Understanding Electron-Optic Biprism Phenomenon in Electron Microscopes

[meteorologist1]

Hi, I'm having trouble with a problem that I'm doing in my electricity and magnetism class. I can't even understand what it's asking and what the picture looks like.

Here is the problem:
In an electron microscope, a beam of energetic electrons, originally accelerated through a potential of Vo (which is typically a few thousand volts), passes a thin charged long wire stretched at right angles to the original beam. Assume the line charge density is lambda, and that the beam is slightly deflected by the wire. Show that the angular deflection of the beam from its original path is independent of the distance of the electron beam from the wire (i.e.: independent of "impact parameter"). Such an arrangement is called an "electron-optic biprism" since electrons passing on opposite sides of the wire are oppositely and equally deflected. [Hint: Assume to first approximation that the beam velocity in the forward direction is unaffected. Find the transverse velocity.]

If anyone can explain this phenomenon to me, that would be great. Thanks.

 

[Gokul43201]

What part don't you understand : the geometry or the way to solve the problem ?

 

[meteorologist1]

Both actually.

 

[Gokul43201]

Geometry : Imagine you have a cross made of two straight lines. Grab the two lines and separate them, so that you have two lines which look perpendicular to each other but are actually offset. One of these lines represents the deflector wire (or line of charge), and the second represents the path of the electron beam, if it were not deflected.

Does this make sense ?

 

[meteorologist1]

Ok I see. What about the part of being accelerated through a potential of Vo? Are there parallel plates or something?

 

[meteorologist1]

Ok, nevermind on the parallel plates. I see that the Vo serves to give the electron an initial energy. But I'm still stuck on this problem.

I have the formula for the electric field due to a line charge: E = (lambda) / [2(pi)(epsilon0)(r)] where r is the distance from the wire. The potential is V = - (lambda) / [2(pi)(epsilon0) ln(r/a)] where a is where we set the zero potential to be. Somehow I have to figure out the velocities in the x and y components and then to get the deflection angle; the deflection angle = arctan(v_y/v_x). And this answer should be indepedent of the impact parameter.

Please help. Thanks.