QUESTION 1 Electrostatics                                                [ 3 + 4 + 3 + 2 = 12 marks ]

Four point charges are arranged as in the diagram below. Charges Q1 and Q2, each with         charge -2q C, are placed a distance, a m, apart at the base of an equilateral triangle. A further negative charge Q3 with charge, -q C, is placed at the top of the triangle and the last charge,  Q4 with charge +q C, is placed a distance, x m, directly below Q3 .

 Q3

x

 Q4

Q1                                                                                     Q2

a

a)  Draw and label all the forces acting on Q3, including relative direction and magnitude.

b)  Find an expression for the net force on Q3 charge at the top of the triangle due to the other three charges in terms of q, a and x.

c)  Determine the direction of the net force on Q3 if X =  a .

d)  Determine the value of x in terms of a and q where the net force on Q3 is zero.

QUESTION 2 Gauss’s Law                                     [ (3 + 4) + 8 + 2 + 5 + 3 = 25 marks ]

A non-uniform electric field, E⃗ , is given by the relationship E⃗ = (2.0X2  − 3.0)NC−1 where x is the distance in metres along the  axis.

A 2 m gaussian cube in the electric field is orientated with one corner at the origin as shown in the diagram below;

a)  Determine the electric flux through the

i)   top and

ii)  right-side

faces of the gaussian cube. (shown by the shaded faces in the diagram)

b)  Using arguments of symmetry, show the total electric flux through the gaussian cube is 32 N m2 C-1 .

c)   Calculate the charge inside the gaussian cube.

d)  Find the potential difference due to the electric field from point A on the left-hand side to point B on the right-hand side of the cube.

e)   If the cube was now replaced by a sphere of radius 1 m and centred at position (1,1,1), would the total flux change? Explain your answer.

QUESTION 3 Electric Circuits                                           [ 8 + 3 = 11 marks ]

A circuit consisting of three cells and five resistors as shown in the diagram below;

10 Ω                       A                      10 Ω

0.05 A

6.0 V

10 Ω

R Ω

10 Ω

B

The left-most cell is 3 V and the remaining two cells are 6 V. The outer resistors have a         resistance of 10 Ω and the central resistor, R, is unknown. An ammeter measures a current of 0.05 A through the right-most cell. The currents I1 and I2 are also unknown.

a)  Determine the value of the central resistor, R.

b)  Calculate the voltage drop across the points A-B as indicated in the diagram using TWO circuit pathways.

QUESTION 4 Electromagnetism                                        [ 3 + 1 + 6 + 2 = 12 marks ]

Choose ONE of the following electromagnetic phenomena and discuss points i – iv. Use no more than one page. Clearly state which phenomena you have chosen on your paper.

a)  Faraday cage.

b)  Meisner Effect.

c)  Displacement current.

i)   Briefly describe the effect using diagrams where appropriate.

ii)  State which of Maxwell’s Equations best describes the phenomenon.

iii) Discuss how the phenomenon occurs in terms of the appropriate electromagnetic theory.

iv) Provide a brief description for a use of this phenomena.

QUESTION 5 Relay Switch                                     [ 4 + 4 + 1 + 1 + 5 = 15 marks ]

Two Physics students, Ivory and Chen, are working on a design for a relay switch for their      electric bike. A simple relay switch consists of a solenoid with iron core. Once a small current is passing through the solenoid through the input circuit by turning the ignition key, the iron   core is magnetised and attracts a metal lever, iron armature, which in turn connects a               secondary circuit by connecting contacts C. The output circuit will then turn on the motor of  the bike. A diagram of a relay switch is shown below;

Initially, Ivory wants to investigate the type of wire they will use to determine the strength of magnetic field the solenoid will be able to generate. The team chooses a long insulated,         current-carrying wire with direct current, I, through the wire of 2.0 mA and the radius, R, of  0.5 mm.

a)   Close examination of the cross-section of the wire shows the wire is hollow with a  radius of 0.1 mm and a uniform distribution of current throughout the conductor.     Draw a graph of the strength of the magnetic field versus the distance to 5 mm from the centre of the wire, r. Include relevant scale and key points on your graph.

r 

0.1 mm  0.5 mm

b)  Ivory arranges the centre of the wire in a semi-circle with radius of 5 mm. Using the Biot-Savart Law, determine the magnitude of the magnetic field and its direction due to the wire at point C as shown in the diagram below.

Question continued next page.

Using Ivory’s wire, Chen now looks at designing the solenoid by winding a long length of wire into n circular loops with a radius of 5 mm.

c)   Chen calculates the magnetic field required to trigger the switch needs to be 5 x 10-4 T. Show that almost 200,000 turns per metre in the solenoid will be required to produce a magnetic field of this size without an iron core?

d)  Calculate the permittivity of the iron core required to create this magnetic field with a more reasonable, 40 turns per metre in the solenoid.

e)  Now that the input circuit has been designed, Chen states that because there is a direct current in the solenoid, Faraday’s Law says a magnetic field won’t be induced in the  iron core and hence the switch won’t close in the output circuit. Ivory disagrees and is convinced the relay switch will work. Who is correct? Justify your answer.

