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prelim2005 - Table of Contents Complete Partially Complete...

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Table of Contents Complete Partially Complete Not Complete Preliminary Examination 2005 2 Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Problem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Problem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Problem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Problem 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Problem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Problem 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Problem 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Problem 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Problem 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Problem 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Problem 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Problem 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Problem 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Problem 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Problem 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Problem 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Problem 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Problem 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1
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Preliminary Examination September , 2005 1. Consider two electric dipoles, each of moment p = 2 aq , whose centers are a distance d apart. One is fixed parallel to the x axis, with its center at the origin. The other has its center fixed at x = d , but is free to rotate in the x - y plane. (a) Find the frequency of small oscillations of the dipole at x = d , if the mass of each charge is m . Assume a << d and neglect any radiation reaction. (b) The amplitude of oscillation will decrease exponentially in time due to energy loss by electromagnetic radiation. Determine the exponential coefficient assuming a << c/ω . 2
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Preliminary Examination September , 2005 2. In n dimensions, find the spherically symmetric solution of the equation 2 φ = ( 2 ∂x 1 2 + 2 ∂x 2 2 + · · · + 2 ∂x n 2 ) φ = n ( x ) where Q is a constant. Assume n > 2 . Note: the surface area of an ( n 1) dimensional unit sphere, x 2 1 + x 2 2 + · · · + x 2 n = 1 , is n 1 = 2 π n/ 2 Γ( n/ 2) 3
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Preliminary Examination September , 2005 3. Consider a pendulum in a uniform gravitational field with a mass m suspended by a rod that can be stretched and compressed. Model the rod by a spring of equilibrium length l and spring constant k . (a) Find the Lagrangian and give the equations of motion in terms of the distance r of the mass from the suspension point and the angle θ the rod makes with the vertical ( θ = 0 corresponds to the downward hanging pendulum). (b) Find the system’s equilibrium positions. Which are stable? Which are unstable? (c) Find the linear equations of motion by expanding about the stable equilibrium. (d) Find the normal coordinates and the eigenfrequencies there. 4
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Preliminary Examination September , 2005 4. In a modern fixed-target particle physics experiment, an outgoing positron enters a region of the spec- trometer containing a uniform magnetic field. (a) In the laboratory frame of reference, the positron is traveling with velocity β in the ˆ z -direction, and the magnetic field strength is B y ˆ y . Calculate the E and B fields in the rest frame of the positron in Gaussian or SI units. (Hint: In Gaussian units, E and B have the same units). (b) If the positron (of mass = 0 . 511 MeV / c 2 ) has energy 50 GeV , and the magnetic field has strength 1 T in the laboratory frame, what is the magnitude of the electric field E obtained in (a)? Use reasonable approximation. 5
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Preliminary Examination September , 2005 5. (a) Find the exponential Fourier Transform of the function f ( x ) = exp( α | x | ) , where α > 0 .
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