Motion of Charged Particles in E&B Fields

[PumpkinCougar95]

Homework Statement

Problem Statement: A system consists of a long cylindrical anode of radius a and a coaxial cylindrical cathode of radius b (b < a). A filament located along the axis of the system carries a heating current I producing a magnetic field in the surrounding space. Find the least potential difference between the cathode and anode at which the thermal electrons leaving the cathode without initial velocity start reaching the anode.

Homework Equations

F = m dv/dt = q( E + v x B )

The Attempt at a Solution

What I have tried: https://drive.google.com/open?id=1sLIz7ZsUwS0wkBW4iGEXWJBrxCenahKs

In the last step, I forgot to put a square on the log() term. But what I have written is given as the correct answer. I do not understand what I have done wrong. Help, please?

 

[TSny]

Hello, and welcome to PF!

Why do you think you've made a mistake?

 

[PumpkinCougar95]

The answer I got is not correct. Apart from that, I don't really understand.

 

[TSny]

The answer I got is not correct. Apart from that, I don't really understand.

In your first post you said that what you had written gives the correct answer. Here you are saying you did not get the correct answer. So, I'm a little confused.
Your answer in the first post looks correct to me (except for possibly a sign issue).

 

[PumpkinCougar95]

Oh sorry for the misunderstanding. In the last step I wrote, you can see I missed a square sign on the log() term from my previous line. What I meant was that the correct answer doesn't have a square on the log() term.

 

[TSny]

I think the correct answer should have a square on the log term.

 

[PumpkinCougar95]

But here it doesn't have a square...

https://drive.google.com/open?id=150Xo6D7BZjyYOkX3oiMmyqtbWZdhIVGA
https://drive.google.com/open?id=1RFOoNg3Pk5N35cuZ9o_Wv5ugz3aMfgIW

 

[TSny]

In deriving the second equation from the first equation above, you should get a square on the log term.
[EDIT: The first equation above should have a square on the log in the second term on the right side. The answer for V in the second equation should also have a square on the log.]

 

[PumpkinCougar95]

Yes, i got it. Thanks for your help!

 

[PumpkinCougar95]

Hmmm, I guess you are right but I mostly use cartesian coordinates because that's the only thing I know how to do properly.

I think that the expressions are correct (in the numerator that is V*x and V*y).

If you like problems like this, you can find lots of really interesting ones here:
https://archive.org/stream/IrodovPr...roblems_in_General_Physics#page/n111/mode/2up

 

[PumpkinCougar95]

I think you would only need to solve the Fz equation to get the answer, so it doesn't make any difference whether you use cylindrical or cartesian coordinates.

I have no idea on how to solve these equations completely.

 

[PumpkinCougar95]

But if you know $V_z$ you can apply conservation of energy between r=a and r=b. See my first post.

 

[rude man]

I'm deleting all my files for this thread because I'm stumped! May revisit later.