Speed, force, magnetic flux of an ion?

In summary, the conversation discusses the use of a mass spectrometer to measure the masses of isotopes. The spectrometer consists of an ion generator, accelerator, velocity selector, and ion separator, all in a vacuum. An experiment with tin ions is described, and the final speed of an ion of tin-120 is calculated to be 177 km s-1. The force on an ion due to the electric field in the velocity selector is found to be 3.2 * 10-15 N. The required magnetic flux density for selecting ions traveling at a speed of 177 km s-1 is 0.11 T. In the ion separator, the ions move in circular paths and the distance between the points of impact on
  • #1
moenste
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Homework Statement


The diagram shows a mass spectrometer used for measuring the masses of isotopes. It consists of an ion generator and accelerator, a velocity selector and an ion separator, all in a vacuum.

af11bacf44df.jpg


In one experiment, tin ions, each of which carries a charge of +1.6 * 10-19 C, are produced in the ion generator and are then accelerated by a p. d. of 20 000 V. Tin has a number of isotopes, two of which are tin-118 (118Sn) and tin-120 (120Sn).

(a) (i) Assuming that an ion of tin-120 is at rest before being accelerated, show that the final speed after acceleration is 177 km s-1. Mass of nucleon = 1.7 * 10-27.
(a) (ii) What will be the final speed of an ion of tin-118?

(b) In practice all ions produced by the ion generator have a range of speeds. A velocity selector is used to isolate ions with a single speed. In the velocity selector the force produced by the electric field is balanced by that due to the magnetic field which is perpendicular to the plane of the paper.
(b) (i) The plates producing the electric field have a separation of 2.0 cm. The potentials of the plates are marked on the diagram. What is the magnitude of the force on an ion due to this electric field in the velocity selector?
(b) (ii) Write down the equation which must be satisfied if the ions are to emerge from the exit hole of the velocity selector. Define the terms in the equation.
(b) (iii) What magnetic flux density is required if ions traveling with a speed of 177 km s-1 are to be selected?

(c) After selection the ions are separated using a magnetic filed on its own, as shown in the diagram.
(c) (i) Explain why the ions move in circular paths in this region.
(c) (ii) Show that the radius of the path is directly proportional to the mass of the ion.
(c) (iii) The ions are detected using the photographic plate P. Determine the distance between the points of impact on the photographic plate of the two isotopes of tin when a magnetic flux density of 0.75 T is used in the ion separator.

Answers: (a) (ii) 179 m s-1, (b) (i) 3.2 * 10-15 N, (iii) 0.11 T, (c) (iii) 0.50 cm.

2. The attempt at a solution
(a) (i) (1 / 2) M v2 = Q V, where Q = 1.6 * 10-19 C, V = 20 000 V, v = ? and M = ?.

We find M first: Relative atomic mass Ar = Mass of atom M / (1 / 12( the mass of a 126C atom u → Ar = M / u → M = Ar * u = 120 * 1.7 * 10-27 = 2.04 * 10-25 kg.
So v = √ 2 Q V / M = √ 2 * 1.6 * 10-19 * 20 000 / 2.04 * 10-25 = 177 123 m s-1 or 177 km s-1.

(a) (ii) Doing the same thing, find M first: M = Ar * u = 118 * 1.7 * 10-27 = 2.006 * 10-25 kg.

Plug in: v = √ 2 Q V / M = √ 2 * 1.6 * 10-19 * 20 000 / 2.006 * 10-25 = 178 617.7 m s-1.

I didn't continue since it's either a typo in my book answer, or I did something wrong. I get 179 km s-1, not 179 m s-1.
 
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  • #2
moenste said:
get 179 km s-1, not 179 m s-1.
I think that is indeed a typo. Final speed of tin-120 is 177km/s. So the answer should be in km/s and not m/s. Also, 179m/s practically seems too small (slower than a bullet shot from a gun).
 
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  • #3
cnh1995 said:
I think that is indeed a typo. Final speed of tin-120 is 177km/s. So the answer should be in km/s and not m/s. Also, 179m/s practically seems too small (slower than a bullet shot from a gun).
Good : ).

I continued the work. For (b) (i) I need to find the force.
moenste said:
(b) In practice all ions produced by the ion generator have a range of speeds. A velocity selector is used to isolate ions with a single speed. In the velocity selector the force produced by the electric field is balanced by that due to the magnetic field which is perpendicular to the plane of the paper.
(b) (i) The plates producing the electric field have a separation of 2.0 cm. The potentials of the plates are marked on the diagram. What is the magnitude of the force on an ion due to this electric field in the velocity selector?
So I used F = e E, where E = V / d so F = e V / d = 1.6 * 10-19 * 200 / 0.02 = 1.6 * 10-15 N. But the answer is two times larger. What's the reason?

Update
Maybe since the potentials are +200 and -200 we can assume that the electric field is being created by a scale of between +200, 0 and -200 or a 400 V in total. F = e V / d = 1.6 * 10-19 * 400 / 0.02 = 3.2 * 10-15 N
 
  • #4
moenste said:
Update
Maybe since the potentials are +200 and -200 we can assume that the electric field is being created by a scale of between +200, 0 and -200 or a 400 V in total. F = e V / d = 1.6 * 10-19 * 400 / 0.02 = 3.2 * 10-15 N
Right.
 
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  • #5
cnh1995 said:
Right.
Great.

