# ps6 - (a Make a 3D Mathematica plot(Plot3D of the effective...

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PHY 504 Problem Set #6 due 15 October 2010 1. For the attractive inverse-square potential (a) Calculate the differential cross section σ ( Θ ) . (b) Calculate the capture cross section. This is defined to be the cross- sectional area of an incoming beam of particles of energy E , which gets captured rather than scattered by the potential. 2. Exercise 3.32. 3. A particle moves in the xy -plane, in a uniform perpendicular magnetic field and in an inverted harmonic-oscillator potential: (a) Rewrite L in polar coordinates. (b) Obtain conservation laws for angular momentum and energy. (c) Describe possible orbits for the case l = 0. Under what circumstances will the orbit be bounded? 4. Consider two masses in circular orbit around each other. Following the discussion in Sec. 3.12,

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Unformatted text preview: (a) Make a 3D Mathematica plot (Plot3D) of the effective gravitational potential of the two bodies in the rotating reference frame in which the bodies are stationary. Take m 1 = 2 m 2 . Neglect the Coriolis term. (b) Show – either explicitly or by blowing up your plot - that Goldstein’s claim that the L4 and L5 points in Fig. 3.30 are local minima of the effective gravitational potential is wrong. Plot a trajectory with a small value of l . (c) In light of problem 3, is it possible that the Coriolis term makes a difference, and stabilizes L4 and L5? Explain. L = 1 2 m ( ˙ x 2 + ˙ y 2 ) + 1 2 eB ( x ˙ y − y ˙ x ) + 1 2 m ω 2 ( x 2 + y 2 ) θ − Λ Ν Μ Μ Μ Ξ Π Ο Ο Ο 5. Derivation 4.10. Exponentiate the matrix...
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ps6 - (a Make a 3D Mathematica plot(Plot3D of the effective...

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