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Path of charged particle in magnetic field

Let's explore how to calculate the path of the charged particle in a uniform magnetic field. Created by Mahesh Shenoy.

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  • stelly blue style avatar for user Gokul Selvaraj S
    At ,Mahesh tries to explain why the speed of the charged particle remains same. I don't understand why. Can someone please explain? Thanks.
    (2 votes)
    Default Khan Academy avatar avatar for user
    • starky tree style avatar for user mclellanrenae
      well its because at every single moment in time the magnetic force on the particle due to the magnetic field will always always always be acting perpendicular to the particle's velocity vector. Therefore, the force vector you can think of as the x axis and the velocity vector as the y axis. You know that the x axis is completely independent of the y axis because they are perpendicular so the force and velocity vector do not have a mixture of both horizontal or vertical components because if they did then the force's vertical component could either increase or decrease the particle's velocity (and hence speed). In reality though, the force and velocity vectors are the horizontal and vertical components themselves.
      (1 vote)
  • blobby green style avatar for user Roza
    Hi,
    my question relates to Practice ex #1, some explanation for this ex is provided in hints, though I struggle to undrstand it:

    The only force on the electron is a centripetal force,
    , that points toward the
    \[z\]-axis.
    Let's split the velocity vector,
    \[\bold v\], into components parallel (
    \[\bold v_\parallel\]) and perpendicular (
    \[\bold v_\perp\]) to the magnetic field,
    \[\bold B\]. We conclude:
    there's no force parallel to
    \[\bold B\], so
    \[\bold v_\parallel\] points in the
    \[+z\] direction, since this is the only component of
    \[\bold v\] that's constant;
    \[\bold v_\perp\] is tangential to the path of the electron, causing the circular part of the helical motion.

    Could you please kindly help to understand - how could we we split velocity vector into components parallel and perpendicular to magnetic field, if we do not know the direction of magnetic field in the first place ..?
    (1 vote)
    Default Khan Academy avatar avatar for user

