When a current flows through a wire is there an electric field inside the wire? Is there an electric field outside of the wire? When a spherical metal shell is given a static charge is there an electric field inside the shell? Is there a field in the metal? Is there an electric field outside of the shell?
Physics and Mathematics
What are the concepts in Physics and Mathematics that students find difficult? In this blog I will post tutorial points that I hope will stimulate thinking and so broaden the knowledge and understanding of students allowing them to write thorough examination responses. Tutorial notes, from a student's point of view, will be posted on the HSC, IB and university level topics of Relativity, Electromagnetism, Mechanics, Quantum Physics and Calculus. The overall purpose of these pages is to encourage and promote learning in Physics and Mathematics and so open new horizons in learning.
The term black body radiation sometimes causes confusion in examination answers. Below is a tutorial sheet on this topic.
- Define the term blackbody. Is the "blackbody" the object, a cavity in the object or a small hole in the wall of the object?
- Define the term radiation as used in this context.
- Is the Sun a perfect blackbody? Is a lump of hot coal a perfect blackbody? Is the filament of a light globe a perfect blackbody? Is a hot oven a perfect blackbody?
- Sketch a graph showing the energy released per unit wavelength on the Y axis and wavelength on the X axis for a blackbody at a constant temperature. Is this graph the same shape as the "normal curve"? Is this graph the same shape as the distribution of particle speeds curve in an ideal gas?
- Sketch a graph showing the number of oscillators (vibrating atoms in the walls of a hot object) versus the energy of each oscillator at a constant temperature. This is a key difference between the quantum theory of light and the wave theory of light. The classical wave theory of light assumes that every oscillator has the same kinetic energy at a given temperature and this (incorrect) assumption leads to the ultraviolet catastrophe.
- Sketch a graph showing the energy released per unit frequency versus frequency for a blackbody at a constant temperature.
- Sketch a graph showing the peak wavelength in the black body spectrum versus the temperature of the object.
- What does the equation E=hf mean?
- Quantum theory predicts that an oscillator can only have certain particular energies. These are given by E=(n+1/2)hf where n=0,1,2,3.... Why is this result different to E=hf?
- *A perfect blackbody has a temperature of 5778 K. What fraction of the total energy released is contained between the wavelengths of 380 nm and 700 nm? This is the range of visible wavelengths coming from a blackbody at the same temperature as the photosphere of the Sun. [39%]
- In the previous question the temperature is doubled. Is the percentage of radiation emitted as visible light increased, decreased or the same? [decreased, 31%]
Here is part of a question on Special Relativity taken from Columbia University's Physics Qualifying Examination in 2016. It was set by renown physicist and author Brian Greene. HL students aiming for a 7 must do extra reading to broaden their knowledge of Physics. Professor Greene's books make a good start. An asteroid is on a collision course with a space station located 5000 light minutes from the Earth. The asteroid is moving away from the Earth toward the space station at a speed 3/5c along a trajectory which is a straight line connecting the Earth and the space station. To save the station, NASA launches a missile from Earth at 4/5c. When the missile is launched, NASA determines that the asteroid is 400 light-minutes from Earth. How many minutes should NASA set on a timer located on the missile so that it will explode just as it catches the asteroid? [1200 minutes]
A common Assessment Task in HSC Physics is to release a ball bearing from rest on a track inclined at a small angle to the horizontal and measure the time taken (t) to move various distances (d). The average speed is then calculated (v=d/t) and a graph of v-t is drawn. A common mistake is to then say that the gradient of this graph is the acceleration of the ball. Why is this incorrect? How do we determine the acceleration of the ball?
When a current flows in a wire a magnetic field is created around the wire. Is there a magnetic field inside the wire?
Two straight parallel wires carrying currents in the same direction attract each other. Two electron beams travelling in the same direction repel each other. Why?
One of the most misunderstood concepts in the quanta to quarks section of the current NSW HSC Physics syllabus is binding energy. Listed below are some tutorial points on this concept.
- The binding energy of a nucleus is the work that must be done on the separate nucleons to assemble them from an initial state where they do not influence each other into the final nucleus.
- Binding energy has a negative sign.
- A lone proton or neutron has no binding energy.
- The total energy of a nucleus is the sum of the rest energy of each component particle and the binding energy of the nucleus.
- Energy is released in a nuclear reaction when the product nuclei are more strongly bound (more stable) than the initial nuclei (less stable).
