(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Two long concentric, cylindrical conductors of radiiaandb(a<b), are maintained with a potential differenceVand carry equal but opposite currentsI.

An electron, with velocityuparallel 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?

2. Relevant equations

[tex]\oint_{C} \textbf{B}\cdot \textbf{dl}=\mu_{0}\int di[/tex]

[tex]\oint_{S} \textbf{E}\cdot \textbf{dA} = \frac{Q}{\epsilon_{0}}[/tex]

[tex]\textbf{F}=q(\textbf{E} +\textbf{u}\times \textbf{B})[/tex]

3. The attempt at a solution

The B field is curling around the inner cylinder.

Using Ampere's Law,

[tex]B=\frac{\mu_{0}I}{2\pi r}[/tex]

for a<r<b

r = distance from centres of cylinders

E field (electrostatics) of concentric cylinders

[tex]E=\frac{Q}{2\pi\epsilon_{0}r}[/tex]

where Q is charge per unit length, assuming length>>r

Following from above

Capacitance

[tex]C=\frac{2\pi\epsilon_{0}}{log_{e}(\frac{b}{a})}[/tex] per unit length

so using Q=CV, get

[tex]E=\frac{V}{r log_{e}(\frac{b}{a})}[/tex]

now forceF=q(E+u^B)

so

[tex]|\textbf{u}|=\frac{|\textbf{E}|}{|\textbf{B}|}[/tex]

and so

[tex]|\textbf{u}|=\frac{2\pi V}{\mu_{0}I log_{e}(\frac{b}{a})}[/tex]

would this look right?

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