# Calculate the speed of the electrons as they enter the gap

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1. Oct 24, 2016

### moenste

1. The problem statement, all variables and given/known data
Two parallel metal sheets of length 10 cm are separated by 20 mm in a vacuum. A narrow beam of electrons enters symmetrically between them as shown.

When a PD of 1000 V is applied between the plates the electron beam just misses one of the plates as it emerges.

Calculate the speed of the electrons as they enter the gap. (Take the field between the plates to be uniform.)

(e / m = 1.8 * 1011 C kg-1.)

Answer: 6.7 * 107 m s-1.

2. The attempt at a solution
I used 0.5 m v2 = e E → v = √ 2 * (e / m) * E = √ 2 * (1.8 * 1011) * 1000 = 18 973 665.96 m s-1.

But this formula does not include the length of the plates and their separation. Not sure what to do next.

2. Oct 24, 2016

### PeroK

What is $v$ in your equation for energy?

3. Oct 24, 2016

### moenste

Speed, velocity. Isn't it?

4. Oct 24, 2016

### PeroK

If you mean the speed as it enters the apparatus, what does that have to do with the strength of the electric field?

5. Oct 24, 2016

### cnh1995

The situation in this problem is like when you throw a ball horizontally (parallel to ground) from top of a building and it hits the ground.
Which force is acting on the electron? In which direction?

6. Oct 24, 2016

### moenste

Don't we need to find the speed?

F = m g vertically downwards? And also the acceleration force which is horizontal F = m a?

7. Oct 24, 2016

### PeroK

Post #5 above has given you a big clue. Do you understand what he is saying?

8. Oct 24, 2016

### moenste

Is this correct:
?

9. Oct 24, 2016

### PeroK

It doesn't say the plates are horizontal. In any case, the electron is moving so fast that gravity is negligible over such a short time.

Or, perhaps more accurately, once you calculate the electric force, you will see that the gravitational force is negligible by comparison.

Last edited: Oct 24, 2016
10. Oct 24, 2016

### lychette

Do you know how to calculate acceleration of an electron in an electric field?

11. Oct 24, 2016

### cnh1995

@moenste, As peroK has pointed out, gravitational pull will be negligible here. The electron is in an electric field. What is the force on the electron? In which direction? Which component of its velocity will be affected by this force? We have discussed a similar problem in another thread recently.

12. Oct 24, 2016

### moenste

a = E e / m = (1.8 * 1011) * 1000 = 1.8 * 1014 m s-2.

F = e E in an electric field. On the picture they are moving downwards, so downwards is directed the force?

13. Oct 24, 2016

### cnh1995

Yes. The force is acting vertically downwards.
You know the acceleration of the electron. So which component of its velocity gets affected by this force? What can you say about the displacement of the electron?

14. Oct 24, 2016

### moenste

Vertical component has that acceleration.

Well, it moves from left to right and is moving closer to the bottom.

15. Oct 24, 2016

### cnh1995

Right. So there is a force in the vertical direction and there is no force in the horizontal direction. What does this tell you about the horizontal component of its velocity? Also, what are the displacements of the electron in vertical and horizontal directions?

16. Oct 24, 2016

### lychette

With this acceleration can you calculate the time taken to travel 10mm

17. Oct 24, 2016

### moenste

s = v t + 0.5 a t2
0.1 = 0 * t + 0.5 * (1.8 * 1014) * t2
t = 3.33 * 10-8 s.

There is no horizontal component of the velocity?

Like y = 0.5 * (e E / m) * t2, x = v t so y = (e E / 2 m v2) x2?

18. Oct 24, 2016

### lychette

the only force is due to the electric field which is at right angles to the 'horizontal' velocity.
You now know the time for the electron to 'fall' 10mm.
can you calculate the velocity of the electron to travel the length of the plates? (watch the units!!)

19. Oct 24, 2016

### cnh1995

There is. How do you think the electron undergoes horizontal displacement then? No force in the horizontal direction means horizontal velocity is constant throughout (Newton's first law).
Right. You know the horizontal displacement and time. You can calculate the horizontal component of velocity using this equation.
This is the velocity of the electron while entering the gap.

20. Oct 24, 2016

### Staff: Mentor

E is the electric field, not the voltage. The potential difference between the plates is not the same thing as the electric field strength. You need to determine E first.