One of the most difficult and misunderstood concepts in the ideas to implementation section of the current NSW HSC Physics syllabus is band theory. Listed below are some tutorial points on this concept.
- Bands are not the same as the orbits of electrons around a nucleus (these are called shells).
- Bands refer to an entire material not a particular atom.
- The valence band is the set of possible energy states an electron can have if it is bonded to an atom. In a full valence band conduction is not possible as there are no states available for an electron to move to (no net flow of electrons).
- The conduction band is the set of possible energy states that an electron can have if it is free of an atom. In a full conduction band conduction is not possible.
- In a conductor at room temperature there is a partial overlap of the valence and conduction bands. The valence band is full and the conduction band is partially filled. Electrons can be moved easily by an applied electric field and enter nearby unoccupied states in the conduction band. In a semiconductor at room temperature there is a small energy gap between the valence and conduction bands. The valence band has some unoccupied states and some electrons occupy states in the conduction band. An electron in the valence band can gain thermal energy and enter the conduction band creating a hole in the valence band that behaves like a positive charge carrier. In an insulator there is a large energy difference between the valence and conduction bands. The valence band is full and the conduction band is empty.
One of the most difficult and misunderstood concepts in the space section of the current NSW HSC Physics syllabus is simultaneity. Listed below are some tutorial points on this concept. Two sentences from Einstein's own popular exposition on relativity, Relativity, the Special and General Theory, Methuen, 1920, are given italics.
- A frame of reference is a measurement system (coordinate system) where time and position values of an event can be determined. A railway platform is a frame of reference. A moving train is a frame of reference. A person on the platform makes measurements in the platform reference frame. A passenger on the train makes measurements in the reference frame of the train.
- Two events are simultaneous in a reference frame if each has the same time coordinate in this reference frame.
- Every reference body (coordinate system) has its own particular time; unless we are told the reference body to which the statement of time refers, there is no meaning in a statement of the time of an event.
- Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity).
- Suppose that two simultaneous events (such as lightning strikes) occur at the different points x1 and x2 in the reference frame of the platform. The times at which these events occur in the reference frame of the train are not the same as those in the reference frame of the platform. The time interval between these events in the platform reference frame is zero. The time interval between these events in the reference frame of the train is -v/c2(1-v2/c2)-1/2(x2-x1). This expression shows that the time interval between two events depends on the location of the events and the relative speed of the reference frames. The factor c is the speed of light in a vacuum, this being the same in all inertial (non accelerating) frames of reference.
During the next 4 weeks I will list the most misunderstood concept in each of the four sections of the current NSW HSC Physics syllabus. Back emf is the most misunderstood concept in the motors and generators section. Here are some tutorial points on back emf.
- Look at a demonstration motor and a generator. Draw diagrams of these labelling the parts.
- When the magnetic flux through a coil changes an emf is induced in the coil.
- Imagine a coil of wire (solenoid) carrying a constant current. As the current flows through the windings it makes a magnetic field that passes through the coil. If the current is turned off it does not drop to zero immediately. For a short time the current keeps flowing at a value that exponentially falls towards zero. During this time interval an emf is induced in the coil due to the changing magnetic flux through it causing the current to "keep going" for a short time, opposing the change that is happening. Sketch graphs showing (A) the current flowing in the coil versus time, (B) the potential difference across the coil versus time.
- As the coil of a motor rotates in a magnetic field the changing magnetic flux through it induces an emf in the coil in accordance with Lenz's law. This emf tends to make an induced current in the opposite direction to the current supplied to the coil and is called a back emf.
- When a motor is first turned on the current in the coils is large as there is no back emf. As the coils gain speed the back emf increases and the current in the coil decreases.
The IB Physics examination papers are on October 31 and November 1. From now until this date I will list questions to help students revise for their Physics examination. This list will be updated over the next six weeks.
Students can increase their marks significantly during this period if they are positive, organised and have a plan. What is required is a balanced approach to study with adequate nutrition, sleep, exercise and recreation.
