Classical Electrodynamics · Problem 2.20

Boundary-Value Problems in Electrostatics: II — John David Jackson

Physics — E&M · electric-quadrupole-four-line-charges

Problem

Two-dimensional electric quadrupole focusing fields for particle accelerators can be modeled by a set of four symmetrically placed line charges, with linear charge densities

\pm\lambda

, as shown in the left-hand figure (the right-hand figure shows the electric field lines). The charge density in two dimensions can be expressed as

\sigma(\rho, \phi) = \frac{\lambda}{a}\sum_{n=0}^{3}(-1)^n\,\delta(\rho - a)\,\delta(\phi - n\pi/2)

(a) Using the Green function expansion from Problem 2.17c, show that the electrostatic potential is

\Phi(\rho, \phi) = \frac{\lambda}{\pi\epsilon_0}\sum_{k=0}^{\infty}\frac{1}{2k+1}\left(\frac{\rho_<}{\rho_>}\right)^{4k+2}\cos[(4k+2)\phi]

(b) Relate the solution of part a to the real part of the complex function

w(z) = \frac{2\lambda}{4\pi\epsilon_0}\ln\left[\frac{(z - ia)(z + ia)}{(z - a)(z + a)}\right]

where

z = x + iy = \rho e^{i\phi}

. Comment on the connection to Problem 2.3. (c) Find expressions for the Cartesian components of the electric field near the origin, 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

E_x

compared to the

k = 0

(

2^2

-pole or quadrupole) term?

Related theory

Electric Field Due To A System Of Charges

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