Electrons being fired at cold atomic hydrogen

[Saketh]

Electrons are accelerated through a potential V into a cloud of cold atomic hydrogen. A series of plates with aligned holes select a beam of scattered electrons moving perpendicular to the plates. Immediately beyond the final plate, the electrons enter a uniform magnetic field B perpendicular to the beam; they curve and strike a piece of film mounted on the final plate. When the film is developed, a series of spots is observed. The distances between the hole and the two most distant spots are measured. Find the magnitude of B, of V, and the total number of spots on the film.​

Basically, an electron gun is shooting electrons into cold atomic hydrogen. Something happens, and electrons come out of the other side, passing through a velocity selector and then a uniform magnetic field.

The first part of the problem (not shown) required me to calculate the energy of an electron, which I derived as E_n = -\frac{2.18 x 10^{-18} J}{n^2}.

I know how to find the velocity of the electrons entering the cloud of cold atomic hydrogen: eV = \frac{mv_i^2}{2}. I also know how to find the magnitude of the magnetic field: \frac{mv_f}{|q|R} = B. What I don't know how to find is v_f.

I don't know what's going on in the cold atomic hydrogen. I figured that since the electrons are all entering with a uniform velocity, then they will all leave with the same velocity (might be different from the initial one, though) and hit the same place on the film after curving around. However, according to the problem, the electrons are "scattered" and hit different points on the film. This makes no sense to me - how can I calculate this scattering?

The closest I got to understanding what's going on was writing this:
\frac{mv_i^2}{2} -\frac{2.18 x 10^{-18} J}{n^2} = \frac{mv_{f1}^2}{2} - \frac{mv_{f2}^2}{2}. But then I had no idea how to find v_{f2} and v_{f1}.

In summary, here are my questions:

1. What is going on in the cold atomic hydrogen?
2. How can the electrons be "scattered"?
3. What is leaving the cold atomic hydrogen?
4. How can I calculate the final velocities of the electrons?

All of these questions are closely interrelated, so I just need a push in the right direction.

Thanks in advance.

 

[Andrew Mason]

In summary, here are my questions:

1. What is going on in the cold atomic hydrogen?
2. How can the electrons be "scattered"?
3. What is leaving the cold atomic hydrogen?
4. How can I calculate the final velocities of the electrons?

All of these questions are closely interrelated, so I just need a push in the right direction.

This is electron-proton scattering. The electrons passing very near a proton will feel coulomb attraction and deflect toward the proton. But if it is traveling fast enough, it deflects but does not strike it. The closer it passes, the more it will be deflected from its straight line path.

Since the proton speed is so much less than the electron speed, you can consider the protons as a stationary target.

Electrons leave the hydrogen cloud traveling in different directions depending on how much they have scattered. The amount of deflection depends on the energy of the electron and the probability of encountering a proton (ie. the scattering cross-section).

The final velocities of the electrons should not be significantly affected. The protons will absorb very little of the electron energy since the proton's relative mass is so great.

AM

 

[Saketh]

Electrons leave the hydrogen cloud traveling in different directions depending on how much they have scattered. The amount of deflection depends on the energy of the electron and the probability of encountering a proton (ie. the scattering cross-section).

The person who gave the problem to me found it online, apparently. http://www.compadre.org/psrc/evals/2004semifinal.pdf" - it's the last one in the document.

Since the electrons all enter with the same energy, then that means all of the scattering is caused by the probability of encountering a proton. That makes sense. Then the same electrons leave the cloud.

But since there is a velocity selector, that means that the electrons that enter the magnetic field either did not get deflected at all, or were deflected into the selector at the proper angle.

The final velocities of the electrons should not be significantly affected. The protons will absorb very little of the electron energy since the proton's relative mass is so great.

How does the proton's mass affect the coulombic attraction?

When I solve for the uniform magnetic field, I have the equation \frac{mv_f}{|q|R} = B. Since the problem gives the distances between the hole of the velocity selector and the point on the film where the electrons struck for two of the electrons, I know the value of R. I also know m. The two unknowns are B and v. I know the velocity of the electrons entering the cloud, but not the velocity of those exiting.

I suppose I should come up with an equation that describes the velocity loss of electrons in the cloud. In that case, finding the total number of spots on the film is just an exercise in finding the velocities of electrons in the cloud that are deflected into the velocity selector properly. But how am I supposed to do this? Is there some way I can determine the "scattering cross-section" from the velocities of two exiting electrons? I still don't know the value of B, so I think there are too many unknowns and too few equations...

Thanks for helping me - I'm starting to understand what's going on.