Where did the expansion for particle accelerators come from?

[steve_rb]

In "Particle Accelerators" book of Stanly Livingston I saw attached potential expansion for the potential of axial traveling beam of particles throght a electrostatic lense. I want to know where this expansion came from? Which expansion is being used and how?

 

[Bob S]

In the book "Particle Accelerators" by Livingston and Blewett on page 101, Eq 5-16 is derived from the Laplace Equation in cylindrical coordinates on page 98 Eq 5-6 with ρ=0.

In a charge-free region, the potential V(r,z) with axial symmetry must satisfy Laplace's equation with charge density ρ =0

Laplace's equation in cylindrical coordinates is then

$$\frac{1}{r}\cdot\frac{\partial}{\partial r}[r\frac{\partial V(r,z)}{\partial r}]+\frac{\partial^2V(r,z)}{\partial z^2} = 0$$

The following approximation is useful for approximating paraxial electric fields:

$$V(r,z)=V(0,z)+A r\frac{\partial V}{\partial z}+ B r^2\frac{\partial^2 V}{\partial z^2} + C r^3\frac{\partial^3 V}{\partial z^3}+ D r^4\frac{\partial^4 V}{\partial z^4}$$

where the partial derivatives are evaluated on the axis. Substituting the second equation into the first yields A = C = 0, and B = -1/4 and D = +1/64.
To see further discussion, look at page 111 Eq 6.2 in the free downloadable ebook by Humphries "Principles of charged Particle Acceleration":

http://www.fieldp.com/cpa.html

Bob S

test Latex equations here http://www.equationsheet.com/textoimage.php

 

[steve_rb]

Thanks Bob. Great job.