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VICTORIA JUNIOR COLLEGE2009 JC2 PRELIMINARY EXAMINATIONS

PHYSICS 9811/01Higher 3

Paper 1

28 Sep 2009 8 am – 11 amMONDAY (3 Hours)

INSTRUCTIONS:

You may use a soft pencil for any diagrams, graphs or rough working.Do not use staples, paper clips, highlighters, glue or correction fluid.

Section AAnswer all questions.You are advised to spend 1 hour 50 minutes on Section A.

Section BAnswer any two questions.

At the end of the examination, fasten all your work securely together.The number of marks is given in brackets [ ] at the end of each question or part question.

For marker’s use

Section A:

1

2

3

4

5

6 (data)

Section B:

7

8

9

10

Total (max. 100):

Percentage:

This question set consists of a total of 13 printed pages.

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Data

speed of light in free space c = 3.00 x 108 m s-1

permeability of free space µo = 4 x 10-7 H m-1

permittivity of free space o = 8.85 x 10-12 F m-1

(1/(36)) x 10-9 F m-1

elementary charge e = 1.60 x 10-19 C

the Planck constant h = 6.6.3 x 10-34 J s

unified atomic mass constant u = 1.66 x 10-27 kg

rest mass of proton mp = 1.67 x 10-27 kg

rest mass of electron me = 9.11 x 10-31 kg

molar gas constant R = 8.31 J K-1 mol-1

the Avogadro constant NA = 6.02 x 1023 mol-1

the Boltzmann constant k = 1.38 x 10-23 J K-1

gravitational constant G = 6.67 x 10-11 N m2 kg-2

acceleration of free fall g = 9.81 m s-2

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Formulae

Lorentz factor = (1 – (v/c)2)-1/2

length contraction L = Lo/

time dilation T = To

Lorentz transformation equation (1 dimension) x’ = (x – vt)t’ = (t – vx/c2)

mass energy equivalenceE = moc2 =

Wien’s displacement law pT = 2.898 x 10-3 m K

Compton shift formula

population distribution of atoms with energy Ex

time-independent Schrödinger Equation

allowed energy states for a particle in a box

normalised wave function for a particle in a box

transmission coefficient, where

Drude model of electrical resistivity

Fermi energy for metals

density of energy states for electrons in a metal

Fermi function

refractive index n = v1/v2

phase difference of a circularly polarised light

Brewster’s angle tan n = n2/n1

attenuation of light intensity

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Section A: Answer all questions in the spaces provided.

1. (a) A physicist describes the net potential energy experienced by each ion in an ionic lattice as

where e = electron charge; = permittivity in vacuum, r = nearest-neighbour distance, and

= Madelung constant

Explain the origin of the term in the expression above. [2]

(b) The physicist feels that Pauli’s Exclusion Principle cannot be ignored and adds another term to the potential energy expression such that it now appears as

where and are constants for a particular lattice.

(i) Suggest the significance of the term . [2]

(ii) Show that at equilibrium when r = r0 , the physicist will obtain the following equation for the potential energy:

[3]

(iii) Explain why it is important for the ratio to be less than one. [1]

2. A pion is an unstable particle with an average lifetime of 2.60 x 10-8 s (measured in the lifetime of the pion). The pion is moving relative to the laboratory and its lifetime is measured to be 4.20 x 10-7 s. Calculate

(a) how fast the pion is moving. [2]

(b) how far the pion will travel in the laboratory frame of reference before it decays. [1]

(c) the pion’s momentum in MeV/c in the laboratory frame of reference, given that the rest mass of the pion is 139.6 Mev/c2. [2]

(d) the pion's total energy. [2]

(e) the kinetic energy of the pion. [1]

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3. (a) A system consists of atoms having only two energy levels. There are N1 atoms per unit volume in the lower energy level E1 and N2 atoms per unit volume in the higher energy level E2. The atoms are irradiated with photons of frequency f that can cause resonant absorption to occur. The system is in thermal equilibrium at temperature T.

(i) Explain what is meant by resonant absorption .State an expression based on this process linking some of the quantities described above. [2]

(ii) The population distribution of atoms with energy Ex at temperature T is given by

where No is the total number of atoms per unit volume.

Deduce an expression for the ratio in terms of f, T, k and the Planck

constant h. [2]

(iii) If the frequency for resonant absorption is 2.46 x 1015 Hz, calculate the temperature at which one in a thousand of the atoms in this system will be in energy level E2. [1]

(b) For laser action to occur, the medium used must have at least three energy levels.

