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L A N C A S T E R U N I V E R S I T Y
2012 EXAMINATIONS
Part II
PHYSICS - Paper 3.C
- Candidates should answer all those sections identified with the modules for which they are registered.
- The time allocated is 60 minutes per section.
- An indication of mark weighting is given by the numbers in square brackets following each part.
- In each section attempted, candidates should answer question 1 (30 marks) and either question 2 or question 3 (30 Marks).
- Use a separate answer book for each section.
PHYSICAL CONSTANTS
Planck’s constant h = 6. 63 × 10 −^34 J s ℏ = 1. 05 × 10 −^34 J s Boltzmann’s constant kB = 1. 38 × 10 −^23 J K−^1 Mass of electron me = 9. 11 × 10 −^31 kg Mass of proton mp = 1. 67 × 10 −^27 kg Electronic charge e = 1. 60 × 10 −^19 C Speed of light c = 3. 00 × 108 m s−^1 Avogadro’s number NA = 6. 02 × 1023 mol−^1 Permittivity of the vacuum ≤ 0 = 8. 85 × 10 −^12 F m−^1 Permeability of the vacuum μ 0 = 4 π × 10 −^7 H m−^1 Gravitational constant G = 6. 67 × 10 −^11 N m^2 kg−^2 Bohr magneton μB = 9. 27 × 10 −^24 J T−^1 (or A m^2 ) Bohr radius a 0 = 5. 29 × 10 −^11 m Gas constant R = 8 .31 J K−^1 mol−^1 Acceleration due to gravity g = 9.81 m s−^2 1 standard atmosphere = 1. 01 × 105 N m−^2 Mass of Earth = 5. 97 × 1024 kg Radius of Earth = 6. 38 × 106 m
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Section A: Module 344 - Synchronization (The time allocated for this section is 60 minutes. Candidates should answer question A1 and either question A2 or question A3.)
Compulsory question:
A1. (a) Consider a one-dimensional dynamical system ˙x = f (x).
(i) Define the term fixed point. (ii) Explain what is meant by stable equilibrium. (iii) Explain how you can distinguish a stable equilibrium from an unstable one for a given f (x). (iv) Consider a function f (x) which also depends on a parameter r. Explain the term bifurcation and the condition necessary for it to occur. (^) [12]
(b) (i) Define the phase of an oscillation. (ii) If an oscillator is called self-sustained, what does this imply? (iii) Explain the concept of a limit cycle and how this concept is related to the self-sustainability of an oscillator. [9] (c) You are given the experimental phases θ 1 and θ 2 of two oscillators. Describe how you would use the stroboscopic technique to investigate any possible syn- chronization. Using the stroboscopic technique, discuss how you would distin- guish between the synchronization orders 2:5 and 2:3. [9]
Answer one of the following two questions:
A2. Consider an oscillator, defined by its phase θ and its eigenfrequency ω, that can synchronize with an external forcing. The phase of the external forcing is Θ and its frequency is Ω. The phase of the oscillator and its interaction with the external forcing are described by θ˙ = ω + ≤f (Θ − θ) where ≤ > 0 is the coupling strength.
(a) Suppose that f is a smooth 2π-periodic function. (i) If f has one minimum and one maximum, show that the condition for synchronization is ≤fmin < Ω − ω < ≤fmax. (ii) Explain with the help of a diagram how the number of equilibrium points, i.e. fixed points for the dynamics of the phase difference, are related to the number of local minima and maxima of the function f. (^) [10]
(b) Now suppose that f is the following function, periodic in [−π; π]
f (φ) =
(φ + π) φ if − π < φ < 0 (π − φ) φ if 0 ≤ φ ≤ π
(i) Find the value of ≤ > 0 for synchronization to occur. (ii) Find the equilibria for the difference of the phases as a function of ≤. (^) [20]
Section B: Module 385 - Advanced Spectroscopy & Microscopy (The time allocated for this section is 60 minutes. Candidates should answer question B1 and either question B2 or question B3.)