- Powers of Ten The radius of a proton is 0.88 fm. What is the density of a proton in kg/m3? (A) 10-6, (B) 102, (C) 1010, (D) 1018, (E) 1026
- Uncertainty Analysis The measured uncertainty in the length of a pendulum is 2%, the uncertainty in its mass is 3% and the uncertaity in the acceleration due to gravity is 4%. What is the percentage uncertainty in its period of oscillation? (A) 1% (B) 2%, (C) 3%, (D) 4%, (E) 5%
- Dimensions Surface tension is force per unit length. What are the dimensions of surface tension? (A)ML-2 (B)MT-2 (C)ML-2T-2 (D)L-1T-2 (E) ML-1T-2
- Relative Velocity At noon a ship S is 10 km west of a tanker T. The velocity of S relative to the Earth is constant at 12 km/h N30°E and the constant velocity of T relative to the Earth is 4.0 km/h S60°W. What is the distance between the vessels at 3pm on the same day? (A)3.2 km (B) 21 km (C) 41 km (D) 48 km (E) 58 km
- Change in Velocity A particle moves at a constant speed v around a circular path. What is the magnitude of its change in velocity after it turns through an angle 2𝜽? (A) vcos𝜽 (B) vsin𝜽 (c)2vcos𝜽 (D) 2vsin𝜽 (E)vsin2𝜽
- Constant Acceleration A stone is thrown vertically upwards from the edge of a building and strikes the ground after a time t1. When the stone is thrown vertically downwards from the same point at the same initial speed it hits the ground after a time t2. The time to fall if the stone was released from rest is (A) t1+t2 (B) t1-t2 (C) (t1t2)1/2 (D) (2t1t2)1/2 (E) (0.5t1t2)1/2
- Resultant Force A marble dropped from a height h above soft sand penetrates a distance d into the sand before comng to rest. If the marble is dropped from a height 2h the distance that it travels in the sand before coming to rest is (A) d (B) 2d1/2 (C) (2d)1/2 (D) 21/2d (E) 2d
- Stretching a Spring An unstretched spring of force constant k has one end tied to a vertical wall and a mass m at its other end. A constant horizontal stretching force F is applied to the mass. The extension of the spring in the horizontal direction when the mass comes to rest is (A) F/2k (B) F/k (C) 2F/k (D) 3F/k (E) 4F/k
- Explosion A mass M is initially at rest. It breaks up into two equal pieces having a total kinetic energy E. What is the magnitude of the relative velocity of the two pieces after explosion? (A)(2E/m)1/2 (B) (4E/m)1/2 (C) (8E/m)1/2 (D) (2m/E)1/2 (E) (4m/E)1/2
- Ideal Gas Boltzmann's constant is equal to (A) R/NA (B) RNA (C) NA/R (D) R+NA (E) R/NA2
- Ideal Gas Particles A cylinder contains a volume V of helium gas at a temperature T. A second cylinder contains a volume 2V of argon gas at a temperature T. (A) the average speed of a helium atom is the same as that of an argon atom, (B) the average speed of a helium atom is less than that of an argon atom, (C) the average speed of a helium atom is greater than that of an argon atom, (D) each container contains the same number of particles, (E) the pressure exerted by each gas is the same
- Change of State A copper block is heated in a Bunsen flame. The copper block is then placed on a large block of ice at 0°C in which it becomes half buried. What was the initial temperature of the copper block? (A) 5°C (B) 22°C (C) 42°C (D) 62°C (E) 82°C
- Specific Heat Capacity One kilogram of copper and the same mass of water are heated using the same Bunsen burner. (A) the rate of temperature increase of both objects is the same, (B) the rate of temperature increase of water is greater than that of copper, (C) the rate of temperature increase of copper is greater than that of water, (D) the rate of internal energy increase is the same for each object, (E) the rate of internal energy increase for water is greater than that of copper.
- Electric Current Two copper wires, P and Q, at the same temperature have the same battery of zero internal resistance connected across each of them. The length of P is twice that of Q. The drift speed of the electrons in P is (A) the same as that in Q, (B) greater than that in Q, (C) less than that in Q (D) is zero (E) depends on the diameter of the wire.
- Resistance Four equal resistors R form the sides of a square. Another resistor R is placed across a diagonal of the square. What is the total resistance between the other two corners of the square? (A)R/4 (B) R/2 (C) R (D) 2R (E) 4R
- Electric Field Strength Point charges of +4Q and -Q are placed a distance d apart in a vacuum. The resultant electric field is zero at a distance (A)2d to the left of -Q (B) d to the left of -Q (C) d to the right of -Q (D) 1.5d to the right of -Q (E) 2d to the right of -Q
- Electrical Power A battery of emf 12V and constant internal resistance is connected to a 6Ω resistor. The power of the 6Ω resistor is greatest when the internal resistance is (A) 0Ω (B) 3Ω (C) 6Ω (D) 9Ω (E) 12Ω
- Multiple Point Source Interference Monochromatic, coherent light is produced by two point sources. The intensity of the maxima produced on a screen is I. The experiment is repeated using 4 point sources with the same source spacing. The intensity of the principal maxima on the screen is (A) I, (B) 2I, (C) 4I, (D) 8I (E) 16I
- Secondary Maxima The number of secondary maxima between the principal maxima in the interference pattern produced by 4 coherent, monochromatic point sources of light is (A) 0, (B) 1, (C) 2, (D) 3, (E) 4
- Interference Pattern Compared to the interference pattern produced by 2 coherent, monochromatic point sources the intereference pattern produced by 8 sources of the same spacing (A) has brighter principal maxima that are closer together, (B) has brighter principal maxima that are farther apart, (C) has brighter principal maxima that are the same distance apart, (D) has principal maxima of the same intensity that are the same distance apart, (E) has principal maxima of the same intensity that are the closer together.
