1.13 Assignments 25
and use this in Eq. (1.41).
Because of spherical symmetry use spherical polar coordinates for thekspace integral.
The only fixed direction is therdirection so choose the polar axis of thekcoordinate
system to be parallel tor. Thusk·r=krαwhereα= cosθandθis the polar angle.
Also,dk=−dφk^2 dαdkwhereφis the azimuthal angle. What are the limits of the
respectiveφ,α,andkintegrals? In answering this, you should first obtain an integral
of the form
φ(r)∼
∫?
φ=?
dφ
∫?
α=?
dα
∫?
k=?
k^2 dk×(?) (1.44)
where the limits and the integrand with appropriate coefficients are specified (i.e.,
replace all the question marks and∼by the correct quantities). Upon evaluation of
theφandαintegrals Eq.(1.44) becomes an even function ofkso that the range of
integration can be extended to−∞providing the overall integral is multiplied by 1 / 2.
Realizing thatsinkr=Im[eikr], derive an expression of the general form
φ(r)∼Im
∫∞
−∞
kdk
eikr
f(k^2 )
. (1.45)
but specify the coefficient and exact form off(k^2 ).Explain why the integration con-
tour (which is along the realkaxis) can be completed in the upper half complexk
plane. Complete the contour in the upper half plane and show that the integrand has a
single pole in the upper half plane atk=?Use the method of residues to obtainφ(r).
- Make sure you know how to evaluate quicklyA×(B×C)and(A×B)×C. A use-
ful mnemonic which works for both cases is: “Both variations = Middle (dot other
two) - Outer (dot other two)”, where outer refers to the outer vector of the parentheses
(furthest from the center of the triad), and middle refers to the middle vector in the
triad of vectors.
- Particle Integrator scheme (Birdsall and Langdon 1985)-In this assignment you will
develop a simple, but powerful “leap-frog” numerical integration scheme. This is a
type of “implicit” numerical integration scheme. This numerical scheme can later
be used to evaluate particle orbits in time-dependent fields having complex topology.
These calculations can be considered as numerical experiments used in conjunction
with the analytic theory we will develop. This combined analytical/numerical ap-
proach provides a deeper insight into charged particle dynamics than does analysis
alone.
BriefnoteonImplicitv.Explicitnumericalintegrationschemes
Suppose it is desired use numerical methods to integrate the equation
dy
dt
=f(y(t),t)
Unfortunately, sincey(t)is the sought-after quantity , we do not know what to use in
the right hand side fory(t). A naive choice would be to use the previous value ofyin
the RHS to get a scheme of the form
ynew−yold
∆t
=f(yold,t)
