Department of Physics PRELIMINARY EXAMINATION 2012 Part II. Long Questions

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1 Department of Physics PRELIMINARY EXAMINATION 2012 Part II. Long Questions Friday May 25th, 2012, 2-5pm INSTRUCTIONS Answer 5 questions out of the choice of 8. This is a closed book exam. Approved calculators may be used (non-programmable ones), though approximate numerical results are valid. If you attempt more than 5 questions, you should clearly mark which ones should be graded. Write your name and student ID on the exam booklet. question number next to each answer. This exam has 10 pages, including this title page. Clearly indicate the Examiners: Prof. A. Cumming (Chair), Prof. K. Dasgupta, Prof. P. Francois, Prof. G. Gervais, Prof. B. Vachon

2 2012 Prelim Long Answers 2 1. Suspended rod. 1 Long Questions 1.1 Relativity Judy emits a flash of light in all direction at t = 0, x = 0 (for simplicity, only consider one dimension of space and time). Her sister Laurie travels at half the speed of light in the +x-direction such that at t 0 = t = 0, x 0 = x = 0. (a) Which observer sees the light signal move outward at the same speed in all directions from her point of view? Prove mathematically your explanation, i.e. whether each observer does or does not see herself in the middle of this light cone. (b) Draw a Minkowski diagram describing this particular problem. A uniform rod of mass 1.2 mclassical and length mechanics L is suspended on one end by a string, while the other end rests on the ground, such that it forms an angle θ with the horizontal. A uniform rod of mass m and length L is suspended on one end by a string, while the other end rests on the ground, such that it forms an angle with the horizontal. (a) What is the tension in the string? (b) Now imagine the string is cut, and the rod starts to fall. Calculate the vertical component of the acceleration of the center of mass of the rod, at teh moment t = 0 immediately following the cut. Ignore friction with the ground. (c) What is the force F N that the rod exerts on the ground at time t = 0, immediately after the cut? (Hint: It is di erent than before the string was cut.) (a) What is the tension in the string? (b) Now imagine the string is cut, and the rod starts to fall. Calculate the vertical component of the acceleration of the center of mass of the rod, at the moment t = 0 immediately following the cut. Ignore friction with the ground. (c) What is the force F N that the rod exerts on the ground at time t = 0, immediately after the cut? (Hint: It is different than before the string was cut.) 1

3 2012 Prelim Long Answers 3 2. Far side of the moon. The goal of this problem is to explain why we always see the same side of the moon. This is a general phenomenon in the solar system called synchronous rotation : most satellites are locked and show the same side to the planet they are rotating around. ~ Figure 1: Synchronous rotation of the Moon around the Earth We will call r the distance between the centers of mass of the moon and of the Earth, φ the angular velocity of the moon rotating around the Earth, m the mass of the Moon, Ω the rotation vector of the moon. [This question continues on the next page.]

4 2012 Prelim Long Answers 4 (a) Consider a satellite of mass m orbiting around the Earth (gravitational potential K ) with no intrinsic rotation. Write down the Lagrangian, and show r that for a circular orbit, the angular velocity of rotation is a constant φ = ω circ so that ωcirc 2 = K where r is the radius of the orbit. mr 3 (b) We now consider a satellite of mass m with rotation vector Ω (see Figure 1), that we model as a spherical homogenous body (moment of inertia I = 2 5 mr2 ). 1. Assume that the angular momentum of center of mass of the moon and Ω are parallel. Give the expression of the total angular momentum L of the moon, as a function of r, φ and I. 2. Show that E can be written as E = 1 2 mṙ2 + V eff (r, Ω) and give an expression for V eff. [Hint: assume L is conserved.] 3. Show that the effective potential energy V eff (r, Ω) is minimized when Ω = ω circ. (c) This locking is often called tidal locking. Explain why the gravitational influence of the Earth is expected to create a bulge on the Moon (very similar to the tides created by the Moon and the Sun on the Earth), and deduce a qualitative scenario explaining how the Moon evolved to show the same side to the Earth. Is the reasoning from part (b) fully consistent with the conservation of angular momentum?