QUESTION 6 Fluids                                                            [ 2 + 1 + 4 + 2 + 2 = 11 marks ]

A water reservoir, A, whose free-surface is kept at a pressure 2 atm, discharges to another reservoir, B, open to the atmosphere. The water free-surface level at the second reservoir is z = 0.5 m above the pressurized reservoir A. The horizontal connecting duct has a constant diameter.

a)  What is the pressure due to water and air between a generic point on the surface A and the duct inlet (point 1)?

b)  What is the pressure at the outlet (point 2) due to reservoir B?

c)  Use now the Bernoulli equation between surface A and point 2 to compute the water velocity in the connecting duct.

d)  Would the velocity at the duct exit (point 2) change if the diameter of the connecting duct (1 to 2) is not constant?  Justify your answer.

e)  What is the pressure difference between the duct inlet (1) and outlet (2) if the duct is horizontal and of constant diameter?

(1 atm = 101325 Pa, water density 1 x 103   kg/m3)

QUESTION 7 Heat and Work                                [(1 + 2 + 2 + 3 + 2) + 3= 13 marks]

a)  A large closed plastic bag contains 0.1 m3 of gas at initial temperature of 15 °C and with  the same pressure of the surroundings, 1 atm. The bag is left in the sunlight and the gas    warms up to 43 °C, expanding to 0.11 m3 after absorbing 3.7 kJ of heat energy. Assuming the pressure does not change and using the ideal gas model, determine

i) the number of moles of the gas contained in the bag.

ii) how much work is done by the gas in the bag.

iii) calculate the internal energy of the gas in this transformation.

iv) is it possible to tell if the gas is mono or diatomic? Explain your answer.

v) Can a gas absorb heat without getting hotter?  Justify your answer.

b)  In the kinetic theory of gasses explain what is meant by the “equipartition of energy” .

QUESTION 8 Particle wave duality                       [3 + (2 + 2 + 2 + 3) + 2 + 2 = 16 marks]

a)   In a photoelectric experiment, light of different wavelengths is shone on an unknown metal. It is observed that when a wavelength larger than 293 nm is used no electrons are emitted. Considering that the potassium, chromium, zinc and tungsten have work functions 2.26, 4.24, 4.42 and 4.49 eV respectively, identify which is possibly the unknown metal. Justify your answer.

b)  X-rays having an energy of 500 keV undergo Compton scattering from a target. The scattered rays are detected at 45o relative to the incident rays. Find:

i)   The Compton shift at this angle.

ii)  The energy of the scattered x-ray.

iii) The energy of the recoiling electron.

iv) The ratio of the photon wavelength to the electron wavelength. What do you learn from their comparison?

c)  Explain why we cannot measure simultaneously with the same accuracy the wavelength and position of a photon.

d)  If the wavelength of a photon is 500 Å and is known with an accuracy of 10-7 . What is the minimum uncertainty in the position of the photon?

QUESTION 9 The black body                                [(2 + 2 + 2) + (2 + 2) = 10 marks]

a)   The universe is filled with relic radiation from the early stage of the universe – the so- called Cosmic Microwave Background (CMB) radiation. When the universe was        about 300,000 years old, the temperature of the CMB was 3,000K. Assuming the        universe is a black body

i)   Estimate 入max of the emitted radiation at this stage of the universe.

ii)  Assuming the universe at that time was a sphere of 4×1021 km2 surface area, how much power does it radiate?

iii) Find the spectral power per wavelength at 入max.

b)  Although the CMB emitted a spectrum of waves all having different wavelengths, model its whole power output as carried by photons of wavelength 入max :

i)   Find the energy of one photon.

ii)  Find the number of photons it emits each second.

QUESTION 10 The atom                                                    [3 + (2 + 2 + 2 + 3)) = 12 marks]

a)   In the Bohr model of the atom, it is hypothesised that stable electron orbits have a quantized angular momentum L=n. Explain briefly why the postulate of stationary states is in contradiction with classical mechanics and electromagnetism?

b)  The muon is a particle equivalent to an electron with a mass about 200 times larger and the same charge. It is possible to form a muonic hydrogen atom, where the muon   takes the place of an electron in the orbit around the nucleus. Calculate

i)         The Bohr radius of the muonic atom.

ii)        The muonic atom energy levels in terms of the principal quantum number.

iii)       The energy of the photon emitted when a muon makes a transition from n=2 to n=1.

iv)       Compare the results obtained with the one for the normal atom. What do you conclude?

QUESTION 11 The nucleus, nuclear decays                     [3 + 3 + 2 + 3 + 2= 13 marks]

A bottle shop has a very expensive bottle of wine that is about 5 years old. The wine        contains a number of different atoms with radioactive isotopes of oxygen, hydrogen and  carbon. A customer wants to buy the bottle but wants to be sure that the year of               production is correct.  The radioactive isotopes in the wine include carbon- 14, 14C with a half-life of 5730 yr, oxygen- 15, 15O with a half-life of 122.2 s and hydrogen-3, 3H           (tritium) with a half-life of 12.33 yr. The activity of each isotope is known at the time the bottle was sealed (14C : 0.23 Bq per gram of carbon, 15O: 7 x 1010 Bq per gram of oxigen, 3H: 3.6 x 1014 Bq per gram of hydrogen).

a)   Calculate the activity for each isotope after 5 years.

b)  Which one of these 3 radioactive isotopes can be used to date the wine, if the activity of each isotope is known at the time the bottle was sealed? Justify your answer.

c)  Find the nuclear radius of the oxygen and carbon isotopes.

d)  How much energy is released when a 146C decays to 147N by beta emission?

e)   Calculate the binding energy of the tritium given that its atomic mass is 3.0160492 u.

(146C mass = 14.003241 u; 147N mass = 14.003074 u)