Though I'm now struggling with the (b) (iii) and (c) (iii) answers.
moenste said:
(b) (iii) What magnetic flux density is required if ions traveling with a speed of 177 km s-1 are to be selected?
I can only think of formulas involving B like r = M v / Q B, where we don't have radius.

moenste said:
(c) (iii) The ions are detected using the photographic plate P. Determine the distance between the points of impact on the photographic plate of the two isotopes of tin when a magnetic flux density of 0.75 T is used in the ion separator.
As I understand we need to find the radius. I used the formula above: r = M v / Q B. So we have: B = 0.75, I took 120Sn so M = 2.04 * 10-25 kg, Q = 1.6 * 10-19 C and v is 177 000 m s-1. So r = 2.04 * 10-25 * 177 000 / 1.6 * 10-19 * 0.75 = 0.3009 m or 30.09 cm. Doesn't stand even close to 0.5 cm in the answer.

What am I missing?
 
  • #6
moenste said:
I can only think of formulas involving B like r = M v / Q B, where we don't have radius.
That is when the ions enter the ion separator. I believe the question is asking about the selection of ions in the velocity selector. In the velocity selector, electric and magnetic forces balance each other. What is the expression describing this? You have solved a similar problem recently.
 
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  • #7
moenste said:
As I understand we need to find the radius. I used the formula above: r = M v / Q B. So we have: B = 0.75, I took 120Sn so M = 2.04 * 10-25 kg, Q = 1.6 * 10-19 C and v is 177 000 m s-1. So r = 2.04 * 10-25 * 177 000 / 1.6 * 10-19 * 0.75 = 0.3009 m or 30.09 cm.
This only gives the radius of tin-120 ion. The question is asking you the distance between two impact points. What is that distance in terms of the radii of the ions?
 
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  • #8
cnh1995 said:
That is when the ions enter the ion separator. I believe the question is asking about the selection of ions in the velocity selector. In the velocity selector, electric and magnetic forces balance each other. What is the expression describing this? You have solved a similar problem recently.
e E = e v B?
B = E / v = (400 / 0.02) / 177 000 = 0.11299435 T = 0.11 T?
 
  • #9
moenste said:
e E = e v B?
B = E / v = (400 / 0.02) / 177 000 = 0.11299435 T = 0.11 T?
Yes.
 
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  • #10
cnh1995 said:
This only gives the radius of tin-120 ion. The question is asking you the distance between two impact points. What is that distance in terms of the radii of the ions?
Hm, maybe I'm calculating wrong. Not sure.

r1 = 0.3009 m or 0.3011091 m when I use 177 123 m s-1 instead of 177 000 m s-1.

r2 = 2.006 * 10-25 * 179 000 (or 178 617.7) / 1.6 * 10-19 * 0.75 = 0.299 m (or 0.298 589 255 m).

If I use rounded values like 177k and 179k I get the difference around 0.1 cm, while if I use 177 123 m / s and 178 617.7 m / s I get 0.25 cm. But still not 0.50 cm.

What am I missing?
 
  • #11
moenste said:
Hm, maybe I'm calculating wrong. Not sure.

r1 = 0.3009 m or 0.3011091 m when I use 177 123 m s-1 instead of 177 000 m s-1.

r2 = 2.006 * 10-25 * 179 000 (or 178 617.7) / 1.6 * 10-19 * 0.75 = 0.299 m (or 0.298 589 255 m).

If I use rounded values like 177k and 179k I get the difference around 0.1 cm, while if I use 177 123 m / s and 178 617.7 m / s I get 0.25 cm. But still not 0.50 cm.

What am I missing?
You are off by a factor of two. This is a big clue. See the diagram carefully. Do you want to calculate the difference between the radii?
 
Last edited:
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  • #12
cnh1995 said:
Do you want to calculate the difference between the radii?
Yes, isn't it the answer?
 
  • #13
moenste said:
Yes, isn't it the answer?
No. Its the difference between the diameters. See the diagram carefully.
 
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  • #14
cnh1995 said:
No. Its the difference between the diameters. See the diagram carefully.
Aaah, yes indeed! It's the difference between the diameters and not the radii. In that case we get 0.5 cm.

Thank you, didn't expect to solve such a large problem so fast.
 
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Related to Speed, force, magnetic flux of an ion?

1. What is the relationship between speed and force of an ion?

The speed and force of an ion are directly related. As the speed of an ion increases, so does the force acting on the ion. This is because force is defined as the product of an object's mass and acceleration, and as an ion's speed increases, its acceleration also increases.

2. How is magnetic flux related to ions?

Magnetic flux is the measure of the strength of a magnetic field passing through a given area. Ions, being charged particles, are affected by magnetic fields and can therefore interact with magnetic flux. The strength of the magnetic field can impact the movement and behavior of ions.

3. Can the speed of an ion be affected by magnetic flux?

Yes, the speed of an ion can be affected by magnetic flux. This is because ions are charged particles that can be influenced by magnetic fields. Changes in the strength or direction of a magnetic field can impact the movement of ions, altering their speed.

4. How do scientists measure the speed of an ion?

Scientists can measure the speed of an ion using various techniques such as time-of-flight mass spectrometry, laser-induced fluorescence, or particle accelerators. These methods involve measuring the time it takes for an ion to travel a known distance or the energy it emits as it moves, allowing scientists to calculate its speed.

5. What factors can affect the speed, force, and magnetic flux of an ion?

Several factors can affect the speed, force, and magnetic flux of an ion. These include the strength and direction of the magnetic field, the charge and mass of the ion, and any external forces acting on the ion. Additionally, the medium in which the ion is traveling, such as air or a vacuum, can also impact its speed and the strength of the magnetic flux it experiences.

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