midtermlastyear - Cornell University Department of Physics...

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Unformatted text preview: Cornell University Department of Physics Classical Electrodynamics P561 Midterm exam October 11, 2007 You have 3 hours to work on the exam. This is an open notes and open Jackson exam: you can use any of your handwritten notes, but no book other than Jackson or anything printed is permitted. You may use calculators. Also NO collaboration is allowed and you may not be in real or virtual contact with a living person. :‘2 Do not open the exam until you are told to do so. .v If something is not clear about the exam, ask Johannes. Each of the four problems carry the same weight (25 points). However not each subpart will be graded equally, the value of the subpartsdis noted everywhere. Do NOT just right down and/ or copy equations from Jackson. Rather explain your calculations and reasoning clearly as if you were explaining it to someone who does not know the problem and its solution. Graded exams will be available on Thursday, Oct. 18 in class. NAME: SIGNATURE: 1. Relativistic collisions a. (10 points) A particle with mass m1 and velocity 171 collides with a stationary particle of mass m2 and is absorbed. Find the rest mass and velocity of the resulting compound system. b. (15 points) A particle of mass m collides elastically with a stationary particle of equal mass. The incident particle has kinetic energy T 0. What will be the kinetic energy of the scattered particle if the scattering angle is (9? (The scattering angle is the angle between the outgoing momentum and the incoming momentum.) “3 E1423 ‘i it ,, l 2/ .———'>. o. m 2. Particle in a magnetic field A particle of charge q, mass m moves in circular radius of orbit R in a uniform magnetic field Be}. a. (5 points) Find B in terms of R, q./ m and w, the angular frequency. Do not neglect relativistic effects. " b. (10 points) The speed of the the particle is constant since the B—field can not do any work on the particle. An observer moving at velocity ficé‘x however does not see the speed as constant. Calculate the 4 vector velocities u“ both in the lab frame and in the frame of the moving observer. 0. (10 points) Calculate duo/ /d7' and thus dPOI /dT. Explain how the energy of the particle can change if the B—field does not do any work on it. Resolve the “paradox” quantitatively. 3. Dielectrics and capacitors A capacitor is made of two co—axial conducting cylinders of length L, inner radius a and outer radius 1). This capacitor is brought to the surface of a fluid With density pf, and the dielectric constant of the fluid is e. The capacitor plates are held at a fixed potential difference U. We find that the fluid climbs to a height h into the capacitor. a. (8 points) Calculate the capacitance of the capacitor with the fluid filling it to height h. b. (2+7 points) Calculate the energy of the capacitor both using the usual formula W = 1/2C’U2 and also by integrating the energy density w = $6E2. c. (8 points) Using the energy found above calculate the force acting on the fluid. Find an expression for the dielectric constant iof the fluid as a function of h. 4. Green’s functions in spherical coordinates A surface charge density n 2 no cos2 6 is distributed along a sphere of radius R. a. (13 points) Expand the potential in appropriate orthogonal functions that solve the Laplace equation outside and inside the sphere. Write down the appropriate matching condi— tions at 7“ 2 R explicitly. Using the expansion find the complete expression for the potential everywhere by solving the matching conditions for the expansion coefficients. b. (12 points) Now verify your result by solving the same problem via integrating the Green’s function GD (f, 33”) for free space in spherical coordinates. 79 C05 2c? ’ a ...
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This note was uploaded on 10/25/2009 for the course ECE 4300 taught by Professor Lipson/pollock during the Fall '08 term at Cornell University (Engineering School).

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midtermlastyear - Cornell University Department of Physics...

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