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# Magnetic field created by a current carrying wire

## Video transcript

so not only can a magnetic field exert some force on a moving charge we're now going to learn that a moving charge or current can actually create a magnetic field so there is some type of symmetry here and as we'll learn later when we learn our our calculus and we do this in a slightly more rigorous way we'll see that electro that magnetic fields and electric fields are actually two sides of the same coin of really electromagnetic fields but anyway we won't worry about that now and I think it's it's enough to ponder right now that a current can actually induce a magnetic field and actually just a moving electron creates a magnetic field and it does it in a kind of a surface of a sphere I won't go into all of that right now because that gets a math gets a little bit fancy there but what you might encounter in your your standard high school physics class that's not getting deeply into vector calculus is that if you just have a wire let me draw a wire that's my wire and it's carrying some current I it's carrying some current I it turns out that this wire will generate a magnetic field and the shape of that magnetic field is going to be Co centric circles around this around this wire and let me see if I can draw that so here I'll draw it just like how I do when I try to do rotations of solids in the calculus video so the magnetic field it would go behind and in front and it goes like that or another way you could think about it is if let's go down here is on the left side of this wire if you say that the wire is in the plane of this video the magnetic field is popping out of your screen it's popping out of the screen and on this side on the right side the magnetic field is popping into the screen it's going into the screen right and you could imagine that right this you could imagine if on this drawing up here you could say this is where it intersects the screen all of this is kind of behind the screen and all of this is in front of the screen and this is where it's popping out and this is where it's popping into the screen hopefully that makes a little bit sense and how did I know that it's rotating this way well it actually does come out of the cross product when you do it with a regular charge and all of that but we're not going to go into that right now and so there's a different right-hand rule that you can use and it's literally you hold this wire or you imagine holding this wire with your right hand with your thumb going in the direction of the current and if you hold this wire with your thumb going into the direction of the current your fingers are going to go in the direction of the magnetic field so let me see if I can draw that I will draw it in blue so if this is my thumb and my thumb is going along the top of the wire and then my hand is curling around the wire right those are my knuckles those are the veins on my hand this is my nail so as you can see if this was if I was holding that same wire let me draw the wire so if I was holding that same wire we see that my thumb is going in the direction of the current so this is a slightly new thing to memorize and what is the magnetic field doing what's going in this in the direction of my fingers right my fingers are popping out on this side of the wire right they're coming out on this side of the wire and they're going in or at least my hand is going in on that side so it's popular that's going into the screen hopefully that makes sense now how can we quantify well before we quantify let's let's get a little bit more of the intuition of what's happening it turns out that the closer you get to the wire the stronger the magnetic field and the further you get out the weaker the magnetic field and that kind of makes sense if you kind of a imagine the magnetic field spreading out you know well yeah I don't want to go into too sophisticated knowledge but if you imagine the magnetic field spreading out as it goes further and further out it kind of gets distributed over a wider and wider circumference and actually the formula I'm going to give you kind of has a taste for that so the formula for the magnetic field and it really is defined with with a cross-product and things like that but for our purposes we won't worry about that you'll just have to no kind of that this is the shape if the current is going in that direction and of course if the current was going downwards then the magnetic field would just reverse directions there would still be in Co centric circles around the wire but anyway what is the magnet the magnitude of that field it is the magnitude of that magnetic field is equal to MU which is a Greek letter which I will explain in a second times the current divided by 2 PI R so this has a little bit of a feel for what I was saying before that the further you go out where R is how far you are from the wire right the further you go out if R gets bigger the magnitude of the magnetic field is going to get weaker and this 2 pi R that looks a lot like the circumference of a circle so that gives you a taste for it I know I haven't proved anything rigorously but it at least gives you a sense of look there's a little formula for circumference of a circle here and that kind of makes sense right because the magnetic field at that point is kind of a circle it's the the magnitude is equal at an equal radius around that wire now what is this mu this thing that looks like a u well that's the permeability of the material that the wire is in so the magnetic field is actually going to have a different strength depending on whether this wire is going through rubber whether it's going through a vacuum or air or metal or water and for the purposes of your high school physics class we assume that it's going through air normally and the value for air is pretty close to the value for a vacuum and it's called the permeability of a vacuum and I forget what that value is I could look it up but it actually turns out that your your calculator has that value so let's do a problem just to put some numbers to the formula so let's say I have this current and it is it is I don't know the the current is equal to I'm going to make up a number two amperes two amperes and let's say that I am that I just pick a point right here that is let's say that that's three meters away from the wire in question so my question to you is what is the magnitude of the direction of the magnetic field right there well the magnitude is easy we just substitute in this equation so the magnitude of the magnetic field at this point is equal to the PERT and we assume that the wire is going through air or vacuum the permeability of free space that's just a constant well that looks fancy times the current times two amperes divided by two PI R what's R it's three meters so 2 pi times 3 so it equals permeability of free space let's see the two and the two cancel out over three pi so how do we calculate that well we we get out our our trusty ti-85 calculator and i think you'll you'll be may be pleasantly surprised or shocked to realize that I delete everything just you can see how I get there that it actually has a permeability of free space stored in it so what you do is you go to second and you press constant which is the for button and say the built-in constants now let's see it's not one of those you press more it's not one of those you press more I'll look at that mu not a permeability of free space that's what I need and I have to divide it by three pi divided by three and then where is pi there it is over the power sign divided by three PI it equals one point three times ten to the negative seventh it's going to be Tesla's the magnetic field is going to be equal to one point three times ten to the minus seventh Tesla's so it's a fairly a fairly weak magnetic field and that's why you don't have metal objects being thrown around by the wires behind your your television set but anyway hopefully that gives you a little bit and you know just just so you know just how it all fits together so we were saying that a a these moving charges not only can they be affected by a magnetic field not only can a current and be affected by a magnetic field or just a moving charge it actually creates them and that that kind of creates a little bit of symmetry in your head hopefully because if that was also true of electric field charge a stationary charge is obviously pulled or pushed by a static electric field and it also creates its own static electric field so it's always in the back of your mind because if you keep studying physics you're going to actually prove to yourself that electric and magnetic fields are two sides of the same coin and it just looks like a magnetic field when you're in a different frame of reference when something is whizzing past you while if you are whizzing along with it then that thing would look static and then it would might look a little bit more like an electric field but anyway I'll leave you there now and in the next video I will show you what happens when we have two wires carrying current parallel to each other and and you might guess that they might actually attract or repel each other anyway I'll see you in the next video