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HOMEWORK PROBLEMS 8 PHYS 532, Spring 2017 Radiation Generation Due date: Wednesday, 4/26/17 — 5pm Given the timeline for this homework, and the need to prepare for exams, a treat is now in store for this homework. You only have to complete one of the two problems below to be eligible to receive full credit. You get to choose the problem! They are of equal value. You will receive no additional credit for attempting both of them, and so focus your efforts on one. 1. Thomson scattering: Consider a monochromatic electromagnetic wave colliding with an electron that can freely respond to its fields. Let it be 100% polarized so that its electric field vector at any position assumes the form E = E0 cos(ωt−k·r+α) , with E0 being a constant vector and k = (ω/c) x̂ . (a) In the non-relativistic dipole approximation, solve the equation of motion for the electron in the wave’s electric field only, deriving the condition for its oscillation to be non-relativistic and of small amplitude. (b) Now add in the magnetic field component B = B0 cos(ωt − k · r + α) for the wave, which is orthogonal to E. Solve for the motion in the x -direction due to the oscillating magnetic field. Determine the maximum speed and the amplitude of the motion in this direction, and the ratio of these quantities to the corresponding ones in E0 -direction. (c) Derive the complete angular distribution of Thomson-scattered radiation due to this compound 2D oscillation for any emission direction n. What is the degree of polarization of the outgoing wave? 1 (d) Now substitute your compound motion for parts (a) and (b) into the Lorentz force equation and solve again for both components to the motion. This develops higher order contributions to the motion. Repeat the process to obtain an infinite series for each component to provide an exact description (perturbation expansion) of the electron’s motion in the coherent wave. If you can sum the series analytically, do so. Interpret your results. (e) Finally, let us consider momentum conservation. Since the “radius of influence” of the electron in its interaction with the wave is r0 = e2 /mc2 , compute the energy of the wave in a cylindrical volume V of cross sectional radius r0 and length equal to the wavelength λ = 2π/k of the wave. The axis of this volume is coincident with the wavevector k. For scattering by an angle θ from the x -direction, what is net momentum change in the wave in this volume? What is the resulting speed of the electron if it recoils with this momentum? Explain physically how the electromagnetic wave can transfer this momentum decrement to the electron. What is the net momentum transfer to the electron from this volume V of the electromagnetic wave when integrating over all scattering angles? [100 points credit] 2. Bremsstrahlung: Here we adapt the unbound trajectory solutions for Coulomb scattering, as studied in Homework 6, to the problem of bremsstrahlung emission by non-relativistic charges. (a) Consider a beam of positrons initially traveling parallel to each other with speed v c in the x -direction, repulsively scattering off the Coulomb field of a massive, stationary charge +Q . Let the beam by uniformly distributed in the plane transverse to the beam axis, within a circle of radius bmax . Write down the expressions derived in lectures for the polarization-dependent energies of bremsstrahlung emission per unit solid angle. These are integrated over the beam cross section and the duration of each unbound trajectory. 2 (b) Write down the unbound Coulomb trajectory solution for non-relativistic positrons in terms of polar coordinates (ρ, θ) , and from these derive the dipole moment d at each point along the trajectory. Resolve this vector into Cartesian components parallel to and perpendicular to the beam direction. (c) Convert the time integrations for the emission results in part (a) to integrals over either ρ or θ , and express the dipole moment acceleration vector d̈ in terms of your chosen integration variable. Simplify your integrations as much as you can, evaluating where possible. More credit will be given for the simplest expressions. Do your results remain finite when bmax → ∞ ? (d) Illustrate your results by plotting the two polarizations, as functions of the bremsstrahlung emission angle θ to the beam axis, for different values of the parameter eQ/(me bmax vc) of your choosing. Is there any direction where the bremsstrahlung is 100% linearly polarized? Explain the physical reasons underpinning your answer. [100 points credit] 3