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# Magnetic force on a proton example (part 2)

## Video transcript

in the last video we figured out that if we had a proton coming in to the right at a velocity of six times ten to the seventh meters per second so the magnitude of the velocity is one-fifth the speed of light and if it were to cross this magnetic field we used this formula to figure out that the magnitude of the force on this protein proton not protein would be four point eight times 10 to the negative 12 Newtons and then the direction we used our right-hand rule because it was a cross product and we figured out that it would be perpendicular well has to be perpendicular both because we're taking the cross product and right what it enters the net force will be downwards but then as think about what happens if you have a downward force right there then the product will be deflected downward a little bit so it's velocity vector will then look something like that but it's still in the magnetic field right and not only is it still in the magnetic field but since the particle is still moving within the plane of your video screen it's still completely perpendicular to the magnetic field and so the Chi the magnitude of the force on the moving particle won't change just the direction wheel right because the if we do the right hand rule here but if we just move our hand down a little bit if we tilt it down then our thumb is going to be pointing in this direction and that just keeps happening it's deflected that way a little bit so the magnitude of the velocity doesn't ever change it always stays perpendicular to the magnetic field because it's always staying in this plane but the orientation does change within the plane and because of that because the orientation velocity changes the orientation of the force changes so when the velocity is here the force is perpendicular so two acts as kind of a centripetal force and so the particle will start moving in a circle so let's see if we can break out our toolkit from what we've learned before in classical mechanics and figure out what the radius of that circle is and that might seem more daunting than it really is well what do we know about centripetal forces and radiuses of circles except let me make some space first see I probably don't need this I don't even need the first few of these field lines let's see well I'll leave everything else I'll get rid of that feline that should give us enough space I'll try to write small but let legibly so what is the formula for centripetal force and we prove it many many videos ago early in the physics playlist well centripetal acceleration is the magnitude of the velocity vector squared over the radius of the circle and since this is acceleration if we want to know the centripetal force it's just the mass times acceleration so it's the mass of the particle or the object in question times the magnitude of its velocity squared divided by the radius of the circle in this case this is the radius of the circle and that's what we're going to try to solve for and what do we know about this interpret force what is causing the centripetal force what's the magnetic field and we figure that out right this is going to be equal to this which we figured out is going to be equal to at least the magnitudes the magnitude of this is equal to the magnitude of this and that the magnitude is 4.8 times 10 to the minus twelfth Newtons and so the radius is going to be let's see if we flip both sides of this equation we get radius over mass velocity squared is equal to one over four point eight times ten to the minus twelve I could just figure out what that number is well I won't worry about it now that we can multiply both sides times this times this MV squared and we get that the radius of the circle is going to be equal to the mass of the proton times its the magnitude of its velocity squared the magnitude of its velocity squared divided by the force from the magnetic field the centripetal force four point eight times ten to the minus twelve Newtons and the radius should be in meters since everything is kind of in the standard SI units and let's see if we can figure this out get our calculator and this is where that constants function is useful again because what is the mass of a proton well that's something that I personally don't have memorized but if we go into the built-in constants on the TI 85 let's see more mass of a proton this is mass of an elected over a little bit this is mass of an electron this is mass of a proton so mass of a proton that's what we care about times the magnitude of the velocity squared what was the velocity was 6 times 10 to the seventh meters per second so times 6 times 10 to the seventh meters per second squared right because I have to take them and then all of that divided by all that divided by the magnitude of the centripetal force which is the force that's being generated by the magnetic field that's 4.8 times 10 to the negative 12 so divided by four point eight e minus 12 let's see hopefully we don't get something funky there we go and that's actually a pretty neat number one point two five meters that's actually kind of a number that we can imagine so if you have a a proton going through in going in this direction at six at one-fifth the speed of light through a what did I say was it was a point five Tesla magnetic field where the vectors are pointing out of the video we have just shown that this proton will go in a circle of radius one point two five meters which is neat because it's a number that I can actually visualize and so this this whole business of magnetic fields making charge particles go into circles this is one of the few times that I can actually say has a direct application into some things into things that you've seen namely your TV or at least the the old-school TVs and the non plasma or LCD TVs your cathode ray TVs take advantage of this where you essentially have a beam of not protons but electrons the end a magnet if you take a part of TV which I don't think you should do because you're more likely to hurt yourself because there's a vacuum in there that can implode and all that but essentially you have a magnet that deflects the this electron beam and does it really fast so it scans your entire screen of different intensities and that's what forms the image I won't go into that detail maybe one day I'll do a whole video on how to v's work so that's one application of a magnetic field causing a beam of charged particles to curve and then the other application and this is actually one where it's actually useful to make the particle go in a circle is these cyclotrons that you read about where they take these protons and they make them go in circles really really fast and then they smash them together well have you ever wondered how do they even make a proton go in a circle it's not like you can hold it and you know guide it around a circle well that's what they do they pass it through an appropriate a strength magnetic field and it curves the path of the proton so that it can keep going through the same field over and over again and and then they can use those you know then they can actually use electric fields I don't know I don't claim to have any expertise in this but then they can keep speeding it up using the same devices because it keeps passing through the same part of the through the of the of the of the collider and once it collides you've probably seen those pictures those pictures right you know that you spent billions of dollars on super colliders and you end up with these pictures and somehow these physicists are able to take these pictures and say oh this is some new particle because of the way it moved well actually what they're actually talking about is you know these are moving at relativistic speeds and since they're at relativistic speeds as they move at different velocities their mass is changing all of that but the basic the basic idea is what we just learned they move in circles they move in circles because they're going through a magnetic field but they go their radiuses are different because their charges and their velocities are going to be different and actually some will move to the you know laughs some will move to the right of that might because because they're positive or negative or you know and then the radius will be dependent on their masses anyway I don't want to confuse you but I just want to show you that we actually are touching on some physics that a physicist would actually care about now with that said what would have happened what would have happened if this wasn't a proton but if this was an electron moving at this velocity at six times the seventh meters per second through a point five Tesla magnetic field popping out of this video what would have happened well this formula would have still been saying the magnet magnitude of the force is the charge but it wouldn't be the charge of the proton would be the charge of an electron times six times 10 to the seventh meters per second times the point five Tesla's so what's the difference between two charge of a proton and a charge of electron well the charge of an electron is negative so if this was an electron then the net force would actually end up being a negative number so what does that mean well when we use the right hand rule with the proton example we said that the at least one the pro when the proton is moving in this direction that the net force would be downwards right but now all of a sudden if we reverse the charge if we say we have a negative charge is the same magnitude but it's negative because it's an electron what happens where now the force is now in this direction using the right-hand rule but it is negative so really it's going to be a positive force of the same magnitude in this direction so if we have a proton it'll go in a circle in this direction it'll go like this but if we have an electron it'll go in a circle of the other direction and I'll let me ask you a question is that circle going to be a tighter circle or a or a wider circle well the mass of an electron is a lot smaller than the mass of a proton and we have the radius is equal to the mass times the velocity squared divided by the centripetal force so this this mass is smaller the radius is going to be smaller so the electrons path would actually it would move up and it would be a smaller radius actually proportional to the difference the difference is in the radius is R is the the difference in their masses actually but that would be the path of the electron anyway I thought you'd be interested in that as well I have run out of time I will see you in the next video