IB Physics Capacitance

A tutorial sheet on capacitance, a section in HL topic 11, is given below.

1. What is the meaning of the term capacitance?
2. For a given capacitor at a constant temperature, is the "capacitance" constant?
3. Write down an expression for the capacitance of a charged sphere of radius R in a vacuum.
4. Express the unit of capacitance in terms of SI base units.
5. An uncharged capacitor of capacitance C is connected in series with an open switch across a 12 V DC battery of zero internal resistance and a resistance R. The switch is now closed. Describe the build up of charge on the capacitor.
6. A capacitor of value C has a charge Q. It is connected in series to a resistor R and an open switch. The switch is now closed. Describe the current through the resistor.
7. Two capacitances C1 and C2 are connected in series across a battery. Find an expression for the total capacitance. Show your working.

8. Two capacitances C1 and C2 are connected in parallel across a battery. Find an expression for the total capacitance. Show your working.

9. An uncharged capacitor is charged by being connected by wires to a battery of constant emf and zero internal resistance. (a) Is the current in the connecting wires equal to the rate of increase of charge on the plates? (b) Is there a current in the space between the plates? (c) Is there a magnetic field in the space between the plates? (d) Is the work done by the battery equal to the energy stored in the capacitor?

10. In the IB Physics Data Booklet a formula is given for the capacitance of a parallel plate capacitor. What are the assumptions for this formula?
11. To take into account the finite area of the plates of a capacitor see the Berkeley Physics Course Volume 2 Electricity and Magnetism by E M Purcell (red version page 97, blue version page 105) for a table of end correction values. Another reference is Physics Education, Vol 17, Number 2, March 1982, page 35.

HSC Physics Bouncing Superballs

A tutorial sheet involving collisions is given below.

1. A tennis ball is dropped from a height h1. It bounces to a height h2. Is momentum conserved in the collision? Is kinetic energy conserved in the collision? Is energy conserved in the collision?
2. A steel ball bearing of mass M moving at a speed U collides in a line with two balls of mass M that are at rest. Why doesn't each ball move at U/3 after the collision?
3. Imagine two superballs, of different mass, placed one on top of the other and dropped from a height H very large compared to the radius of each ball. Assuming an elastic collision with the ground, what is the maximum height to which the top superball can bounce? [9 H]
4. For more multi-highball problems see Anthony Anderson, Physics Education, vol 34, number 2, March 1999, page 76.

IB Physics Phase Difference

A tutorial sheet on phase difference is given below.

1. What does the term "in phase" mean?
2. What does the term "180 degrees out of phase" mean?
3. How do we define phase difference?
4. Are points on a progressive wave all in phase?
5. A progressive wave has a wavelength of 1.0 m. What is the phase difference between two points on the wave that are 0.8 m apart?
6. Are points on a standing wave on a string all in phase?
7. The wavelength of the standing wave on a string is 1.0 m. What is the phase difference between two points on the string that are 0.8 m apart?
8. In an AC transformer are the primary and secondary voltages in phase?
9. In an AC transformer are the primary and secondary currents in phase?
10. For application to transformer problems see S J Osmond, Physics Education, vol 17, no 5, Sep 1982 p236

IB Physics Why is it Hotter in Summer than in Winter?

A tutorial sheet of true/false questions on this work in topic 8 is given below.

1. Summer occurs when the Earth in its orbit is closest to the Sun.
2. The angle of the Sun above the horizon causes less energy to strike the surface of the Earth in winter.
3. The low Sun angle causes a bundle of Sun's rays to be spread over a larger area of the surface of the Earth in winter than summer.
4. The Sun is lower above the horizon in winter causing a lower amount of energy to enter the atmosphere.
5. The Sun is above the horizon for a shorter time in winter reducing the time for a bundle of Sun's rays to heat the surface of the Earth.
6. The atmosphere is not heated by the Sun but by reflected longer wavelength radiation from the Earth. In winter each square metre of surface reflects less energy and so the surrounding atmosphere is at a lower temperature.
7. The variation in the maximum angle made by the Sun above the horizon during the year is due to the inclination of the Earth's spin axis at 66.5° to the plane of the Earth's orbit.
8. People at the equator experience 12 hours of night and day every day of the year.
9. Sunset in Melbourne on January 1 is before sunset in Sydney.
10. Sunrise in Melbourne on January 1 is after sunrise in Sydney.
11. See Roy L Bishop, Journal of the Royal Astronomical Society of Canada vol 87 No 5 p346 1993

HSC Physics Does ∆U always equal mg∆h?