Compulsory question:
B1. (a) (i) Briefly explain the meaning of magnification and spatial resolution in mi- croscopy. (ii) State what determines the highest lateral resolution of an electron micro- scope and an atomic force microscope. (iii) Explain with the help of a ray diagram why the axial resolution of a confo- cal optical microscope is superior to that of a wide field optical microscope. [10]
(b) (i) Name the main components and explain the principles of operation of a scanning tunneling microscope (STM). Explain the difference between the constant height and constant current STM modes. (ii) Calculate the minimal current detection bandwidth required for STM imaging of 1 nm Au spheres if the scan width is 100 nm and the scan rate is 5 lines per second. [10]
(c) (i) Sketch the Jablonski diagram for a free fluorescent dye, labeling all states, and indicate the absorbance, radiative and non-radiative transitions. Ex- plain the differences between the characteristic transition times of fluores- cence and phosphorescence. (ii) In a laser fluorescence experiment, two markers A and B have the same absorbance cross-section, concentration and quantum yield for a certain excitation line. Marker B has a lifetime in the excited state which is 50 times larger than that of marker A. Estimate the laser power at which the fluorescence for marker B becomes saturated if marker A is saturated at 5 mW. Explain your reasoning. [10]
Answer one of the following two questions:
B2. (a) Sketch a diagram of a Raman microscope labeling the main components and explaining its principles of operation. Explain why laser excitation is essential for a Raman microscope. [10]
(b) Raman micro-spectroscopy is used to distinguish between single layer graphene and bulk graphite using the position of a certain peak in their Raman spectra. (i) For a laser excitation wavelength of 532 nm, find the wavelength of the Raman peak for single layer graphene, given that the value of its Stokes shift is 2676 cm−^1. (ii) Discuss how changing the laser excitation towards the red or blue end of the spectrum would change the signal-to-background and signal-to-noise ratios of the Raman spectra for graphene. Discuss if similar conclusions are valid for biological samples. [8]
(c) Consider a fluorescein isothiocyanate (FITC) labeled cell studied in a fluores- cence microscope with a CCD array detector. Each pixel of the CCD detector has a dark current of 6 electrons per second and a full well capacity of 4096 electrons. In fluorescing areas the FITC marker produces 10 electrons per second per pixel. (i) Calculate the maximum exposure time that does not saturate the CCD neglecting readout noise. (ii) Calculate the maximum signal-to-noise ratio, SNR, for measurements in the fluorescing areas neglecting readout noise. (iii) Estimate how the SNR of the fluorescing areas will change if the readout noise of 16 electrons rms per pixel is taken into account. Suggest methods that would improve the SNR explaining your reasons. [12]
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Section C: Module 388 - Energy (The time allocated for this section is 60 minutes. Candidates should answer question C1 and either question C2 or question C3.)
Compulsory question:
C1. (a) (i) Briefly explain what is meant by the War of Currents. (ii) Briefly explain why electricity is distributed by the National Grid at very high voltage. (iii) State three main advantages of high-voltage DC as compared to AC for long distance transmission of electrical power. [16]
(b) (i) Name the four main designs of hydro turbines. State in each case which of the two classes of turbines they belong to. (ii) Write down an equation for the maximum power input to a hydro turbine as a function of effective head and water flow. (iii) Taking the density of water to be 1000 kgm−^3 and g to be 10 ms−^2 , rewrite the equation to give the power in units of kW. [8]
(c) Niagara Falls comprises the American Falls on the USA side of the border and the Horseshoe Falls on the Canadian side. On average 13400 m^3 s−^1 of water flows along the Niagara towards the falls. Both, the Americans and the Canadians, use the water from the Niagara to generate hydroelectricity. (i) Estimate the flow that must be diverted from the 57 m Horseshoe Falls to run the Canadian Sir Adam Beck power station to generate 1600 MW of hydroelectricity, assuming an efficiency of 95%. (ii) The average height of the American Falls is a little smaller at 30 m, but the output of the US Robert Moses power station is 2500 MW. Estimate the flow required to achieve this, assuming an efficiency of 95%. (iii) A pact between the Americans and the Canadians forbids hydroelectricity generation having a visual impact on the falls during daylight hours in the summer by guaranteeing a minimum flow over the falls of 2800 m^3 s−^1. How do the Americans resolve the contradictory requirements of high day- time demand for electricity and the terms of the pact? [6]
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Answer one of the following two questions:
