Magnetic and electric field between current carrying coaxial cables

[disillusion]

Homework Statement

Two long concentric, cylindrical conductors of radii a and b (a<b), are maintained with a potential difference V and carry equal but opposite currents I.
An electron, with velocity u parallel to the axis, enters the evacuated region between the conductors and travels undeviated. Find an expression for |u|.

I am not sure I understood the question very well and would like to see what others make of the question.
Since the current is flowing so there would be an E field along the direction of the current, which would not affect the electron. Should I just treat the potential difference as a separate E field along the radial direction?

Homework Equations

$$\oint_{C} \textbf{B}\cdot \textbf{dl}=\mu_{0}\int di$$
$$\oint_{S} \textbf{E}\cdot \textbf{dA} = \frac{Q}{\epsilon_{0}}$$
$$\textbf{F}=q(\textbf{E} +\textbf{u}\times \textbf{B})$$

The Attempt at a Solution

The B field is curling around the inner cylinder.
Using Ampere's Law,
$$B=\frac{\mu_{0}I}{2\pi r}$$
for a<r<b
r = distance from centres of cylinders

E field (electrostatics) of concentric cylinders
$$E=\frac{Q}{2\pi\epsilon_{0}r}$$
where Q is charge per unit length, assuming length>>r

Following from above
Capacitance
$$C=\frac{2\pi\epsilon_{0}}{log_{e}(\frac{b}{a})}$$ per unit length
so using Q=CV, get
$$E=\frac{V}{r log_{e}(\frac{b}{a})}$$

now force F=q(E+u^B)
so
$$|\textbf{u}|=\frac{|\textbf{E}|}{|\textbf{B}|}$$
and so
$$|\textbf{u}|=\frac{2\pi V}{\mu_{0}I log_{e}(\frac{b}{a})}$$

would this look right?

 

[LowlyPion]

Should I just treat the potential difference as a separate E field along the radial direction?

I would think so.

The result looks OK.

 

[nlightner]

I would say that your solution looks correct. Your approach of using Ampere's Law and the equations for electric fields between concentric cylinders is a good way to solve this problem. However, I would suggest double-checking your calculations and units to ensure accuracy. Additionally, it might be helpful to provide a more detailed explanation of your thought process and reasoning for each step in your solution. This can help others, who may be struggling with the same problem, to better understand the concepts and techniques involved. Overall, your solution appears to be a solid response to the given problem.