A tutorial sheet of gravitational potential energy questions is below.

1. What do the symbols in the equation ∆U=mg∆h mean?
2. When is it correct to use the equation ∆U=mg∆h?
3. What is the meaning of the equation U=-GMm/r?
4. When should we use the equation U=-GMm/r?
5. An object of mass m is lifted through a vertical height ∆h from the surface of the Earth. If the radius of the Earth is R shew that ∆U=mg∆hR/(R+h), where g is the acceleration due to gravity at the surface of the Earth.
6. An apple of mass 85 g falls from a height of 2.5 m to the ground. What is the change in gravitational potential energy of the apple? [0.99999961mg∆h ≅ mg∆h, -2.08 J]
7. Mt Everest is 8848 m sbove sea level. If the radius of the Earth is 6400 km show that the increase in gravitational potential energy of a climber of mass m in going from sea level to the top of Mt Everest is 0.9986mg∆h.
8. The International Space Station (ISS) orbits the Earth at an average altitude of 408 km. If its mass is 419,700 kg determine the gain in gravitational potential energy in lifting this mass from the surface of the Earth to its final height.[0.9401mg∆h, 1.58x10¹² J]

IB HL Physics Resolution

Image resolution is a poorly understood topic in IB HL Physics. A tutorial sheet is given below.

1. A double star is examined through a telescope. The stars appear too close together to be resolved. To resolve the image of the stars a coloured filter is placed in front of the telescope. Which coloured filter could allow the images to be resolved? Red or violet?
2. Two stars are observed through a telescope and appear too close together to be resolved. Does increasing the magnification of the telescope resolve the images?
3. The headlights of a stationary car subtend an angle of 1" at a distant point. The human eye can distinguish between two images that are 1' apart. If the wavelength of the light is 550 nm, determine the minimum diameter of a telescope that can allow the headlights to be resolved as separate images.
4. A satellite is in orbit 350 km above the surface of the Earth. Newspint has a size of 3.0 mm. Determine the diameter of the aperture of a camera in the satellite that will allow a newspaper on the Earth to be read from orbit in light of wavelength (a) 600 nm, (b) 400 nm. The resolution of the human eye is 1'.
5. If we increase the amount of light entering a telescope, keeping the diameter and the wavelength constant, does this improve the image resolution? Why?

HSC Physics Rutherford's Nuclear Atom

A tutorial sheet of miscellaneous questions on Rutherford’s nuclear model is given below.

1. What is a spinthariscope?
2. At which university was Rutherford a professor when his model was proposed (1911)?
3. Why were alpha particles used in the famous experiment?
4. Why was a gold foil used?
5. Do the alpha particles hit the gold atoms in the foil?
6. Where are the electrons in Rutherford's model of the atom and what are they doing?
7. Did most of the alpha particles pass through the gold foil with small amounts of scattering? Give a reason.
8. Some alpha particles were scattered through angles greater than 90°. Give the approximate number and give a reason for this.
9. In an alpha particle scattering experiment the speed of the alpha particles is doubled. Will the number of scattered alpha particles at a particular angle increase, decrease or stay the same?
10. In an alpha particle scattering experiment the thickness of the foil is doubled. Will the number of scattered alpha particles at a particular angle increase, decrease or stay the same?
11. An alpha particle scattering experiment is repeated with a pure thin metal sheet of double the atomic number of the first case. Will the number of scattered alpha particles at a particular angle increase, decrease or stay the same?
12. Draw the famous diagram showing the scattering of an alpha particle by a positively charged gold nucleus. (Many people draw this diagram incorrectly, be careful).
13. In question 12 draw the diagram for a negatively charged particle fired at a gold nucleus. Is the path of a negatively charged particle, fired with the same initial kinetic energy, the same as that of a positively charged particle of the same magnitude?
14. Is the scattering of alpha particles proof that the nucleus is positively charged?
15. Rutherford's scattering equation was deduced using classical physics and then later quantum mechanics, an example of classical physics principles giving the correct result for an atomic process. State this equation and find out who derived it for Rutherford. Use the equation to sketch the graph of s(𝜽), where 𝜽 is the angle through which an alpha particle is deflected.
16. How did Geiger determine the number of alpha particles scattered through each angle?

