that the vector sum of the weight force and the contact force is always directed towards the centre of the circle. At the top of the circle both forces point towards the centre of the circle and so the contact force has its minimum value at this position. Therefore the answer is D.

Speed of Light

This is a phrase that often appears in student responses in Physics examinations. Let's look at various versions of it and see if they are correct. We will start from the most basic.

1. Light is constant. This is incorrect as the property of light has not been identified.
2. The speed of light is always constant. This is incorrect. The speed of light in water is less than the speed of light in a vacuum.
3. The speed of light in a vacuum is a constant value. This is not fully correct as the frame of reference has not been included.
4. The speed of light in a vacuum is the same value in any inertial frame of reference. Correct. This was one of Einstein's postulates in 1905.
5. The speed of light in a vacuum is the same value in any frame of reference. Incorrect. In accelerating (non-inertial) frames of reference the speed of light in a vacuum is not always c.

Here are some other misconceptions about the speed of light.

1. The speed of light in a vacuum is given the symbol c because c is the first letter of constant. Incorrect. The c comes from the Latin word celeritas for speed.

2. The Michelson and Morley experiment showed that the speed of light was always constant. Incorrect. There is no mention of the constancy of the speed of light in their paper describing the experiment. Einstein proposed this in 1905. M and M were not able to detect the motion of the Earth relative to the aether (this is a null result)

3. Light waves are different to sound waves. The speed of sound in air at 20 degrees C is 343m/s. Imagine that you move towards a source of sound at 20 m/s. What is the speed of the sound waves relative to you? [363 m/s]
4. The Michelson-Morley experiment found that the light waves in their interferometer always arrived in phase. Incorrect. The rays of light produced an interference pattern after travelling on perpendicular paths. This pattern did not change when the apparatus was rotated through 90 degrees. This is called no fringe shift.
5. The first determination of the speed of light was made by the Dutch physicist Christian Huygens in 1677 using astronomical measurements made by Olaus Roemer. The value obtained was 2.3x10⁸ m/s. The French physicist Hippolyte Fizeau using an Earth based method determined the speed of light as 3.1x10⁸ m/s in 1849.

“To Tom...for always and in all ways”

— Elizabeth, Catherine and Stephen

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Damping of Forced Vibrations

In the May 2015 TZ1 IB Physics Paper 1 SL and HL there was a question (13SL,10HL) on the effect of damping on an oscillating system experiencing a driving force of variable frequency. The IB Examination Report states that "as damping is increased and friction is introduced into the system so the time per oscillation will increase, answer is B". This may not be intuitively obvious so a mathematical route to the solution may be helpful.

Let the frequency of vibration of the system when no damping occurs (usually just called the natural frequency) be F. The frequency of the driving force that acts on the system is f. The amplitude of the forced oscillations (this is the greatest displacement of the system from its equilibrium position) caused by the action of the driving force is A. Damping means that a resistive force is acting on the system as it vibrates causing its energy to decrease.

1. Driving force with no damping. When damping is not present a huge vibration known as resonance ocurs when the frequency of the disturbing force equals the natural frequency of vibration. As the amplitude is huge we can predict that the amplitude of oscillation must be inversely proportional to the difference in the frequencies. When f - F approaches zero A approaches infinity. A mathematical solution of the differential equation describing the system without damping gives A = 1/(f²-F²). How can a catastrophic vibration be avoided when f = F? Including a damping term in the equations removes some energy from the system allowing it to be unavailable to participate in a large vibration.

2. Driving force with damping. When a damping term is present the differential equation describing the system contains an additional term 2kv, where k is the damping coefficient and v is the velocity of the mass. The solution for A with damping is A = 1/√((f²-F²)²+ 4k²f² ). Notice the additional term in the denominator. When f = F we have A = 1/(4k²f²) giving a finite amount of vibration when f = F. Catastrophe is avoided! Let us examine the equation

   A = 1/√( (f²-F²)²+ 4k²f² )

If we plot A versus f we obtain the graph shown on the 2015 examination paper. By differentiating A with respect to f we find that A has a maximum value of A₀ when f has the value f₀

f₀ = √( F²-2k² )

A₀ =1/( 2k √(F²-k²) )

To answer the question, as k increases A₀ decreases and f₀ decreases so alternative B is the correct answer.

