Electron moving through various fields

[theowne]

Homework Statement

An electron initially at rest is accelerated through a potential difference of

V and directed into a region with 1) two parallel plates separated by 20 mm with
a potential difference of 20 V between them and 2) a uniform 0.5 T magnetic field
which is parallel to the plates. (a) Calculate the minimum
V required such that the
electron feels no force as it moves through the region between the plates. (b) Why is
this just a minimum?

Homework Equations

Fm= qvb
Fe=qE

The Attempt at a Solution

My initial thought was that the magnetic force and electric force would oppose each other and we could solve qE = qvB for v and thus the delta V required to accelerate to V. However then it asks why is this a minimum. But I don't think it could work if it was just a minimum because increasing v would increase and give an excess magnetic force, so the electron would feel a force, leading me to think this reasoning is wrong.

Could I get some advice for this? For example, it says the electron is directed into the region, does that means its moving parallel to B?

 

[berkeman]

Homework Statement

An electron initially at rest is accelerated through a potential difference of
V and directed into a region with 1) two parallel plates separated by 20 mm with
a potential difference of 20 V between them and 2) a uniform 0.5 T magnetic field
which is parallel to the plates. (a) Calculate the minimum V required such that the
electron feels no force as it moves through the region between the plates. (b) Why is
this just a minimum?

Homework Equations

Fm= qvb
Fe=qE

The Attempt at a Solution

My initial thought was that the magnetic force and electric force would oppose each other and we could solve qE = qvB for v and thus the delta V required to accelerate to V. However then it asks why is this a minimum. But I don't think it could work if it was just a minimum because increasing v would increase and give an excess magnetic force, so the electron would feel a force, leading me to think this reasoning is wrong.

Could I get some advice for this? For example, it says the electron is directed into the region, does that means its moving parallel to B?

Weird. I'm not seeing the trick either. Does seem like you just want a zero net Lorentz force. Will think about it more...


EDIT -- is that exactly how the problem is worded?

 

[buffordboy23]

My initial thought was that the magnetic force and electric force would oppose each other and we could solve qE = qvB for v and thus the delta V required to accelerate to V. However then it asks why is this a minimum. But I don't think it could work if it was just a minimum because increasing v would increase and give an excess magnetic force, so the electron would feel a force, leading me to think this reasoning is wrong.

Could I get some advice for this? For example, it says the electron is directed into the region, does that means its moving parallel to B?

First, about the "minimum". This doesn't make sense at all. There is only one potential difference V that corresponds to the scenario where the net force on the electron is zero. You already said why, the magnetic force is equal in magnitude and opposite in direction relative to the electrostatic force by the conducting plates. Specifically, you provided the correct equation qE = qvB, and according to the given data these values are all constant, which implies that only one value for the velocity will lead to equal forces. Where did this question come from?

No, the electron cannot be moving parallel to B. The magnetic force equation is actually a cross-product, and therefore, if the velocity vector and the magnetic field vector are parallel, then the magnetic force is equal to zero. The problem statement does not appear to give you a coordinate axes, nor any indication of which plate is at a higher potential. Therefore, the axes that describe the scenario is arbitrary along the with the direction of the electric field vector between the plates. However, you can determine the relative orientation of the magnetic field vector to the electric field vector.

 

[theowne]

You can find it on #5 http://www.hep.yorku.ca/menary/courses/phys2020/2009/p5.pdf" , if that helps. I think that I copied it exactly. No diagram is given.

 

[buffordboy23]

I would suggest drawing the scenario using some chosen coordinate system.

For example, if you orient the parallel conducting plates to be parallel to the xy-plane and let the top-most plate be at the higher potential, then the electric field vector points in the negative z-direction. Since the particle is an electron, then the electrostatic force acts in the positive z-direction. For there then to be no net force on the electron, the magnetic force must be oriented in the negative z-direction and have magnitude equal to the electrostatic force. Based on these considerations, the electron must then move in a straight line trajectory that is parallel to the xy-plane. If it had any other orientation, say, at some small angle theta with respect to the xy-plane, then the magnetic force would have two components (even three perhaps), one of which we could still find suitable conditions so that we can cancel the electrostatic force, but the other force component would still exist, meaning that the net force is not zero. The velocity vector and the magnetic field vector must form their own plane that is parallel to the xy-plane for the net force to be equal to zero.

 

[theowne]

Yes, but what about the minimum delta V part? If the V which it is accelerated through were to increase and q remains the same, doesn't that mean the resultant velocity would be greater? And if the velocity is greater, doesn't that mean the magnetic force would be greater? That's what i don't understand, when it asks "why is this a minimum". I don't understand how it could be a minimum. If magnetic force is to cancel out the electric force, I believe that this should only be happening for one velocity, right?

Or am I making a mistake here?

 

[buffordboy23]

No, I completely agree with you and I touched on the reasoning in a previous post. There is only one value of the velocity v that can result in a cancellation of forces. I would ask your professor if this is supposed to be a trick question and explain your logic for why this is so, b/c it certainly seems like one.

 

[theowne]

Ah, I see. That was my main hangup. I considered if it was a trick question, but the way it's worded seems to just take it as a fact.

Thanks for the help.

 

[theowne]

How about thinking about it this way?

The plates and magnetic field are parallel to each other as the question says, and the velocity vector is in the same plane as the magnetic field. So they interact on say, the xy plane, and the electric field from the plates is always in the z direction, and the magnetic force produced by the B field and velocity is always in the -z direction. The minimum part of it comes from when the angle between the B field and the velocity in the 2D xy plane is 90. At this point a specific velocity is needed to produce a magnetic force opposing the electric force in the z component. But if the angle is less than 90, and B=qvBsin(theta), the same magnetic force can be achieved by increasing v. But it always results in a single magnetic force only in the -z direction.

Does that make any sense or completely wrong?

 

[buffordboy23]

I see what your saying and this makes sense now. Then this is the equation for zero net force:

qE = qvBsin(theta)

There is an infinite set of pairs (theta, velocity v) that would satisfy the equation. I was thinking for some reason that the velocity vector and the magnetic field were at right angles to each other, but really what matters is that they are in the same plane.

 

[berkeman]

Ohh! The angle! Brilliant. Thanks.