IB Physics Voltage

A tutorial sheet on voltage and potential difference is given below.

1. Is voltage the same as potential difference?
2. Define electric potential.
3. Define electric potential difference.
4. Point A in an electric field has a potential of +15V. Point B in this field has a potential of -12V. (a) What is the potential difference between A and B? (b) What work is done by an external force in moving a +1C charge at a constant speed from A to B? (c) what work is done by the field when a proton moves from A to B?
5. Is voltage the same as energy? Explain.
6. Is potential difference the same as energy? Explain.
7. Sketch the equipotential lines around the point charges +Q and -Q.
8. Sketch the equipotential lines around the point charges +4Q and -Q.
9. Can electric field lines ever intersect?
10. Can equipotential lines ever intersect?

HSC Physics Fundamental Particles

1. What is meant by the term fundamental particle?
2. Is a neutron a fundamental particle?
3. List three fundamental particles.
4. Is a photon a fundamental particle?
5. Is a positron a fundamental particle?
6. Do an electron and a positron have opposite spins?
7. In beta minus decay a neutron "converts" into a proton. How can a neutron convert into a proton?
8. Radioactive materials emit gamma rays. Where does the energy of a gamma ray come from?
9. Does a positron have a negative energy? [No, Dirac said that this would make the "dynamical relations all wrong", see Physics Today, June 2001, p63]
10. Does a fundamental particle have a radius? [no]
11. This year the radius of a proton was measured as 0.831±0.014 fm. This is given in Nature , 575, 61-62, 2019

Angle Questions

A tutorial sheet on HSC/IB Physics questions involving angles is given below.

1. A metal bar of length 1.5 m points to the north. The bar can turn about an axis through the origin. A rope is tied to the end of the bar. Determine the torque exerted by the rope when (a) the rope pulls to the north with a force of 6.0 N, (b) the rope pulls to the east with a force of 4.0 N, (c) the rope pulls in the direction E40°S with a force of 3.0 N, (d) the rope pulls in the direction N16°W with a force of 5.0 N.
2. A uniform magnetic field of 3.0 mT comes out of the page. A straight wire of length 30.0 cm carries a current of 600 μA in the direction N23°E. Determine the magnetic force acting on the wire.
3. A uniform rod of length 2.0 m is placed with its upper end against a smooth vertical wall and its lower end on smooth horizontal ground. The rod's upper end initially makes an angle of 45° with the wall. The rod is released from rest. What is the angle at which the rod loses contact with the wall as its slips?
4. A cannon can fire a projectile at 150 m/s. The cannon is situated on the horizontal ground level. The target is on a horizontal plateau 800 m above the ground and is located at a distance of 100 m from the edge of the plateau. At what angles to the horizontal can the projectile be fired if it is to hit the target? Neglect air resistance and take g=9.81 m/s². [87.7° at 41.5m from the cliff down to 61.4° 1168 m from the cliff]
5. In the previous question the cannon can now fire at any speed. What is the least speed of firing if the target is to be hit, what is the angle of firing to the horizontal and the distance from the cliff where this occurs? [132 m/s, 72.4°, 465 m]
6. A wall is 20 m tall and 10 m wide. A ball is thrown from ground level. What is the least speed of projection of the ball if it is to pass over the wall? What are the angle of projection to the horizontal and the distance of the point of projection from the wall? [22.7m/s, 63.8°, 15.9 m]
7. A wall has the form of a semi-circle of radius 10 m. What is the least speed at which a ball can be thrown from ground level and pass over the wall? What is the angle of projection and what is the distance from the wall at which it is thrown? [16.7m/s,57.8°,12.9m ]
8. A wall has the shape of a circle of radius 5.0 m. What is the least speed of a particle if it is to pass over the wall from ground level? [15.7 m/s, 63.7°, 10.1 m]

HSC Potential Difference

A tutorial sheet on the concept of potential difference is given below.

1. Two wires, one of resistance R and the other of resistance 2R are connected in parallel to a 12 V battery of zero internal resistance. Compare the potential differences across each wire.
2. Two wires are made of the same material and have the same diameter. One wire has a length L and the other wire a length 2L. Thw wires are connected in series with a 12 V battery. (a) Which wire has a greater potential difference across its ends? (b) Which wire releases the most heat energy in one second?