(i) Describe the nature of each of the three energy levels. [1]

(ii) Explain why the minimum number of energy levels is three. [2]

4. (a) Write down Schroedinger’s equation for a quantum harmonic oscillator of mass m and oscillating with an angular frequency of . Let E be the total energy of the oscillator. [2]

(b) Show that the following wave function

where C is a normalisation constant, is a solution to the Schroedinger’s equation for the quantum harmonic oscillator. . [2]

[Hint: ]

(c) Explain the meaning of the term normalization constant. [1]

(d) Using your working in (b), determine the energy E of the excited state represented by the wave function n. [2]

(e) If the wave function n represents the nth excited state of the harmonic oscillator, state the value of n. [1]

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5. Light with a wavelength in vacuum of 546.1 nm falls perpendicularly on a biological specimen that is 1.000 m thick. The light splits into two beams polarized at right angles, for which the indices of refraction are 1.320 and 1.333.

(a) Calculate the wavelength of each component of the light while it is traversing the specimen. [4]

(b) Calculate the phase difference between the two beams when they emerge from

the specimen. [2]

(c) Show that the minimum thickness d for the sample to produce circularly polarized light is

where is the wavelength of the light, and n1 and n2 are the refractive indices. Calculate this thickness. [2]

6. (a) (i) Explain what is meant by Fermi energy of a metal. [1]

An investigator plotted a graph (shown in Fig. 6.1) of data relating the Fermi energies EF of some metals and their electron number densities n. He knew

that EF varies with n according to where C is a constant. Show

that this relationship is valid using data from Fig. 6.1, and calculate a value for the constant of proportionality C, giving your answer with its units. [4]

(ii) 1. Zinc has a Fermi Energy of 9.47 eV. With reference to the graph in fig 6.1, estimate a value for its electron number density . [3]

2. It is known that zinc has a density of 7140 kg m-3 and its molar mass is 65 g mol-1. Calculate the number density of zinc atoms, and hence determine how many free electrons each zinc atom contributes to the “sea” of electrons in the metal lattice of zinc. [4]

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(b) (i) The graphs in Fig. 6.2 are plotted for Fermi function f(E) against E/EF where E is the energy of electrons and EF is the Fermi energy of a metal.

Explain the shapes of the graphs obtained for different temperatures T1 and T2

. [2]

(ii) Refer to the graph plotted for temperature T1 in fig 6.2. Estimate the value for the ratio E/EF when f(E) = 0.1. Hence determine a value of T1 in terms of EF and k, the Boltzmann constant. [3]

(c) In intrinsic semiconductors, the Fermi level is between the valence and conduction band, as shown in Fig. 6.3.

Sketch how the Fermi function f(E) varies across the band gap at a high temperature and hence explain how conductivity in the semiconductor increases as a result. [3]

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Section B: Answer 2 out of the following 4 questions.

7. Experiments using the technique of low energy electron diffraction (LEED) give information about the arrangement of atoms very near the surface of a crystal. Fig. 7.1 illustrates an experimental arrangement.

EMBED Word.Picture.8

crystal

vacuum Electron gun Incident electron

beam

Scattered electron beams

Fluorescent screen

Fig. 7.1

A beam of electrons from an electron gun is directed at the crystal. Scattered electron beams strike a fluorescent screen, producing bright spots at the points of impact. The screen is observed through a window in a vacuum chamber.

(a) A formula for calculating the de Broglie wavelength λ associated with an electron beam that has been accelerated from rest through a potential

difference V is

In this formula λ is measured in nm and V in V.

(i) Show how the formula is obtained from the de Broglie equation. [4]

(ii) Suggest a suitable de Broglie wavelength for the electron beam in a LEED experiment. Explain why this wavelength is suitable. Hence deduce a suitable accelerating potential difference for the electrons in the beam. [3]

(b) Suggest two reasons why a LEED experiment must be carried out in a vacuum chamber. [2]

(c) Fig. 7.2 illustrates low energy electron diffraction from a single row of atoms PQ in the surface of a crystal.

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A beam of electrons with de Broglie wavelength λ is incident normally on the row of atoms, which have regular spacing a. Constructive interference (strong scattering) takes place at an angle θ to the normal to the surface.

(i) Explain the condition for strong scattering to take place. Hence derive an equation, in terms of a, λ, and θ, expressing this condition. [4]

(ii) These strongly-scattered beams are to be observed on the fluorescent screen. Describe and explain what happens to the directions of the strongly-scattered beams as the accelerating potential increases. [2]

(d) Fig. 7.3 illustrates the row of atoms PQ in the surface of the crystal, together

with the row RS immediately below it. Each atom in the second row is a distance a below the corresponding surface atom.