Velocity-Time Graph

Question 16 in the 2016 HSC Mathematics examination involved calculating the distance travelled by a particle given its velocity as a function of time. The velocity during the first second of the motion was negative. This trapped many students. To find the total distance travelled we must add up the absolute values of all of the areas under the v-t graph. Be careful! If you have done it correctly the answer is 10-4ln2. If you obtain 14-12ln2 this is the change in displacement or the final position of the particle measured from the starting point; this is the length of the arrow from the starting point to the finishing point and is not the distance travelled which is the total amount of ground covered.

Relativity

Question 5 in the 2003 HSC Physics examination was a multiple choice question on time dilation and length contraction. Unfortunately, no correct answer was given as an alternative. Students doing revision often ask about this question. As the mark for the question was apparently not included in the final mark it could probably be ignored. However, an answer involves taking into account the travel time of the light coming from the Earth as the question says "when seen from the astronaut's spaceship"...seen implying making an observation using light .  From the point of view of the astronaut in the spaceship the Earth is moving away at 0.8c. The time for the journey in the reference frame of the spaceship is 10 years. The distance of the journey in the reference frame of the spaceship is 8 light years and so a ray of light would take a time interval of 2 years in the spaceships reference frame to reach the spaceship. To determine the corresponding time interval shown by the clock on the Earth we solve the time dilation equation for t0 putting tv as 2 and v as 0.8c. This is because the astronaut considers the Earth to be moving away carrying its clock with it. This works out to be 1.2 years.

Simple Pendulum

The simple pendulum is one of the oldest Physics demonstrations and examination questions. A simple pendulum consists of a mass tied to one end of a string, the other end of which is fixed, and the mass is allowed to swing freely in a vertical plane. The important physical concept involved is energy. At any point of its motion the energy (meaning the "total" energy) of the pendulum is constant, provided frictional forces are negligible. Energy is said to be a constant of the motion. In Physics problems we always look for constants. Constants allow us to determine many properties of the motion of a system. Here is a list of some pendulum problems that students usually find difficult.

1. Determine the magnitude of the acceleration of the mass when it is at the lowest point of its swing. Is it zero? Is it g?
2. What is the direction of the acceleration vector of the mass at the lowest point of its swing?
3. Determine the magnitude of the acceleration of the mass at the highest point of its swing. Is it zero?
4. Imagine that a simple pendulum of mass m and length L is set moving so that it just reaches the vertical position over the point of support. Determine the energy of the pendulum in terms of g, L and m. Neglect frictional forces.[2.5mgL]
5. Imagine that the mass is set moving and the string becomes slack before it reaches the vertical position. The mass then falls on a path that passes through the point of support. Determine the energy of the mass in this situation in terms of g, L and m. Neglect friction forces. [1.86603mgL]
6. As in question 5 but now the path of the falling mass passes through the lowest point of the swing of the pendulum.[1.75mgL]
7. As in question 5 but now the path of the falling mass passes through the horizontal through the point of support at a distance L from the point of support. [2.29904mgL]

Motors and Generators

The Motors and Generators in the NSW government syllabus is usually answered poorly in Year 12 examinations. What are the reasons for this? Firstly, magnetic fields are abstract things....we cannot see them but we can measure their effects when we do experiments. Secondly, the direction of the magnetic force vector acting on a current carrying conductor is perpendicular to both the magnetic field vector and the current vector and this presents challenges in thinking. Finally, students' exam responses sometimes become confused due to a lack of understanding of the basic terms used in this topic. Here is a list of some of the basic facts in this topic.