Video transcript

let's explore how to calculate the path taken by a charged particle in a magnetic field and then we'll be able to answer why these beautiful auroras are only seen by people at the poles that's right at times these people can see spectacular light shows in the sky but only happens at the poles we'll be able to answer that question later all right so let's start with a case where you have magnetic field which is into the board nice and uniform into the board and let's consider a charged particle let's say a proton that's moving perpendicular to the field that's what we'll start so let me just write that down let's consider a case where particles are moving let me just let me just draw it over here so let's say we have a proton that is moving perpendicular to the field so the field is inwards let's consider the proton going upwards so the case we're dealing with is velocity of the charged particle is perpendicular to the magnetic field i want to now know in this case what's going to happen to the proton how will it move will it keep moving straight or will it turn what's going to happen what is the path that it'll be taking how do we calculate that well to calculate the path of any object we'll have to figure out the force acting on it and we've seen before that the force acting on any charged particle in a magnetic field is given by the lorenz equation the lorenz force is f equals q the charge of the part the charge of the particle times v cross b and if you're not familiar with this we've talked about this in our previous videos on lawrence force feel free to go back and check that out so let's figure out the direction of the force acting on this using the cross product and then from there let's see what will be the part taken so why don't you pause the video and see if you can figure out the direction of the force remember to calculate cross product you use your right hand rule so why don't you try using a right hand rule and see the direction of the force all right if you're giving it a shot let's see i'm going to bring in my right hand i orient my right hand to make sure that my four fingers the four fingers are in the direction of the velocity because velocity is my first vector and then i curve them in the direction of the magnetic field now the magnetic field is inwards so i've curve my four fingers also inwards see and so the thumb now represents the direction of the force so my force acting on the charged particle will be this way this is the direction of the magnetic force so what's going to happen to this particle well because it's pulled inwards that particle will sort of turn it's now new velocity after some time is going to be this way it's important to notice that the speed will be unchanged why because the velocity because the force is perpendicular remember vectors in vectors don't have a component in the perpendicular direction so this force will and be unable to change the speed so the speed remains the same okay what's going to happen now what direction will the force act now will the direction of the force be the same well let's see again if we do a v cross b let me uncurl the fingers okay now i'll have to align my finger again in the direction of the velocity and now again i will curve them this way and notice now my thumb is pointing at an angle so the force now is going to be pointing this way interesting so again it's going to turn a little bit inwards and this will continue now as a result you can kind of see that the force is always going to be pointing at this at a some common point which is going to be the center and as a result you can kind of predict now the the charge particle is going to go in a circular path so let me draw that circle for you so charge particle is going to go like this in a circular path this lorentz force is going to act as the centripetal force of the charged particle and so what we now find is when the velocity of the charged particle is perpendicular to the magnetic field we find that particle travels in a circle goes in a circle and it's going to go this way in this example so that's how you figure out the direction let me give you one let me give you an example what if we have magnetic field to the right like this is our magnetic field and let's say we have a proton that is moving downwards again notice in the same case we have velocity perpendicular to the magnetic field but now i want you to use your right hand rule think about which direction the force is is going to be and think about how that circular path is going to look like this is going to be in three dimensions so you'll have to visualize a little bit so why don't you pause the video and visualize what will be the part taken by the particle all right hopefully you've tried let me bring in my beautiful right hand and i'm going to orient my right hand this way again notice my four fingers must be pointing in the direction of the velocity and i'm pointing in such a way that i'm going to curve my four fingers in the direction of the magnetic field magnetic field is like this so now i'm going to curve like this can you see that here we go curve this way and as a result notice the thumb will point out of the screen and so from what we learned over here this means my this this particle is also going to go in a circle but the center of that circle is going to be somewhere out of the screen so can you visualize what the circle is going to look like let me get rid of this hand so this particle is going to go somewhat like this let me draw it's going to go this way it's going to come out of the screen let's go like this and again it's going to go into the screen it's going to go in a circle like this it doesn't look like a circle but that's what it is so here's will be the center of that circle so again circular path particle will go in a circular path so this is how you figure out the path taken but now let's consider a more general case okay let me draw over here what if i have a magnetic field like this and let's consider that same proton a charged particle proton whatever whatever that is but it's not perpendicular to the magnetic field let's say it's moving at an angle um interesting what's gonna happen now how do we figure this out well the way i like to think about it is i already know that if the particle was going perpendicular i know it's going to go in a circular path so what i'm going to do is divide this velocity into two components one that is perpendicular to the field and one that is parallel to the field and look at what what will be the path for each of the component so we already know for the one that is perpendicular is going to be a circular path let's now figure out what happens to the path when the charged particle is moving parallel to the magnetic field and again we're going to do the same thing we're going to first figure out what the direction of the force is and then from that we're going to figure out the path taken and again why don't you pause the video use the lawrence force and think about this what would be the force acting on a particle if it was going parallel to the magnetic field pause and try all right let's see because we're doing a cross product remember that cross product becomes zero when vectors are parallel or anti-parallel to each other so because this is going parallel to the magnetic field it experiences no force so this component will just go in the straight line with the same speed unaffected this component will try to make the particle go in a circle so what's going to be the net effect well one is trying to make it go in a circle another one is trying to make it go forward like this in the opposite direction and so together you're going to have a circle that's going forward let me try and draw that so together we're going to have a circle let's use blue okay a circle that is going forward something like this in other words this is going to be a helical motion so in general what we find in general what we find is particles tend to go in a helical path and look at that helical path the axis of the helical path is the magnetic field can you see that and the circle the plane of that circle is going to be perpendicular to the magnetic field just like what we found over here all right let's do one last example what if we have magnetic field this way and imagine there's now an electron an electron is going in some random direction let's say electron is going this way can you pause and figure out what would be the path taken by this electron all right let's see we're going to do the same technique i'm going to decompose this velocity into two parts one perpendicular to the magnetic field one parallel to the magnetic field i know this parallel component tries to keep this electron going in the same direction let's see what the perpendicular component tries to do well velocity is upwards magnetic field is towards the right and so if you use your right hand rule i hope you can do that yourself now i don't have the picture of that available but if you use the right hand rule the force with the thumb would point inwards okay but remember this is an electron and so if any if you have an electron there'll be a negative charge over here and you have to flip the force that means the force will act out of the screen and so the so the center of that circle is also going to be out of the screen and as a result the way i visualize this the circle is going to be like this the electron will go come out of the screen go into the screen come out of the screen but at the same time it's going to go forward and so the net is going to be the net motion will be a helical path that will look like this this is not a great helix but you get the point this is how it's going to go all right now let's see why we only get at the poles it's got something to do with the magnetic field of the earth so turns out that sometimes your arson everybody's sun it this it launches charged particles and these charge particles once they enter into the magnetic field they get trapped because once they enter the magnetic field they start following the field in a helical path like this so they will start following the field like this in a helical path and eventually they will follow all the way towards the poles and notice that's where they enter into the atmosphere the same thing could happen on the other end as well so if the charge particle gets trapped over here it'll follow the field and enter into the atmosphere only at the poles because that's where the magnetic field starts and ends and when these charged particles enter the atmosphere they interact with the atmosphere causing these beautiful auroras and that's why we only see them at the poles beautiful isn't it