Some of the electrons directed normally at the surface are scattered by atoms in row PQ, while others are scattered by atoms in row RS. This scattering takes place in addition to the scattering by atoms in the surface, as described in (c).

(i) Considering the interaction of the electron beam with atoms in rows PQ and RS, draw Fig 7.3 and add lines to show how strong scattering may take

place at angles to the normal. [2]

(ii) Assume that the value of the de Broglie wavelength is such as to allow these strongly-scattered beams to be observed on the fluorescent screen. Describe what happens to the directions of the strongly-scattered beams as the accelerating potential increases. [3]

9

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8. (a) Fig. 8.1 shows the experimental emission spectrum of a black body at equilibrium temperature T. Rayleigh-Jeans’ Law is plotted next to the observed spectrum.

(i) Explain what is meant by a black body. [2]

(ii) Cavity radiation at an equilibrium temperature T is a model of black body radiation. The radiation exists in the form of electromagnetic stationary waves in various modes. Classically, the number of modes n in the range of wavelength to ( + ) is given by

where V is the volume of the cavity.

Show that the energy density (i.e. energy per unit volume) at wavelength , in the small range , in the blackbody spectrum should therefore be given by

.

[3]

(iii) With reference to the expression in (ii) and experimental observations of black body radiation in reality, explain what is meant by the ‘ultra-violet catastrophe’. [3]

(b) X-rays with = 1.00 x 10-10 m are scattered from a carbon block. When viewed at an angle of 45o to the incident direction, the emerging x-rays appear to have a shift in wavelength.

(i) Explain how the classical theory failed to explain the observed shift in wavelength. [2]

(ii) With reference to the particulate nature of X-rays, derive the Compton equation which relates the shift in wavelength and the scattering angle .

[6]

(iii) Calculate the kinetic energy imparted to the recoiling electron. [4]

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Wavelength

9. A particle of mass m is trapped inside a hollow cube with sides of length L, equivalent to a 3-dimensional infinite square well. One corner of the cube is taken to be the origin (0, 0, 0) with the sides parallel to each of the three cartesian axes x, y and z, as shown in Fig. 9.1 below.

The overall 3-dimensional wave function of the particle inside the cube is the product of the separate wave functions , and along the x-, y- and z-axes respectively, and is given by

where A is the normalization constant, and n1, n2 and n3 are integers.

(a) Deduce a value for the normalization constant A in terms of L. [3]

[Hint: ]

(b) By using the standing wave condition for the wave functions along the three axes, show that the kinetic energy E of the particle inside the cube is given by

[4]

(c) Using the expression for the kinetic energy in (c), determine the lowest four energy levels that the electron inside the cube can possess. Express your answers in terms of m, L and h. [4]

Recently, scientists have been making minute structures called quantum dots, which are conglomerations of between 100 to 100,000 atoms. Such quantum dots

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have diameters on the order of nanometers. We can simplistically treat such a quantum dot as a 3-dimensional cube treated in parts (a) to (c).

(d) If L = 2.0 nm, determine the minimum kinetic energy that an electron must have for it to exist inside such a quantum dot. This corresponds to the ground state for the wave function . (Hint: use an expression obtained in (c).) [2]

(e) Determine the energy that the electron possesses when it is in the excited state in which . [2]

(f) If the electron transits from the excited state in (e) to the ground state, calculate the wavelength of the light that will be emitted by the quantum dot.

[3]

(g) State and explain what will happen to the wavelength of the light emitted if the size of the quantum dot is reduced. [2]

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10. (a) (i) Explain why scuba divers need to use goggles to see more clearly underwater.

[3]

(ii) A diver standing by the edge of a lake sees a fish which appears higher than it actually is. Explain this observation by including a ray diagram in your answer. [4]

(iii) Using your diagram in (a)(ii) derive an expression for the refractive index of water in terms of the real and apparent depths of the fish for the case of the diver looking downwards almost perpendicularly to the water surface.

[4]

(b) (i) Explain how signals are transmitted via a step-index multimode type optical fibre and how information is distorted at the receiving end. [3]

(ii) Explain why the continuous-index multimode type is preferred over the step-index type in long-distance signal transmissions. [2]

(iii) Fig. 10.1 below shows a step-index glass fiber (refractive index nf) surrounded by a lower optical density cladding (refractive index nc).

EMBED Word.Picture.8

nc

no

nf

c

t

i = max

Fig. 10.1

There is a maximum incident angle i = max such that any ray impinging on the face at i max will arrive at an internal wall at an angle less than c, the critical angle, and will not be totally internally reflected. Show that

sinmax = (nf2 – nc

2)/no

where no is the refractive index of air. [4]

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End of Paper

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