In Fig. 28-58, an electron of mass m, charge -e, and low (ncgligible) speed enters the region be

In Fig. 28-58, an electron of mass m, charge -e, and low...

Key Concepts

Lorentz Force

This concept refers to the net force experienced by a charged particle moving in electric and magnetic fields. It combines the electric force (proportional to the electric field, E) and the magnetic force (proportional to the particle’s velocity, v, and the magnetic field, B) according to the equation F = q(E + v × B). This law is fundamental for determining the trajectory of a charge in environments where both types of fields are present.

Electric Field Derived from Potential Difference

An electric field is created in a region with a potential difference between two points or surfaces. In configurations with parallel plates, the electric field is uniform and its magnitude is given by E = V/d, where V is the potential difference and d is the separation between the plates. This field influences charged particles by exerting a force that accelerates them, a concept that is critical for analyzing their motion in such environments.

Energy Conservation in Electric Fields

When a charged particle moves through an electric field, it undergoes a conversion between potential energy and kinetic energy. For an electron accelerated by a potential difference, the gain in kinetic energy is equal to the loss of electrical potential energy, typically expressed as (1/2)mv² = eV. This relationship is essential for determining the particle’s speed as it enters regions with additional forces, such as magnetic fields.

Circular Motion in a Uniform Magnetic Field

A charged particle moving with a velocity perpendicular to a uniform magnetic field undergoes circular motion due to the magnetic force acting as a centripetal force. The radius of this circular path can be found using the relation r = mv/(|q|B). This concept is key when analyzing how magnetic fields can be used to alter the trajectory of a particle, particularly when balancing or overcoming other forces such as those produced by an electric field.

Transcript

00:01 Okay, folks, so in this video we're going to be talking about this problem.

00:06 An electron of mass charge, i mean, an electron of mass m charge negative e, and low speed enters the region between two plates of potential difference v and place separation d, initially headed directly toward the top plate.

00:22 A uniform magnetic field of magnitude b is normal to the plane of the figure.

00:29 We're going to be looking for the minimum value of b such that the electron will not strike the top plate.

00:41 Before we start, i just want to point out that this problem is going to be relatively speaking more conceptually difficult than some of the other problems that we have done.

00:57 So i would like for you to bear with me for this problem because it's going to take us a little bit of time to do it.

01:05 Okay, but anyway, in order for you to solve this problem, you need to have a very kind of like intuitive understanding, you know, of this problem.

01:19 You can't just go ahead and just start solving it right away.

01:23 You have to have an intuitive feeling for what's going on in this problem.

01:30 So what's going on in this problem is that you have two, let's say, capacitor plates.

01:41 You know, there's a potential difference between these two plates, and there is a magnetic field that is uniform, and it is pointing straight into the page.

01:51 And then on top of that, you have a particle that is negatively charged entering this region.

02:00 So as you can probably imagine in your head, as soon as this particle enters this region from here, if there is no electric field between these two plates, right? if there's no electric field between these two plates, if there's no potential difference, it's just vacuum with a surrounding b field.

02:24 Then that particle is going to be swirling around.

02:29 It's going to do this, right? it's going to be moving in a circular motion.

02:33 I think you all are pretty familiar with that.

02:38 But because of the fact that we have not only a b field, we also have a potential difference between these two plates.

02:46 That means there is going to be an electric field that wants to push this particle from one plate to the other.

02:58 There's going to be an electric field between these two plates.

03:02 And as we know from previous, you know, derivations or previous results or formulas that you might have remembered from your textbook, that electric field between two parallel plates with a potential difference fee, that field is constant everywhere.

03:19 But anyway, so we have a very, you know, not very, but pretty complicated situation here.

03:29 We have, on the one hand, we have a b field that is constantly trying to bend the trajectory of this particle.

03:39 And on the other hand, we have an e field that wants to push this particle from one plate to the other.

03:48 So this is a very, you know, interesting and hard to understand the situation, but as soon as you understand it intuitively, conceptually.

04:01 Then it really becomes not that hard to solve.

04:05 But anyway, imagine, here's a good way to think about it.

04:12 Imagine you have a machine or something that controls, you know, the magnitude of the b field.

04:20 You have a machine that you can, you know, tune up or down in order to make the magnitude of the b field stronger or less strong.

04:32 And imagine you tune it up really, really high.

04:35 Imagine you make the b field really, really high.

04:38 Then what happens when this particle enters this region? what happens if the b field is really high? right.

04:46 Well, if it's really high, then as soon as the particle enters this region, it's just going to, no, it's not even going to bother moving forward.

04:55 It's just going to bend bend back its trajectory is going to be so curved that that this particle is not even going to move forward at all it's just going to do something like this right because the b field is so strong that it literally bends the particle out of its trajectory completely so the particle is going to do something like this now imagine you make the machine less strong right you make the b field less strong you make it weaker what happens then? well, it's going to, you know, the particle is going to move forward a little bit, bend, but still bends a little bit, and still moves back, right? and if we keep doing this, if we keep making the b field weaker and weaker, the trajectories for each of the possible scenarios, it's just going to look less and less and less bent, you know? so eventually, if you make the b field weak, so weak to the point where the particle can literally move to the other side of the plate, i mean to the other plate, and then it achieves kind of an equilibrium, and by that i mean, the y component of the e field and the y component of the b field, cancel out.

06:26 Then that is the situation that we want.

06:29 We want this particle to be in a trajectory where it can go from one.

06:36 Plate to the other plate completely.

06:38 You can it can travel through the entire distance of d.

06:43 You know, the distance between the two plates is d.

06:46 We want this particle to be able to travel through the entire path of d and then achieves kind of an equilibrium, meaning it does not have a wide component of velocity anymore.

06:59 That is where the b field is at a minimum.

07:03 That is where the b field has been turned down for so much that that is not strong enough to bend the particle away from its original trajectory anymore.

07:15 Well, that's not entirely correct, but you can't, but you know what i mean.

07:19 We want the b -field to be so weak, but not too weak to the point where the particle hits the top plate.

07:26 We want the b -field to be weak, but only to a certain degree.

07:31 And, you know, i encourage you to think about this a little bit in your head.

07:36 But basically, the situation that i have drawn here is the situation that i'm trying to describe to you.

07:43 As you make the b field weaker and weaker and weaker, the particle is going to travel more and more and more percentage of the distance d between the two plates.

07:57 But anyway, assuming you have understood what i wanted to say, assuming you have understood that part let's let's start solving this problem the idea here is we are assuming that b field is so weak that the particle that by the time the particle has has zero y component of velocity that particle is already at the other end so so originally this particle has is at potential v meaning it is energy is qv.

08:40 And then by the time, by the way, i just want to remind you that the magnetic force does not do any work on this particle.

08:46 So all of the energy on this particle is being manipulated by the electric field.

08:52 So all of the energy that this particle has originally when it's here, it's this qv.

08:59 And by the time it gets to the other end, its energy, it doesn't have any potential energy anymore.

09:06 And all it has is this...

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