- Single Rectangular Slit Diffraction Monochromatic light passes through a single narrow slit. The intensity pattern is observed on a screen at a large distance from the single slit. Why does the intensity not stay at zero after the central maximum?
The HSC Physics examination is on October 30. From now until this date I will list questions to help students revise for their Physics examination. This list will be updated during the next two months.
Students can increase their marks significantly during this period if they are positive, organised and have a plan. Remember that the HSC Physics examination is not "difficult Physics". What is required is a steady approach to study with adequate nutrition, sleep, exercise and recreation.
- Accelerating Reference Frame A train moving in a straight line on horizontal ground is accelerating to the east. A ball is released from rest (relative to the floor of the train) from a height of 2.0 m. Sketch the path of the ball in the reference frame of the (i) train, if the train is moving to the east (ii) Earth, if the train is moving to the east (iii) train, if the train is moving to the west, and (iv) Earth, if the train is moving to the west.
- AC Motor An AC generator is reversed so that AC is now fed into the coil through the brushes that rub against the slip rings. Does the coil spin?
- Photocells and Solar Cells Describe the differences between a photocell and a solar cell.
- Cloud Chamber Describe how a cloud chamber detects the presence of radiation. Which radiation produces the (i) thickest tracks? (ii) thinnest tracks? (iii) straightest tracks? (iv) Can gamma rays be directly observed in a cloud chamber?
- Earth Satellite A satellite is in orbit above the Earth's equator. Determine the altitude of the satellite if it (i) appears stationary above the equator, (ii) appears to pass overhead (as seen by a person on the equator) from west to east every 8 hours, (iii) appears to pass overhead from east to west every 8 hours.
- Magnetic Field of Two Currents Two long straight parallel wires at a distance d apart carry currents of I and 4I in opposite directions. Where is the resultant magnetic field strength zero?
- Heinrich Hertz Describe how Hertz discovered the photoelectric effect.
- Strong Nuclear Force Sketch graphs showing the force between two nucleons and the potential energy of two nucleons as a function of their distance apart.
- Artificial Gravity A space station far away from any planet or star has the shape of a torus of outer radius 500 m. It rotates about an axis through its centre and perpendicular to its plane at constant rate of 10 rev/min. An astronaut of mass 80 kg is standing on the inside of the outer surface of the space station (i) describe the force/s acting on the astronaut, (ii) determine the weight of the astronaut (iii) an apple is released from rest inside the spacestation at a distance of 1 m from the outer wall. Describe the subsequent motion of the apple.
- EMF and Back EMF What is the difference between an induced EMF (as in a generator) and a back EMF (as in a motor)?
- CRO Describe how an electron beam produces a two dimensional image on the screen of a cathode ray oscilloscope.
- Electron in the Hydrogen Atom According to quantum physics, how do we describe the electron in the hydrogen atom?
- Maximum Range A basketball is thrown at a speed U an angle 𝜽 to the horizontal from of a height h above the ground. Show that the angle that gives the maximum range on the ground is given by csc2𝜽max = 2(1+gh/U2).
- Magnetic Force Two electrons are moving at the same velocity v relative to the laboratory side by side in parallel paths a distance d apart in the laboratory reference frame. Determine the magnitude of the resultant force between the electrons in the (a) reference frame of the electrons, (b) laboratory reference frame.
- Planck and Einstein Outline the political views of Max Planck and Albert Einstein. How did this affect their scientific work?
- Enrico Fermi Assess the contribution of Enrico Fermi to Physics.
- High Speed Electron An electron is accelerated from rest through a potential difference of 100,000 V using a particle accelerator. Find the final speed of the electron.