C2. (a) Explain what is meant by a combined-cycle gas turbine. Draw a simple Carnot diagram showing energy and work flow for a combined-cycle gas turbine. Hence show that the Carnot efficiency depends only on the initial input temperature and the final output temperature. [16]
(b) In August 2009 a British team set a new world record for the fastest steam car. The car was powered by a propane-heated boiler at a temperature of 400◦C, combined with a turbine of diameter 0.8 m. (i) Calculate the Carnot efficiency of the car, assuming an ambient tempera- ture of 20◦C. The record was actually set during a heat wave in the flats of the Mojave desert, California, when the temperature reached 50◦C. Com- ment on the effect of this on the efficiency. (ii) A jet of steam that is released through a perfect nozzle has a speed of 1100 ms−^1. Calculate the rotation rate of the turbine in revolutions per minute, assuming that it has just one set of blades. [14]
C3. Write a short article suitable for a popular science magazine on the subject of solar power for electricity generation. In your article you should describe the charac- teristics of solar radiation impinging on the Earth and how it varies, the different technologies for the capture of solar radiation for electricity generation and the future prospects for solar power. Your article should start with a well-written paragraph that grabs the attention of the reader. Include appropriate factual information. [30]
Answer one of the following two questions:
D2. (a) Let {X 1 ,... , Xn} be an orthonormal frame with coframe {e^1 ,... , en} and ∇ is a Levi-Civita connection. (i) Show that
∇Z Y = Z(Y a)Xa + ωba(Z)XbY a
where ωab(Z) = ea(∇Z Xb). (ii) By expanding ∇Z
g(Xa, Xb)
calculate ωab +ωba where ωab = g(Xa, Xc)ωcb. [10] (b) Let M be the Schwarzschild black hole spacetime with metric
g = −
2 m r
dt ⊗ dt +
2 m r
dr ⊗ dr + r^2
dθ ⊗ dθ + sin^2 θdφ ⊗ dφ
Consider a spaceship near the event horizon with worldline C : R → M given by
t(C(τ )) =
2 m r
τ , r(C(τ )) = r 0 ,
θ(C(τ )) = π/ 2 and φ(C(τ )) = 0
where r 0 is a constant. (i) Calculate C˙ and show that it is normalised. (ii) Calculate the 4-acceleration A = ∇ (^) C˙ C˙ using
∂t
∂ ∂t =^
m r^2
2 m r
∂r
(iii) Calculate g(A, A) and discuss the magnitude of the force required to keep the spaceship just outside the event horizon. [10] (c) Show that
L(f U )α = f LU α + α(U ) df
for α ∈ ΓΛ^1 M , f ∈ ΓΛ^0 M and U ∈ ΓT M. Hint: contract both sides of the above equation on a vector field V ∈ ΓT M and use the identity
[f U, V ] = f [U, V ] − V (f )U [10]
D3. (a) (i) Draw the Penrose diagram for Minkowski spacetime indicating timelike future and past infinity, spacelike infinity and lightlike infinity. (ii) On the above diagram, draw a timelike curve that approaches lightlike infinity. (iii) On the above diagram, show the region of spacetime which cannot send a signal to the curve in part (ii). (iv) If neutrinos do travel faster than light, draw a neutrino worldline on the above diagram. [10] (b) Let M be Minkowski space with coordinates (t, x, y, z) and metric
g = −dt ⊗ dt + dx ⊗ dx + dy ⊗ dy + dz ⊗ dz
Let the electromagnetic 2-form be given by
F = F 01 dt ∧ dx − F 31 dz ∧ dx
and let V = γ( (^) ∂t∂ + v (^) ∂x∂ ) be the 4-velocity of a particle, where γ = (1 − v^2 )−^1 /^2. (i) Determine the electric and magnetic fields for an observer with 4-velocity (^) ∂t∂. (ii) Calculate the Lorentz force q i˜V F on a charge q. (iii) Explain briefly the meaning of the (^) ∂t∂ term of the Lorentz force. [8] (c) On R^3 with coordinates (x, y, z) and alternative coordinates (t, u, v) with co- ordinate transformations x = u, y = v and z = f (t, u, v) let
α =
(1 + x^4 )(1 + y^4 )
dy ∧ dz
(i) Express α in the coordinate system (t, u, v). (ii) Setting
f (t, u, v) = u^3 + u + t
calculate ∫
S
α
over the surface S = {(x, y, z)|z = u^3 + u}. You may use the result ∫ (^) ∞
−∞
1 + s^4
ds =
−∞
s^2 1 + s^4
ds =
π √ 2 [12]
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E3. (a) Briefly describe the construction and mode of operation of a Nd:YAG laser. Explain how population inversion is achieved in this laser. Give reasons why the Nd:YAG laser stops operating at high temperature. [10] (b) Consider a laser cavity of length L with mirror reflectances R 1 and R 2. By considering the round trip gain and losses γ show that the threshold gain in a Nd:YAG laser is given by
kth = γ +
2 L
ln
R 1 R 2
Briefly account for the main loss mechanisms. [8] (c) Determine the number of Nd ions in the excited state at threshold in a Nd:YAG laser with a 6 mm diameter rod of length 10 cm with mirror reflectances of 0. and 1.0. Assume that the cavity loss coefficient is 1 × 10 −^4 cm−^1 and that the excitation cross section for Nd ions is 6 × 10 −^18 cm^2. [6] (d) Using the above information calculate the output power of a Nd:YAG laser operating at 1.064 μm if the lifetime of the upper laser level is 1.5 μs. Is this laser suitable for interferometry and holography experiments? Give reasons. [6]
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