IB Physics Internal Resistance

A sheet on topic 6.3 is given below

1. A battery of emf 6.0 V is connected to an 8 Ω resistor causing a current of 0.5 A to flow. What current flows when a 4 Ω is placed in the circuit?
2. A battery of emf 12 V has an internal resistance of 2 Ω. The battery is connected to a 6 Ω load resistor. (a) What is the power of the battery? (b) What is the power of the load resistor? [13.5W, 13.5W]
3. A battery of emf 12 V has an internal resistance r. When the battery is connected to a 8 Ω resistor the power of the 8 Ω is 11.5 W. What is the value of r? [2 Ω]
4. Is the power of the load resistor always equal to the power of the battery?
5. Is the power of the internal resistance always equal to the power of the load resistor?

Mathematical Tripos Dynamics and Relativity 2019

Listed below are my answers to the MT Dynamics and Relativity questions in Paper 4 of Part 1A held in June 2019. Answers to MT D&R questions will be added each October to correspond to the start of Michaelmas Term.

1. Question 3A. uln2
2. Question 4A. t = √ (m/(ɣg) ) e^-2ɣh/m tanh^-1( √(1-e^-2ɣh/m) )
3. Question 9A. A=-1/4, B=3/4
4. Question 10A. (a) x = 1-r+rcos(⍵t), y = rsin(⍵t), z = vt, where r = mv/( qB √ (1-2v²/c² ) ) and ⍵=v/r.
5. Question 10A. (b) (i) (0,0), three saddle points also exist where the equilibrium is stable in one direction (ii) stable along x and y axes. At (0,0) ∂V/∂x=0, ∂V/∂y=0, ∂²V/∂x²>0, ∂V/∂y=0,∂²V/∂y²>0 (iii) A=3^5/4/(2^3/2√𝜋). At saddle point ⍵=1.389q/√(m𝜀) taking a=1

IB Physics Series and Parallel Resistors

A tutorial sheet on series and parallel resistors is given below.

1. A 2 Ω and a 4 Ω are place in series with a battery. Which resistor has the greater power?
2. A 2 Ω and a 4 Ω are placed in parallel across a battery. Which resistor has the greater power?
3. A 2 Ω and a 4 Ω are connected in parallel. In series with this parallel group is a 1 Ω. When a battery of zero internal resistance is connected across this combination, which resistor has (a) the greatest power? and (b) the least power? [1 Ω,4 Ω]
4. Two 5 Ω resistors are placed in parallel. In series with this combination is a 10 Ω. The power of the 10 Ω is 20 W when the entire combination is connected across a battery of zero internal resistance. What is the power of the 10 Ω when one of the 5 Ω resistors is removed? [13.9 W]

IB Physics MeV in Relativity Option

A tutorial sheet of IB questions on units is given below.

1. An electron has a momentum of 4.0 MeVc^-1. What is the energy of the electron?
2. A particle has a rest energy of 2.0 MeV. The momentum of the particle is 3.0 MeVc^-1. What is the total energy of the particle?
3. A particle has a mass of 2.0 MeVc^-2. What is its momentum if its speed is 0.90c?
4. A particle of mass 0.8MeVc^-2 is accelerated from rest through a potential difference of 2.0 GeV. What are the final energy and momentum of the particle in the laboratory reference frame?
5. A proton has a speed of 0.995c relative to the laboratory. What is the energy of this particle in the rest reference frame and the laboratory reference frame?

IB Physics Electric Current

A tutorial sheet of electric current questions is given below.

1. The density of copper is 8960 kgm^-3. If the molar mass of copper is 63.5 g, determine the number of free electrons per cubic metre of copper if each atom provides one free electron. [8.5x10²⁸]
2. Determine the electron drift speed in a copper wire of diameter 2.0 mm carrying a current of 1.0 A.
3. List some common metals that have a free electron density greater than that of copper.[aluminium, iron, magnesium, tin, lead, zinc, magnesium]
4. What is meant by a free electron?
5. Explain why a wire heats up when a current passes through it.
6. A student in an answer states that "resistance in a copper wire is due to collisions between the electrons and the positively charged copper atoms". Analyse this statement.
7. Two metal wires are made of the same material and are at the same temperature. The potential difference across each wire is the same. If the length of the second wire is one-half that of the first wire and the diameter of the first wire is two times that of the second wire in which wire does a greater current flow?

