How Do Ions Achieve Their Velocity in a Mass Spectrometer?

[Emspak]

Homework Statement

The whole problem statement is a bit involved, but it starts with a figure illustrating a mass spectrometer. You have the chamber the gas you want to study is pumped into, and an anode and cathode. An electron beam ionizes the gas, and the ions are accelerated towards the cathode.

The problem says that after that, the ions go through a slit and enter another chamber with a magnetic and electric field, and those two fields accelerate them. The ions go through a second slit, and make circular tracks that depend on their mass.

I understand all that, but what I want to know is how one gets the velocity to the second slit -- I understand that once an ion is in a magnetic field it gets accelerated (and I know which equations to use, at least partway -- I just need to know the charge q of the ion to figure the force exerted by a given B field). So if I start with an ion at rest, zap t with an E and B field, I will get an acceleration (and per my classical E&M class I should get a helical trajectory).

But what stumps me a bit is what happens with the first chamber. Do I assume that the gas ions start from rest before being accelerated towards the first cathode? In that case the velocity to that first slit would be related to F= q_ionq_cathode / 4πε₀r² if I remember right. Knowing the F = ma I can work out the acceleration and the velocity when it hits the first slit. Once i know the velocity to that point I just have to apply the relevant equations to get the velocity through the E and B chamber (the second one) to get the v through the second slit.

So that's what I want to know. Do I assume the ions start from rest in the very first chamber, and go from there? I also noticed that one method of determining momentum of a particle in the electric field involves using a potential difference, which I assume would be related to the distance between the initial anode and cathode, correct? (p = √2mK = √2mV₀e is the one I am thinking of).

Sorry to be so long about it. I just want to make sure I am not losing the plot.

 

[BvU]

I had to make do without a nice figure, so I asked google to show me a few. I agree that this is involved: you have a nice combination of mechanics, electrostatics, electrodynamics and what else. That's why it is a favorite for exams. I accept that you don't want to use the template, but don't tell anyone else or we both get kicked out of PF...

The second chamber has a special function: acceleration there does not mean speeding up because both B and E are perpendicular to the velocity vector. Work it out or look it up: if the speed is right, nothing happens, otherwise the particles are deflected away from the second slit. So it is a velocity selector.

In the first chamber you may assume the gas ions are at rest. There is some kinetic energy from the temperature being above 0 K, but you can calculate how small that is compared to e.g. a 1kV accelerating voltage.

Your electrostatic force equation there is OK, but it is more economical to look at energy equations: electrostatic energy is converted to kinetic energy. As you already mention at the end of your post.

There is some spread in |velocity| because not all the ions start at the same distance from the cathode, but that's where the second chamber comes in so useful.

After the second slit the particles get to see a magnetic field perpendicular to $\vec v$ so they describe a circular trajectory (helix if you don't have the $\vec B$ parallel to the slit).

I think it is good you ask these questions before going to work. As uncle Alfred said: it's formulating the right problem that takes more work and time than solving it.
OK, now to your question

 

[Emspak]

the only reason i didn't use the template was I wasn't sure if the way I was asking fit. Not every question fits so neatly :-) and I wasn't really asking a specific mathematical question in the way people often do.

Anyhow, if I understand you right, I can start with an E field and say there is a particle of +q in there, and say that the energy it gets increases by the same amount as the work done so I can

use this:

$\Delta W = q \int E dl$ and just use the distance between the cathode and anode -- whatever it is, call it L, and say, since $E = V_0 / L$:

$\Delta W = q \int E dl = q \int^L_0 E dl = qV_0$

which is my change in work, and thus the change in KE. Knowing the KE, which is really momentum times velocity, I can say that $pv = qV_0$ and $v^2 = \frac{qV_0}{m}$ and $v= \sqrt \frac{qV_0}{m}$ which tells me how fast the ion is moving once it gets to slit number one. Knowing that it isn't such a big deal to figure out how much it is accelerated by the E and B field, since the next E field will accelerate it, the B field is perpendicular to the ion path AND the E field (so you get straight-line motion if it's going a certain speed). Seems straightforward enough, yes?

 

[BvU]

Two things: I was taught EK = mv²/2 and in a speed selector $\vec E$ , $\vec v$ and $\vec B$ are all perpendicular to each other. (Can 't make out if you concuded that already, perhaps you have).

All pretty straightforward, yes. But involved.