two-dimensional electric quadrupole focusing field for particle accelerators can

Question

two-dimensional electric quadrupole focusing field for particle accelerators can be modeled by a set of four symmetrically placed line charge ,with linear charge densities ,the charge density in two dimensions can be expressed as ?(?,phi)=lambda/asum_{n=0}^{3}(-1)^n ?(?-a)?(?-n?/2) a)Using the Green function expansion G(?,?;?`,?`)= -ln(?^2>)+2sum_{m=1}^{infty}1/m((?<)/(?>))^m.cos?[m(?-?`)] Show that the electrostatic potential is ?(?,?)=lambda /?? sum_{k=0}^{infty}1/(2k+1) ((?<)/(?>))^(4k+2).cos[(4k+2)?] b)relate the solution of part a to the real part of the complex function w(z)=2lambda /4?? ln[((z-ia)(z+ia))/((z-a)(z+a))] Where z=x+iy=?e^i?. c)find expressions for the Cartesian components of the electric field near the orgin ,expressed in terms of x and y .keep the k=0 and k=1 terms in the expansion .for y=0 what is the relative magnitude of the k=1 (2^6-pole)contribution to Ex compared to the k=0(2^2-pole or quadrupole)term?

Explanation / Answer

this can help you http://www.phy.duke.edu/~rgb/Class/phy319/phy319/node97.html http://cccbdb.nist.gov/quadrupole.asp please rate if helpful

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