IB Coulomb's Law with Dielectrics

A tutorial sheet of problems with dielectrics separating the charges is given below

1. Two point charges, +2.0 nC and +2.0 nC are placed 5.0 cm apart in a vacuum. Draw the electric field around the charges and determine the electric force between the charges.
2. The point charges in question 1 are now placed in a material of relative permittivity 4. Draw the electric field lines. Give a reason why the electric force between the charges increases/decreases/stays the same.
3. A point charge -4.0 nC is at a height of 6.0 cm above a deep, still, freshwater lake. Draw the electric field lines in air and water. If the dielectric constant of water is 81 determine the electric force on the charge.[See p184 Electromagnetism by Lorrain and Corson, attraction, k(K-1)/(K+1)q²/(4d²), 9.75x10^-6N]
4. Two point charges +6.0 μC and -6.0 μC are placed 12.0 cm apart in a vacuum. An infinite rectangular slab of dielectric material of thickness 6.0 cm and relative permittivity 4 is placed with its surface 3.0 cm from each charge perpendicular to the line joining the charges. Draw the electric field lines and determine the electric force acting on each charge.

HSC Physics Atoms to Universe

1. Atoms emit discrete wavelengths of light. Explain why a heated iron bar changes colour as its temperature changes.
2. The energy of an electron in the ground state of a hydrogen atom is -13.6 eV. (a) How can an electron have a negative energy? (b) What is the energy of an electron in the first excited state? (c) determine the value of the three longest wavelengths in the Balmer series for hydrogen.

3. A proton moves at a constant speed of 0.90c. The proton moves once around the tube of a particle accelerator of circumference 20.0 km. (a) What are the momentum and the energy of the proton in the laboratory reference frame? (c) What is the distance travelled by the proton in its own reference frame?

4. more to come

HSC Physics Photoelectric Effect

A tutorial sheet of problems on the photoelectric effect follows.

1. What factors determine the intensity of light shining on a surface?
2. If the light shines at 30° to the surface of the metal does this change the number of photoelectrons released compared to when the light shines perpendicular to the surface?
3. The frequency of light is increased but the intensity of the light is kept constant. Describe how this affects the rate at which electrons are released from the metal surface on which the light falls.
4. Sketch a graph showing the maximum kinetic energy of the escaping photoelectrons versus the wavelength of the incident light.
5. Why is the maximum kinetic energy calculated in the photoelectric effect? Do some escaping electrons have more kinetic energy than others?

IB Energy Topics

A tutorial sheet on topic 8.1 energy sources and 8.2 thermal transfer is given below.

1. The intensity of the Sun's radiation at the Earth's orbit is I. If the Earth's albedo is 𝛼 determine the average intensity over the surface of the entire Earth. [I(1-𝛼)/4]
2. An Earth-like planet is discovered orbiting a distant star at an average distance of 2.7x10¹²m. If the power of the star is 1.3x10²⁸W, calculate the surface temperature of the planet. Take the albedo and emissivity of the planet as 0.3 and 0.61 respectively. [163.7K]
3. The solar constant is 1361 Wm^-2. The albedo of the Earth is 0.33 and its emissivity is 0.61. Determine the surface temperature of the Earth if the albedo increases by 10% and the emissivity decreases by 10%. [292.5K]
4. The solar constant is 1361 Wm^-2 at the Earth's distance from the Sun. At what distance from the centre of the Sun is the solar constant 15 Wm^-2? What planet is at this distance? [Saturn]
5. A lake full of water has an area of 5.1x10⁶m and a depth of 40 m. The base of the lake is 34 m above the turbine in a power station. Determine the greatest possible output power of the station if 1000 kg of water flows through it every second. [1.08x10¹⁴ W]
6. In a nuclear power station 7.64x10¹⁹ fission reactions of U-235 occur per day. Each reaction releases 190 MeV of energy. Determine the specific energy and energy density of U-235. Assume that the sample is pure U-235 of density 19.1 gcm^-3 and atomic mass 235.0439 u. [7.78x10¹³Jkg^-1, 1.49x10¹⁸ Jm^-3]
7. In the previous question U-235 only occupies 3% by volume of the fuel rod which has a uniform density throughout. Determine the specific energy and energy density of the fuel rod.[2.33x10¹²,4.46x10¹⁶Jm^-3]