1. Magnetic field and magnetic force are not the same thing.
2. Current is the rate of flow of charge through a conductor.
3. Current is measured in ampere (A). Charge is measured in coulomb (C)
4. The potential difference between two points is the work done in moving a +1 C charge between the points.
5. Potential difference is measured in volts (V).
6. Power is the rate at which work is done.
7. Work is measured in joules (J). Power is measured in watts (W).
8. Current direction is the opposite to the dirction of electron flow.

9. A current carrying conductor experiences a magnetic force when it is placed in a magnetic field.

10. When a conductor moves through a magnetic field a potential difference is induced across the conductor.

11. When a current carrying conductor is placed in an external magnetic field the interaction of the external magnetic field and the magnetic field of the conductor does not exert a force on the wire.

The Value of Physics

For a moment, let us put to one side the value of Physics as a science in its own right that describes the natural world. By studying Physics we pick up skills that flow over into other subjects. A course in Physics teaches us problem solving techniques and ways of thinking that can be applied in other areas. Physics teaches us how to collect and analyse data and then how to explain our measurements in terms of fundamental laws. Physics is highly rated as a "facilitating" subject by the major universities. Many university courses require particular subjects to be studied in the final years of secondary schooling. These are the facilitating subjects. The top 24 universities in the UK, known as the Russell group, publish a document called Informed Choices, giving information for UK university entry. This document states that the top three facilitating subjects in preparation for university study are Mathematics/Further Mathematics, English Literature and Physics. 

Lenz's Law

A very important law that connects electricity and magnetism is Lenz's law. When written as an equation, this law involves two mathematical concepts, a negative sign and a first derivative. Students can perform mathematical operations involving both of these concepts. The difficulty arises, and this happens in many equations, when we try to put a physical interpretation on what the negative sign and the derivative are trying to "tell us". In Physics we need insight into what an equation is saying. To a physicist an equation is a connection between quantities that have physical meaning; an equation describes how nature behaves. As Lenz's law is a common topic in HSC, IB, Cambridge International and first year university examination papers it is worthwhile to compile a list of the statement of Lenz's law as given by the current popular Physics textbooks to see how experts interpret it.

1. Halliday, Resnick and Walker Fundamentals of Physics (14th edition) An induced current has a direction such that the magnetic field due to the current opposes the change in the magnetic flux that induces the current

2. Serway and Jewett Physics for Scientists and Engineers (8th edition) The induced current is in the direction that creates a magnetic field that opposes the change in magnetic flux through the area enclosed by the loop.

3. Sears, Zemansky, Young and Freedman University Physics (14th edition) The direction of any magnetic induction effect is such as to oppose the cause of the effect

4. Knight Physics for Scientists and Engineers (4th edition) There is an induced current in a closed, conducting loop if and only if the magnetic flux through the loop is changing. The direction of the induced current is such that the induced magnetic field opposes the change in flux.

5. Cutnell and Johnson Physics (9th edition) The induced emf resulting from a changing magnetic flux has a polarity that leads to an induced current whose direction is such that the induced magnetic field opposes the original flux change.

6. Tipler and Mosca Physics for Scientists and Engineers (7th edition) The induced emf is in such a direction as to oppose, or tend to oppose, the change that produces it.

Which do you think is clearest?

Friction.....Physics in the Real World

In many textbook problems we see the phrase "neglect air resistance" or "assume that all surfaces are smooth" in  projectile problems or mechanics problems. These assumptions allow problems to be "solvable"  using the well known equations of uniformly accelerated motion. Now let us step into the real world where dissipative agents that remove energy from a system act. What causes the frictional force between two surfaces? How can we explain friction at an atomic level? And, why is it "harder" to push start a heavy object than to keep pushing it along?  Finally, here is a discussion question on air friction. Imagine that a ball is thrown vertically upwards and the air resistance has the same size during the journey. Which stage of the flight, upward or downward, takes the longer time?