How Can You Calculate the Necessary Acceleration for Ion Separation?

[Amrator]

Homework Statement

Ok, so this question is not really a homework problem but just something I'm trying to solve for fun and to improve my problem solving skills.

You must separate fast moving ions from slow moving ones. To do this the ions are brought into the device in a narrow beam so that all of the ions are going in the same direction. The ion beam then passes between two parallel copper plates. Each plate is 5.0 cm long, 4.0 cm wide, and the two plates are separated by 3.0 cm. A high voltage is applied to the plates causing the ions passing between them to have a constant acceleration directly toward one of the plates and away from the other plate. You boss asks you to calculate the magnitude of acceleration between the plates necessary to separate ions with a velocity of 100 m/s from those in the beam going 1000 m/s by 2.0 cm when they hit the ion detector 50 cm away. Before the ions enter the gap between the plates, they are going directly toward the center of the gap parallel to the surfaces of the plates. After the ions leave the gap between the plates, they are no longer accelerated during the remaining 50 cm to the ion detector. You look at the ion detector and find its face be circle with a radius of 7 cm.

Homework Equations

$X_f = X_0 + V_0t + (1/2)at^2$

The Attempt at a Solution

Ignore the 48.04 cm. I realize that's wrong. Same thing with the 2.3. That shouldn't be there.

So I figured that a lot of the information they gave was irrelevent, such as the dimensions of the plates and the radius of the detector. I used Pythagorean theorem to find the velocity of $X_{2c}$ after it left the plates,

$X_{2c} = \sqrt(50^2 + 2^2)$

I just don't know where to go from here. I've been staring at this problem for hours.

 

[mfb]

If something is moving with 1000 m/s to the left, how much time does it take to travel 50 cm? To get deflected by some distance (let's say 5 cm as example, you don't know the value you need yet), what is its vertical velocity?

 

[Amrator]

If something is moving with 1000 m/s to the left, how much time does it take to travel 50 cm? To get deflected by some distance (let's say 5 cm as example, you don't know the value you need yet), what is its vertical velocity?

0.0005 seconds. But the velocity is 1000 m/s before going through the plates.

 

[mfb]

Do you expect the horizontal velocity component to change?

 

[Amrator]

Good point.

$X_{f_y} = X_{0_y} + V_{0_y}t + (1/2)at^2$
0.05 m = 0 + 1000 m/s(0.0005s) + (1/2)a(0.0005)^2
a = 3600000 m/s^2

$V_{f_y} = V_{0_y} + at$
= 1000 m/s + (-3600000 m/s^2)(0.0005 s)
= -800 m/s

 

[mfb]

There is no acceleration over the 50 cm distance.
There is acceleration over the 5 cm of plates before, however.

This acceleration does not happen in the direction of the 1000 m/s velocity, however. You are mixing the two components.

 

[Amrator]

I just realized that. a = 400000 m/s^2
Vy = 200 m/s

Edit: never mind, that's wrong too.

 

[mfb]

Now you can include the displacement that happens within the plates if you like. It is a small correction (~5%).

 

[Amrator]

You mean the values in my last post were not wrong?

 

[mfb]

Oh right, 200 m/s is wrong.
There is no acceleration in the 50cm drift distance. Acceleration gets relevant in the step afterwards.

 

[Amrator]

0.05 m/0.0005 s = 100 m/s. That's vertical velocity, right?

 

[Amrator]

0.05 + 0.02 = 0.07 m

 

[Amrator]

Using Pythagorearn theorem, $\sqrt(1000^2 + 100^2)$ = 1005 m/s

Now I have to find the time between the plates.

 

[Amrator]

(1005 m/s - 1000 m/s)/0.00005 s

I think I did it! a = 100000 m/s^2 Is that right?

 

[mfb]

You can edit your posts instead of making a new one if you want to add something.

0.05 m/0.0005 s = 100 m/s. That's vertical velocity, right?

Correct if you neglect the deflection that happens between the plates. Which is fine if you just want to find a rough estimate for yourself, it would be wrong in a test.

0.05 + 0.02 = 0.07 m

What does that mean?

The total velocity does not matter.

(1005 m/s - 1000 m/s)/0.00005 s

That calculation doesn't make sense.

Also keep in mind that I introduced the 5 cm deflection purely as an example. You'll have to find the actual deflection later.

 

[Amrator]

Alright so I found time for the 50cm drift which is 0.0005 s. The vertical velocity is 100 m/s. I'm not sure where to go from here.

Edit: Time during plates is 0.00005 s.

 

[mfb]

Well, let's keep that approximation of no deflection between the plates.

Edit: Time during plates is 0.00005 s.

During that timescale, the vertical velocity changes from 0 m/s to 100 m/s. What is the acceleration?

 

[Amrator]

a = (0 - 100 m/s)/0.00005 s = -2000000 m/s^2

 

[mfb]

Writing it as 2*10⁶ m/s^2 is probably easier.

What is the next step?

 

[Amrator]

Find deflection which would be change in vertical position.

(1000)(0.00005)-.5(2*10^6)(0.00005)^2 = 0.0475 m

So velocity is 950 m/s?

 

[mfb]

I have no idea what you are calculating in this post.

 

[Amrator]

$x_{f_y} = .5at^2$
(0.0475 m/s)/0.00005 s

Oh I see the error.
Deflection is actually 0.0025m.

No, that doesn't make sense.

 

[mfb]

Right.
Combine it with deflection afterwards and you get the total deflection in this example.

Okay, you know the acceleration necessary to get this total deflection. Now you need the opposite direction: given a specific acceleration, you have to find the total deflection both for 100m/s and 1000m/s, and find the difference.

 

[Amrator]

Deflection afterwards?

 

[mfb]

At the point where they leave the plates, they deviate by 0.25cm from the center, 50 cm (horizontal) later they got an additional 5 cm vertical deflection.
At least in the toy model discussed so far.

 

[Amrator]

At the point where they leave the plates, they deviate by 0.25cm from the center, 50 cm (horizontal) later they got an additional 5 cm vertical deflection.
At least in the toy model discussed so far.

Well the vertical velocity for the other object Is 10 m/s, right? It makes sense since both object are separated by a vertical acceleration of the same magnitude. So by taking the ratio of velocity components of the first object,

100/1000=.1

Multiply it by the horizontal velocity component of the second object to get its vertical velocity component. .1 * 100 = 10

Can I do that?

OK so I found the deflection of the second object which is .25m.

The difference is -.197m. Is this correct?

 

[mfb]

It would be, but your capacitor is not that large.

Yes you can take the ratio, and indeed the deflection will be 10 times the other value.

You have one beam that is deflected 10 times more than the other one. The difference between the two is 2 cm. What is the deflection of beam 1 and beam 2? Note: this question is completely independent of everything done so far.