- Loudspeaker Describe how a loudspeaker uses the motor effect to produce sound waves.
- Blackbody Radiation Curve Explain why the blackbody radiation curve has a peak at a certain wavelength. Does a photon of this wavelength have the greatest energy?
- Linear Accelerator Describe how a linear accelerator produces a high speed particle.
- Projectile Practical A marble is projected horizontally at a known constant speed from a spring gun at various heights above the floor and the horizontal range is measured in each case. (i) Plot this data so that a straight line graph is produced, (ii) how can the graph be used to determine g?
- Step-Up Transformer A step-up transformer is used to increase the voltage of an AC supply with respect to the ground. Explain how this reduces reduces the heat loss along the transmission line from the power station.
- Phonons A student in an answer to a question on the BCS theory mentions a "phonon". Outline what a phonon is.
- Forces of Nature Outline the force that holds together, giving the carrier particle in each case (i) the nucleons in an alpha particle, (ii) the quark combination that produces a proton
- Michelson and Morley Describe the result of the Michelson and Morley experiment.
- Torque and Work Describe the difference between torque and the work done by a force.
- n-type Semiconductor Is an n-type semiconductor negatively charged?
- Binding Energy The mass of the two protons, two neutrons and two electrons is greater than their combined mass when they form a helium atom. Describe why this is so.
- Gravitational Potential Energy When is the gravitational potential energy of two masses least? When they are close together or far apart? Explain.
- Cathode Rays Describe Thomson's experiment to measure the charge to mass ratio of cathode rays.
- Bragg Diffraction Using a labelled diagram, explain using wave concepts how constructive interference occurs when x-rays strike a solid object.
- Line Spectrum Explain how the presence of a line spectrum indicates the existence of energy levels in the hydrogen atom.
- Orbital Decay In an answer a student states that the speed of an orbiting satelite decreases when it is subjected to air drag. Is this correct?
- Back EMF Does the back emf eventually stop the coils of a DC motor from spinning? Explain.
- Electromagnetic Waves Describe how Heinrich Hertz experimentally produced radio waves and identified these as belonging to the same group as light waves.
- Moderator A neutron given off in a fission reaction has a kinetic energy of 6.0x10-13 J. This is reduced to 6.0x10-21 J by causing the neutron to make a series of collisions with carbon nuclei in the moderator. The fractional loss of kinetic energy of a neutron at each collision is 0.14. Find the number of collisions involved in this process. 
- Polar Orbit A satellite is placed in an orbit that passes over points near the north and south poles of the Earth. Does this satellite possess more energy than a similar satellite in orbit at the same altitude above the equator?
- Power Which combination has the higher power? A large resistance R connected to a battery or a smaller resistance r?
- Carbon Carbon has 4 electrons in its outer shell. (a) is the outer shell the same as the valence band? (b) is its valence band full? (b) is carbon a semiconductor?
- Electron Wavelength Compared to an electron in the ground state, the wavelength of an electron in the first excited state of the hydrogen atom is (A) 1/4, (B) 1/2, (c) the same, (D) twice as large (E) 4 times as large
- Acceleration A student doing a physics problem calculates the acceleration of an electron as 2.0x1014 ms-2. Is this possible?
- DC Motor A DC motor has a split-ring commutator and the coils are in a radial magnetic field. An open switch is in the circuit. The switch is now closed. Sketch graphs showing (A) the current flowing in the coils in terms of time (B) the torque exerted by the magnetic field on the coil in terms of time.
- Band Structure A conductor and a semiconductor are at room temperature. Choose the correct statement.(A) the conduction band of the conductor is empty (B) the conduction band of the semiconductor is empty (C) the conduction band of a semiconductor is partially filled (D) the conduction band of a conductor is full (E) the valence band of a semiconductor is full
- Neutron A neutron (A) is not affected by a magnetic field (B) has no spin about its axis (C) has a magnetic moment causing it to align itself with a magnetic field (D) has zero magnetic moment since it has zero charge (E) is made of a proton and an electron.
- Orbit A satellite of total mass m moves at a constant speed v in a circular orbit around a planet. It suddenly fires a projectile of mass 0.5m at a speed of 0.5v in the opposite direction to its initial velocity. After it launches the projectile the satellite (A) continues in the same orbit with the same speed, (B) moves to a higher orbit (C) escapes from the planet, (D) moves to a lower orbit, (E) continues in the same orbit with a shorter period
- Soft Iron Core The role of the iron core in a transformer is to (A) reduce the eddy currents, (B) connect together the primary and secondary coils, (C) increase the magnetic flux passing through the coils, (D) carry all of the magnetic field of the primary coil through the secondary coil, (E) change the voltage of the primary coil.
