Max distance of rectangle that electron beam can pass through

[kkcolwell]

Homework Statement

A beam of electrons is fired into a rectangular region of space that contains a uniform magnetic field in the -z direction. The electrons are moving in the +x direction, as shown. The speed of the electrons in the beam is 6.00 × 106 m/s. The mass of an electron is me = 9.11 × 10−31 kg. The magnitude of the magnetic field in the rectangular region of space is 1.50 × 10−2 T. The rectangular region has a width d.


What is the maximum value of d for which the electron beam will make it through to the other side of this rectangular region, and continue on to the right of the region?

Homework Equations

|⃗v|=|E⃗|/|B⃗1|

(Equations provided)
|F⃗m| = |q||⃗v||B⃗ || sin(θ)|
|F⃗m| = IL|B⃗ || sin θ| r = m | ⃗v |
|q||B⃗ | r = m | ⃗v ⊥ | |q||B⃗ |
|F⃗E|=|F⃗B| −→ q|E⃗|=q|⃗v||B⃗1| −→ |⃗v|=|E⃗|/|B⃗1| | m⃗ | = N I A
τ = NIA|B⃗||sinθ| = |m⃗ ||B⃗||sinθ|
| B⃗ | = μ 0 I 2πr
| B⃗ | = μ 0 N I 2r
| B⃗ | = μ 0 N I L
Φv =|⃗v|Acos(θ)
ΦB =|B⃗|Acos(θ)
E = N|∆Φ| ∆t
I = V/R
I = E/R (linear DC generator)
E = BvL (linear DC generator)
I = E/R = BvL/R (linear DC generator)
Fm = ILB = (B2vL2)/R (linear DC generator)
Pmechanical = Fappl v Pelectrical = IV = IE
ΦB = BA cos(ωt) (rotary AC generator)
E = (NBAω)sin(ωt) (rotary AC generator)
I = E/R = (NBAω/R) sin(ωt) 1Tesla(1T)=1 N
μ0 =4π×10−7 Tm A

The Attempt at a Solution

E=6.00 x 10^6m/s x 1.50 x 10^-2 T
90000 T x m/s

Ok the problem I am having is relating distance to any equation I have. In previous problems I have worked, none of them had distance as part of the problem. If someone could just give me a hint or something about how to relate distance to the problem, that would be much appreciated.

 

[frogjg2003]

Magnetic fields make moving charged particles move in circles. If the radius of the circle is too small, the electrons will remain in the region.
The relevant equations are F=qvB and a=v²/r, and of course, F=ma

 

[kkcolwell]

I attached a picture to go along with my problem, did you happen to look at it? I am not sure if I attached it right.

 

[frogjg2003]

The picture clarified it a bit, but nothing has changed.
The electrons will follow a circle with radius r. If r is smaller than d, then the electrons will turn around and come back the way they came.

 

[asteriatic]

As a scientist, it is important to always start by carefully reading and understanding the problem. In this case, we have a beam of electrons moving through a rectangular region with a magnetic field. The goal is to find the maximum distance, d, that the beam can travel through the region and still continue on to the right.

To solve this problem, we need to use the given equations and information to determine the forces acting on the electrons and how they relate to the distance d. Let's start by looking at the forces acting on the electrons. We know that the electrons are moving in the +x direction, and there is a magnetic field in the -z direction. This means that there will be a force, Fm, acting on the electrons in the +y direction, according to the equation |F⃗m| = |q||⃗v||B⃗ || sin(θ)|.

We also know that the beam of electrons has a certain momentum, p, which is equal to the mass of an electron, me, multiplied by its velocity, v. This momentum will be conserved as the electrons travel through the region, so we can use the equation p = m⃗v to relate the momentum to the velocity of the electrons.

Next, we need to consider the distance, d, and how it relates to the magnetic field and the force acting on the electrons. We can use the equation τ = NIA|B⃗||sinθ| to relate the torque, τ, to the magnetic field, B, and the distance, d. This torque will affect the motion of the electrons and contribute to the overall force acting on them.

Using these equations and information, we can set up an equation that relates the force acting on the electrons to the distance d. Once we have this equation, we can solve for the maximum value of d that will allow the electrons to pass through the region and continue on to the right. It may also be helpful to draw a diagram and use trigonometric functions to help visualize the problem.

In conclusion, in order to determine the maximum distance that the electron beam can pass through the rectangular region, we need to carefully consider the forces acting on the electrons and how they relate to the distance. Using the given equations and information, we can set up an equation and solve for the maximum value of d. It is important to always think critically and carefully about the problem at hand and use the