prelim1988 - Table of Contents Complete Partially Complete...

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Table of Contents Complete Partially Complete Not Complete Preliminary Examination 1988 2 Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Problem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Problem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Problem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Problem 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Problem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Problem 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Problem 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Problem 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Problem 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Problem 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Problem 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Problem 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Problem 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Problem 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Problem 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Problem 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Problem 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Problem 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Problem 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Problem 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Problem 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1
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Preliminary Examination September , 1988 1. A cylindrical, parallel-plate capacitor is filled with a homogeneous, linear dielectric with permittivity ε and permeability μ 0 (SI units). The separation of the plates is L and the radius of the capacitor is R . There is a uniform field, D ( t ) inside the capacitor which increases linearly with time due to a constant current flowing onto the capacitor: D ( t ) = t a (where a is a constant vector) In the following neglect the fringe field outside the dielectric and leakage currents inside the dielectric. (a) Find the H field as a function of r , the radial distance from the central axis inside the dielectric. (b) Find the Poynting vector, S , as a function of r inside the dielectric. (c) Use the Poynting vector to find the energy flux into a cylinder of the dielectric of length L and radius r on the central axis (see figure). (d) Starting from the expression for the field energy density, calculate the rate of increase of teh field energy inside the cylinder of Part (c), and verify that energy is conserved. 2
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Preliminary Examination September , 1988 2. A particle of mass M is constrained to move without friction on the surface of a sphere of radius R , (the particle is always a distance R from a fixed point). Gravity is acting vertically downward. The particle is projected horizontally with a speed V 0 at a point which is at a height that is B below the center of the sphere. Show that when the particle is next moving horizontally, it will be at a height that is X below the center of the sphere, and find the equation for X . Note, it is not necessary to solve for X . 3
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Preliminary Examination September , 1988 3. Obtain the first two terms in the power series expansions of e x 1 + cos x (a) in powers of x , (b) in powers of ( x πi ) , and then (c) give the values of x for which (b) converges uniformly. (A proof is not expected, but you should briefly show your work.) 4
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Preliminary Examination September , 1988 4. An important problem in atomic, nuclear, and particle physics is the calculation of the energy lost by ions as they traverse matter: e.g., protons through multiwire proportional chambers or uranium nuclei through ionization chambers. Assume the ion of charge ze and rest mass M is traveling with velocity v = v ˆ x . Consider the "collision" (electromagnetic interaction) between M and an electron, rest mass m and charge e . Assume further that v is much larger than the characteristic velocity of the electron in its atomic orbit, and neglect the electronic binding energy; i.e., assume the electron is free and at rest, and remains so during the collision time. The ion’s distance of closest approach is b .
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