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# Magnetic force on a current carrying wire

## Video transcript

let's explore the repercussions of this equation some more so let's say that what was the equation it was that the force of a magnetic field on a moving charge particle is equal to the charge that's not what I wanted to do is equal to the charge of the particle and that's just a scalar quantity times the velocity that the cross product of the velocity of the particle with the magnetic field now isn't the velocity vector just the same thing as the distance vector divided by time so the velocity vector is equal to let's call the distance that the electron travels L distance divided by time so we could rewrite that the force vector is equal to the charge times and I'm doing this on purpose 1 over time write times the distance vector taken that you take the cross product with the magnetic field all I did is I rewrote velocity as per time times distance or distance per time and this is a scalar quantity at least for our purposes time only has a magnitude maybe we could call it change in time but doesn't have a direction we're not going at an angle in time so we could take the scalar quantity out it doesn't affect this vector cross product so what we get left with is force is equal to charge per time times and this is just a regular time because this is just a number it's not a vector times the cross product of the distance vector and the magnetic field and what is what is charged per time that coulombs per second well that's just current so we get that force is equal to current times the distance that the current is flowing along the distance that the current is flowing along taken and you take the cross product of that with the magnetic field and sometimes this is written as a capital L because it's a vector and all that but we started the lowest case L so we'll stay with the lower lower case L so let's see if we can apply this formula which is really the same thing as this we just took the divided the division by time and took it out of velocity so we get distance and we took it and we divided the coulombs or we took the charge divided by that so we took charge divided by time or charge per unit of time you get current so this is really just a other way of writing this it's not even a new formula you could almost prove it to yourself if you ever forget it but let's see if we can use this to figure out the effect that a magnetic field has on a current carrying wire so let me actually I probably want to put this up at the top just so that I have space to draw a current carrying wire so let me rewrite it in green so you're familiar with the formula in all colors so now our new our new derivation is that the force of a magnetic field on a current carrying wire is equal to the current in the wire and that's just a scalar quantity although it could be positive or negative depending on the direction or well current is always a positive number but if this current is going in the opposite direction as our distance vector then it might be negative but actually I wouldn't worry about that for now let's just assume this is a current in the direction of the distance vector so it's a scalar quantity current times our distance vector L or maybe the length of the conductor and you take the cross product of L with the magnetic field vector so let's see if we can apply that let's say that we have a wire actually let me let let's do the magnetic field first because that's central let's say the magnetic field I've been doing a lot of magnetic fields that pop out of the screen let's do a magnetic field that goes into the screen and those are even easier to draw they're just X's right now why is it an X because you're looking at the rear end of an arrow that's why it's an X and that's why a circle with a dot means a field or a vector coming out of the window because if an arrow was shot at you all you would see is the tip of the arrow it would maybe a little circle around it but anyway this shows us a vector going into the screen so this is our magnetic field that is B I don't know let's assign some value let's say that the magnitude of B is equal to one Tesla one Tesla and let's say I have a wire going through that magnetic field let's say the wire the wire is going along or it's in the plane of your computer monitor so this is let me so let me just draw a wire going through the magnetic field and my question to you oh I let me let me tell you a little bit of information about this wire let's say the wire is carrying a current so I is going in that direction and it is carrying a current of I don't know I'm just making up numbers five amperes or five coulombs per second now my question to you is what is the net force of this magnetic field on a section of this wire and let's make this section of the wire I don't know let me sit let's say it's a two meter section of wire it's a two meter section of wire so what so obviously the more wire you have the more charged moving particles you'll have so the larger section you have the more of a every force you'll have on that longer piece of wire so we have to kind of pick our length so we want to know what is the force of Magnus section of wire from here to here so let's just go to this formula the force is equal to the current well that's five amperes and remember just from we learned about electricity the current is the direction that notional positive charges would travel and if that suits us fine because when we did the the first equation we cared about the direction of positive charge would go in and if it was an electron or a negative charge we would we put a negative sign there so that works fine but if you ever have to visualize things as they maybe are in reality but when you talk about electrons it's hard to say that they really are reality because they're almost more an idea than an object but it's always good to remember that when the current is flowing in this direction that would be true because if they were positive charges moving but we know it's electric it's negative charges moving in the opposite direction or you could think of it as maybe you know holes well I don't want to get into that but anyway the current you could view you can visualize it if you like as positive charges going in this direction so the the current is going this direction so you could view this distance vector that we care about it's magnitude is 2 meters right because that's the length of wire in question and this direction is the direction of the current so let's select me this is L sometimes I get a little carried away on tangents so that is L it's 2 meters in that direction I is 5 amperes and we just we already figured out that the magnetic field is 1 Tesla so what's what's this going to be equal to so the force is going to be equal to where we're using all SI units so we don't have to convert anything 5 amperes times 2 meters in that direction I won't specify right now we can say that's a magnitude of L actually let's just let me write it well 2 meters time's the magnetic field one Tesla and so when you take a cross product of something this is just a reminder L cross B that's equal to the magnitude of L times the magnitude of B times the sine of the angle between them times some unit directional vector that we figure out what the right-hand rule so we already did the magnitude of the distance vector that was 2 meters we did the magnitude of the magnetic field and what's the sine of the angle between them well if the magnetic field is going into the screen right it's going straight into the screen you could imagine a bunch of arrows just shooting into the screen those are the vectors while our our distance vector or our this L is in the screen they actually are perpendicular in three dimensions so this angle is 90 degrees so this actually just becomes one so in terms of the magnitude we're done the L cross B magnitude is two times one Tesla and then we multiply that times the current and then we actually have the magnitude of the force the magnitude of this force is going to going to be equal to 5 amperes times 2 meters times 1 Tesla which is equal to 10 Newtons and then the only question left is what is the direction of the force that the magnetic field is exerting and this is where we break out the right hand rule and it's no different you can you know you can just imagine one of the one of the positive particles moving in that direction and just use the right hand rule so let's take our hand out and if we let me draw a hand a right hand so this is my right hand if I have my thumb sticking out like that so the L is going to be my index finger the first thing in the cross-product and then the B is the magnetic field let go that's going into the screen so you can't see it all you can take my word for it is that my middle finger is pointed downwards into the screen and then my other fingers are just doing something else and there you have it the your thumb is actually the direction of the force right that your index finger is the direction of you could either say well we'll say L for these purposes and then the magnetic field is going into it so you can't see my middle finger but it's pointing downwards I could draw a little X there to show it's going downwards and then the force is what my thumb is doing so the force on this wire or at least on that section of wire is going to be perpendicular to the direction of the current in that direction is going to be a 10 Newton force anyway I've run out of time