5 2012 Prelim Long Answers 5 3. Charged bowl. (a) Imagine I have a spherical ball of charge with charge density +ρ and radius R. I cut the sphere in two halves, and for the top half I scoop out the charges so that I m left with an inverted hemispherical bowl of radius R carring an uniform surface charge density σ. Alternatively one may view this as though the volume charge density is completely neutralised, leaving us only with the surface charge density. Show that the potential difference between the north pole and the center is given by: V = Rσ ɛ Rσ ɛ 0. (b) On the other hand, I can also do the following. I bring another spherical ball of charge, but now with negative charge density ρ, and overlap this with my previous spherical charge configuration with charge density +ρ partially. In the overlap region the situation is somewhat similar to our previous configuration namely, the total charge density is completely neutralised. This is shown in the following figure: d + _ Call the vector from the positive center to the negative center as d. Show that the field in the region of overlap is now constant and determine its value.

6 2012 Prelim Long Answers 6 4. Scattering by a dielectric sphere. (a) A dielectric sphere of radius a is placed in a uniform static electric field. By writing down the potentials inside and outside the sphere or otherwise, show that the polarizability of the sphere is ( ) ɛ 1 α = 4πɛ 0 a 3, ɛ + 2 where the dielectric constant of the sphere is ɛ, and the polarizability α is the ratio of the induced dipole moment to the applied electric field. (b) Now consider an electromagnetic wave incident on the small dielectric sphere with wavelength long compared to the radius of the sphere, so that the sphere is subject to an oscillatory uniform electric field E e iωt. Argue that the sphere radiates electromagnetic radiation in response to the applied electric field, scattering the incoming radiation. What is the polarization of the scattered radiation? (c) How does the scattering cross-section scale with (1) the wavelength of the incoming radiation, (2) the size of the sphere, and (3) the dielectric constant?

7 2012 Prelim Long Answers 7 5. Particle in an infinite box of potential. (a) Show that allowing n = 0 (the ground state of the system) for a particle in a one-dimensional infinite box of potential violates the uncertainty principle x p h/2. (b) An electron is contained in a one-dimensional infinite box of potential of width nm. Draw an energy-level diagram for the electrons for all levels up to n = 4. Find the wavelengths of all photons that can be emitted by the electron in making transitions that would eventually get it from the n = 4 state to the n = 1 state. (c) A proton is confined to moving in a one-dimensional box of width nm. Find the lowest possible energy of the proton. What is the lowest possible energy of an electron confined in the same box? How do you account for the large difference in your results between the electron and the proton?

8 2012 Prelim Long Answers 8 6. Squeezing fermions. A spherical shell of radius R is kept inside a three dimensional box. The shell is suspended by springs of spring constant k such that the pressure of the springs would squeeze the shell to zero size. In the configuration below: I fill the spherical shell with N massive fermions of masses m each. Clearly now due to the Pauli Exclusion Principle, the fermions will try to avoid being squeezed to zero size. This in turn should create a pressure that will balance the pressure created by the springs. Using your understanding of fermions and particles in a three-dimensional box, can you estimate the minimum volume of the spherical shell (in terms of the variables specified in the problem) when stability is achieved? [Hint: first write down an estimate of the Fermi energy.]

9 2012 Prelim Long Answers 9 7. Einstein solid. One of the simplest models for a solid was proposed by Einstein in In this model, atoms are placed on a regular cubic lattice, and interactions between atoms are modelled as identical quantum harmonic oscillators, with energy levels E n = hω( 1 + n), n integer. For simplicity, in the following, we set ɛ = hω. 2 (a) Explain why in 3D we can model a solid made of N/3 atoms by an Einstein solid with N independent harmonic oscillators. (b) Compute the partition function of one oscillator, then deduce the partition function for the collection of N oscillators. (c) Compute the average energy U and entropy S of the Einstein solid in the high temperature limit. (d) Compute C V and plot it qualitatively as a function of temperature T. What known thermodynamical laws do we recover in the limit T 0 and T?

10 2012 Prelim Long Answers Judy and Laurie. Judy emits a flash of light in all directions at t = 0, x = 0 (for simplicity, only consider one dimension of space and time). Her sister Laurie travels at half the speed of light in the +x-direction such that at t = t = 0, x = x = 0. (a) Which observer sees the light signal move outward at the same speed in all directions from her point of view? Prove mathematically your explanation, i.e. whether each observer does or does not see herself in the middle of this light cone. (b) Draw a Minkowski diagram describing this particular problem.