Electric and magnetic field between concentric, conducting cylinders

[lar49]

Homework Statement

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Two long, concentric, conducting cylinders of radii and b (a<b) each carry a current I in opposite directions and maintain a potential difference V.
An electron with velocity u (parallel to the cylinders) travels undeviated through the space between the two cylinders.
Find an expression for |u|

Homework Equations

F=q(E+u^B)

The Attempt at a Solution

All I've managed is to say that there must be no net force, so
E=-u^B
E=-|u||B|sinθ
I'm not sure how to work out the electric of magnetic field.

 

[dauto]

Use Faraday's law to find the electric field and the relation between voltage and electric field to find the electric field.

EDIT: Sorry, I meant to say use Ampere's law to find the magnetic field

 

[lar49]

OK so using amperes law I get
B=μI/2πr
Is this right?
And what's the relationship between electric field and potential difference?

 

[vanhees71]

Your magnetic field is right for the gap between the inner and outer cylinder (coax cable).

To answer your other question, you should think a bit. The equations are those for stationary fields, i.e., (in SI units)
$$\vec{\nabla} \times \vec{E}=0, \quad \vec{\nabla} \cdot \vec{E}=\rho, \quad \vec{\nabla} \cdot \vec{B}=0, \quad \vec{\nabla} \times \vec{B}=\mu \vec{j}, \quad \vec{j}=\sigma \vec{E}.$$
This holds in non-relativistic approximations for the movement of the electrons in the, and this is a damn good approximations for all household currents :-)).

In addition you need appropriate boundary conditions for the fields at the surfaces of the conductors. These you should find in any textbook of electrodynamics, e.g., Jackson, Classical electrodynamics.

 

[paranormal]

I would approach this problem by using the equations for electric and magnetic fields in a cylindrical coordinate system. The electric field between the two cylinders can be found using the equation E = V/r, where V is the potential difference and r is the distance from the center of the cylinders. The magnetic field can be found using the equation B = μ0I/2πr, where μ0 is the permeability of free space, I is the current in the cylinders, and r is the distance from the center of the cylinders.

We know that at the point where the electron is traveling, the electric field must be equal in magnitude to the magnetic field, so we can set the two equations equal to each other and solve for |u|.

|u| = E/B = (V/r)/(μ0I/2πr) = (2πV)/(μ0Ir)

Therefore, the speed of the electron, |u|, is directly proportional to the potential difference V, the permeability of free space μ0, and the current in the cylinders I, and inversely proportional to the distance from the center of the cylinders r. This expression can be used to calculate the speed of the electron as it travels through the space between the two cylinders.