A proton is projected into a magnetic field that is directed along the positive x axis. Find the

A proton is projected into a magnetic field that is directed...

Key Concepts

Effect of Velocity Orientation Relative to the Magnetic Field

When the velocity vector is perpendicular to the magnetic field, the cross product is maximized, resulting in the largest force. Conversely, when the velocity is parallel to the magnetic field, the cross product becomes zero, implying that no magnetic force is exerted on the particle. This concept is critical in understanding the behavior of particles in uniform magnetic fields.

Right-Hand Rule

This is a method used to determine the direction of the magnetic force on a positive charge. By pointing the fingers of the right hand in the direction of the velocity and curling them towards the magnetic field, the thumb indicates the direction of the force. This mnemonic is essential for solving problems involving vector cross products in electromagnetism.

Magnetic Force on a Charged Particle

This principle explains that a charged particle moving through a magnetic field experiences a force described by the Lorentz force equation, F = q(v × B), where q is the charge, v is the velocity of the particle, and B is the magnetic field. The magnitude of the force depends on the charge, the speed of the particle, the strength of the magnetic field, and the sine of the angle between the velocity and the field vectors.

Cross Product in Vector Calculus

The magnetic force is determined by the vector cross product of the velocity and the magnetic field. The cross product produces a vector that is perpendicular to both the original vectors, thereby setting the direction of the force. Its magnitude is given by |v| |B| sin(?), where ? is the angle between the velocity and the magnetic field.

Transcript

0:00 All righty.

00:01 So we have a situation where we got a magnetic field that's pointing along the positive x direction.

00:08 And they're going to give us different situations where a particle positive recharge proton in particular is going to be moving.

00:16 It's going to have motion in certain directions with respect to this magnetic field.

00:20 Our job is determine the direction of that magnetic field force that's going to be associated with the direction of the particle.

00:27 So let's jump right into this.

00:29 What i'm going to do first, i'm going to sketch over here the axis system.

00:34 So we can have a clear idea of where everything is pointing, et cetera, during this problem.

00:41 All righty.

00:46 So this is my positive x direction here.

00:50 Positive x.

00:52 And up here is my positive y direction.

00:55 You guys don't see it, but my positive z direction comes out toward us.

00:59 And my minus c direction goes into the page.

01:02 Okay.

01:04 And i'm going to now write or just sketch some of these vectors for the magnetic field so you get an idea of the direction.

01:14 It's perpendicular or excuse me, it's parallel to the x -axis.

01:18 I just drew a couple of the magnetic field lines and there it is in red there.

01:25 And the first problem, first situation we have is we got to write it right here part a.

01:31 We have we have the motion of the particle in the plus y direction so i'll write it like this write it in blue plus y plus y direction okay so when we can sketch that particle and just draw the vector of the velocity right here and there's the plus particle there's my little circle indicating the proton and little plus sign in there so you guys can see it's a lovely positively charged particle that's moving in the positive wide direction oh boy oh boy sometimes it is tough to write this stuff out there we go plus okay now take a look at something important here this vector right here this velocity vector is perpendicular to the magnetic field direction okay and we have to recall one thing real quick and that's the main equation we have to use what is that equation for the magnetic magnetic field force f sub b, force of the magnetic field, and that's equal to the charged particle.

02:47 So whether that's the charge on the particle essentially, so if it's plus, minus, you know, whatever the charge is.

02:54 And, you know, exactly what is like the charge of an electron has a specific value.

02:59 So you use that there if this was an electron, whatever.

03:01 Anyway, but we don't have to give exact values, so don't stress about, too much about the q.

03:04 But that will determine the direction of this vector.

03:07 I'll get to that in a second.

03:10 And b.

03:11 Cool.

03:12 So this is the formula where we will be using it.

03:14 It's essentially just part of the lorenz formula.

03:17 But we have no electric field here.

03:23 And because of that, it only includes the portion about the magnetic field.

03:29 All right.

03:31 And i was saying earlier about this part right here, about this q part.

03:36 So what i mean is if this was a negative particle, for example, you may have a velocity vector pointing in this direction.

03:42 But that being, negative will change the direction of that vector so it'll actually be pointing downward, which will affect the whole problem.

03:48 But we don't have to worry about that here because it's positive.

03:51 So that makes our life a little bit easier.

03:53 Anyway, let's continue on with our problem at hand here.

03:55 So let's use that right -hand rule and it's discussed in the textbook, but just to give you guys a general idea of how it works.

04:02 First thing you do is just stick your right hand up.

04:05 And then you want to have your fingers pointing toward the direction of this velocity vector.

04:13 Right here.

04:14 So you want to make sure your fingers are all pointing upward.

04:17 So essentially, i mean to unbox the equation here, but really poor quality picture of someone's hand here, whatever.

04:28 There's their fingers...

Please give Ace some feedback