- Type 1 and 2 Superconductors Superconducting magnets use (A) type 1 superconductors since B=0 inside them, (B) type 1 superconductors since B≠0 inside them, (C) type 2 superconductors since B≠0 up to a certain value, (D) type 2 superconductors since B=0 inside them, (E) both type 1 and type 2 superconductors.
- Bohr's Postulate In 1913 Niels Bohr proposed that mvr = n h/(2𝜋), where n is the principal quantum number, m is the mass of the electron, v is the speed of the electron and r is the radius of the stationary state. (A) the original Bohr theory is still used to explain the hydrogen atom, (B) Bohr's equation gives the magnitude of the electron angular momentum that agrees accurately with modern experiments, (C) Bohr's equation was replaced since quantum mechanics predicts that for each value of n there are n possible values of the orbital angular momentum, (D) the electron is now considered to have non-zero orbital angular momentum in its ground state, (E) the electron is now considered to have zero orbital angular momentum for every value of n.
- Projectile1 A ball is thrown at 49m/s at 60° to the horizontal. Neglecting air resistance, the times after projection when the velocity vector makes an angle of 30° with the horizontal are (A)2.16s, 6.50s (B) 2.33s, 6.33s (C) 2.89s, 5.77s (D) 3.66s, 5s (E) 4.00s, 4.66s.
- Projectile2 A ball is thrown at 49m/s at 60° to the horizontal. Neglecting air resistance, the times after projection when the speed of the ball is 35m/s are (A) 0.48s, 8.18s (B) 1.10s, 7.56s (C) 1.78s, 6.88s (D) 2.56s, 6.10s (E) 4.00s, 4.66s.
- Air Resistance The drag force acting due to air resistance on a projectile is proportional to the square of the speed of the projectile. A projectile is thrown vertically upwards at 49m/s. The maximum height reached is 60m. The projectile is now thrown at the same speed at an angle to the vertical. The maximum possible range of the projectile on the horizontal is (A) 30m (B) 60m (C) 94m (D) 120m (E) 180m
- Speed A projectile is thrown at 49m/s at 60° to the horizontal in the absence of air resistance. The gradient of the distance travelled-time graph for the projectile has a maximum value of (A) 98m/s, (B) 75m/s, (C) 49m/s, (D) 42m/s, (E) 24.5m/s.
- Average Speed Two identical projectiles, P and Q, are thrown on level ground at the same speed. P is thrown at 30° to the horizontal and Q is thrown at 60° to the horizontal. Neglecting air resistance, (A) the range of P is greater than Q, (B) the average speed of each projectile is the same during their flight, (C) the average speed of P is greater than the average speed of Q, (D) the average speed of P is less than the average speed of Q, (E) the average velocity of each object is the same during the flight.
The HSC Physics topic Quanta to Quarks includes the concept of the wave nature of an electron.
- When we draw an electron as a wave what does the height of the wave represent?
- Is the electron mass spread out along the wavelength?
- In quantum mechanics an electron is described by a wave function. Is this the same as the de Broglie matter wave?
- What is the physical interpretation of the wave function?
- According to quantum mechanics what actually is the electron in the hydrogen atom? Is it a particle moving in an orbit?
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.
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:
- Reliability An experiment is performed to measure the acceleration due to gravity. Outline how we can make the result reliable.
- Ceramic Insulators Outline the function of the ceramic insulators on electricity transmission lines.
- Silicon and Germanium Describe the advantages of silicon over germanium in semiconductor devices.
- Zeeman Effect Describe the Zeeman effect. How is it explained?
- 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.
- 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.
- 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.
- Matter Waves A student states that an electron is described as a particle moving along a standing wave. Explain why this is incorrect.
- 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.
- 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
- Resistance A copper wire is maintainted at a temperature of 2K. Describe how an electric current flows through the conductor.
- Uncertainty Principle An observation is made of the position of an electron. How does this affect the momentum of the electron?
- 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.
- 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.
- 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.
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.
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).
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.
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.
[2𝛑, 2𝜋, 𝜋, 2𝜋, 2𝜋, 𝜋, 𝜋, 𝜋/2, 4𝜋, 2, 6𝜋, 2𝜋, 𝜋, 4𝜋, 2𝜋, 2𝜋]
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.