Download Homework 8 - spacibm configuration notes

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?
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(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.
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(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]
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