Charged particles moving through electric fields

[Nissen, Søren Rune]

I'm having some trouble trying to derive an equation for the movement of a positive ion through a quadropole.

The problem is that my primary source in this is "Physics for Scientists and Engineers with Modern Physics" by Raymond A. Serway and John W. Jewett, Jr. which has an excellent part on the subject of charged particles through electric fields*, but doesn't cover the subject of quadropoles, where the magnetic field is in flux.
(Or if it does, and you have the book, please point me at it, although that would make me feel very silly )

*pp. 725

I don't want you to derive it for me (at all. I want to learn this, yes?) but a short pointer on where to start would be nice

 

[OlderDan]

I'm having some trouble trying to derive an equation for the movement of a positive ion through a quadropole.

The problem is that my primary source in this is "Physics for Scientists and Engineers with Modern Physics" by Raymond A. Serway and John W. Jewett, Jr. which has an excellent part on the subject of charged particles through electric fields*, but doesn't cover the subject of quadropoles, where the magnetic field is in flux (Or if it does, and you have the book, please point me at it, although that would make me feel very sill )

*pp. 725

I don't want you to derive it for me (at all. I want to learn this, yes?) but a short pointer on where to start would be nice

Give this a try as a starting point. Then use what you know about the field from a single loop of current

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magquad.html#c2

If you don't know the single loop, try this

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/curloo.html#c1

 

[Nissen, Søren Rune]

Thank you, I'll read it and go from there.

 

[Nissen, Søren Rune]

Hah, it seems like I have to go all the way back to the Biot-Savart Law to go anywhere near where it looks like something I recognize. Seems like a long night of reading is ahead :(

 

[Nissen, Søren Rune]

Possibly, we are speaking of different subjects, it seems. When I say "Quadropole" I refer to something that looks a little like the graphic below: Four charged rods, where the charged particle moves through them along the z-axis.

I'm trying to find out how the particles will move along the rod, which function I can apply with variables q, m, and AFAI remember, also U and v

 y
/|\
 |      (-)
 |   (+)   (+)
 |      (-)
 |
 +-----> x

 

[OlderDan]

Possibly, we are speaking of different subjects, it seems. When I say "Quadropole" I refer to something that looks a little like the graphic below: Four charged rods, where the charged particle moves through them along the z-axis.

I'm trying to find out how the particles will move along the rod, which function I can apply with variables q, m, and AFAI remember, also U and v

Your original post made reference to magnetic fields and "flux". Now you are talking about an electric quadrapole, which IS a totally different thing. You are right about that. Based on your diagram, You can calculate the fields along the path of the particle on the z axis. It sounds like the particle is constrained to move on that axis, so only the z component of the electric field is needed. Are the rods of finite length, or infinite, or are they really just charged particles? I suspect the latter, based on the use of the term quadrapole.

 

[Nissen, Søren Rune]

Your original post made reference to magnetic fields and "flux". Now you are talking about an electric quadrapole, which IS a totally different thing. You are right about that. Based on your diagram, You can calculate the fields along the path of the particle on the z axis. It sounds like the particle is constrained to move on that axis, so only the z component of the electric field is needed. Are the rods of finite length, or infinite, or are they really just charged particles? I suspect the latter, based on the use of the term quadrapole.

The reason my original post refers to "magnetic field" and "flux" is because I am, in fact, a damn fool. Sorry.

I meant "electric field" and "varies"

The rods are technically of finite length (It's a practical problem), but I've been informed that the results will be "close enough" if rods of infinite length are used. The particle has an (approximately) constant v in the z-axis direction.

I'm trying to find out how the particle moves along the X axis as a function of the voltage over the positively charged rods, as well as the mass and charge of the particle. I'm also trying to find the same for the Y axis, as a function of the voltage over the negatively charged rods, as well as the mass/charge of the particle.

I know it's either a sinus or co-sinus function, but I'm having trouble finding out where to start.

(Basically, I've been tasked with describing exactly what makes our mass-spectrometer work, and I'm having some trouble with the particle selector part, ie: quadropole.)

 

[OlderDan]

Can you assume a uniform charge distribution on each rod? That may not matter either as long as they have the same distribution and are symmetric relative to the motion. I assume, since you are talking about a mass spectrometer that you are looking at deflection forces that will take the particle off axis.

 

[Nissen, Søren Rune]

Can you assume a uniform charge distribution on each rod? That may not matter either as long as they have the same distribution and are symmetric relative to the motion. I assume, since you are talking about a mass spectrometer that you are looking at deflection forces that will take the particle off axis.

I believe the charge distribution will be uniform, yes, so the particles position along the Z axis can be ignored. They are symmetric along the Z axis. If no charge is applied to the quadropole, all values of m/q will move through the quadropole with no problems. However, by varying the voltage applied, only specific m/q values will be allowed through the quadropole, the rest will hit the quadropole and discharge.

 

[OlderDan]

OK. If you assume infinitely long uniformly charged rods, the electric field for each rod is proportional to the charge density and inversely proportional to the distance from the rod. Adding up the electric fields from four rods of equal charge density becomes a vector addition problem. The way people usually do this is the treat the quadrapole as two dipoles, but that is probably only useful at some distance from the quadrapole. It might be easier for you to look at the potential, which will vary as the log of inverse distance from each rod. There is a section on the potential of an infinite rod here

http://www.pa.msu.edu/~duxbury/courses/phy294H/lectures/lecture10/lecture10.html

 

[Nissen, Søren Rune]

http://www.pa.msu.edu/~duxbury/courses/phy294H/lectures/lecture10/lecture10.html

This looks very much like the exact thing I'm looking for. Thank you, OlderDan.