Lecture 13 Magnetic Field from Current Loop / I hate my Alarm Clock

we're in chapter 18 which is about magnetic fields now last time we talked about how to calculate a cross-product mathematically and we've also been discussing how to know the direction of the cross products using the right-hand rule we discussed the difference between electron current and conventional current from now on in the course when we say current we're just going to be in conventional current okay so current means conventional current although inside of a real material the current is actually carried by negatively charged electrons right all electrons are negatively charged so the current you know the current being carried by electrons is carried by negatively charged particles conventional current is where we pretend we pretend that the current is actually carried by positively charged particles moving in the opposite direction partly that's just convention and partly it makes the math easier because there's one less minus sign to worry about so we calculated the electron current and compared it to the conventional current and of course our conventional current isn't quite true it's just that it's useful because it's the convention and it's also easier to track positive charges than negative charges we talked about the abuse of our law and a wire the biot-savart law was well paired it's the biot-savart law for a point charge we discussed first which is where we have a point charge that's moving with respect to you okay so it's got a velocity with respect to you the observer and then to find out the direction of the magnetic field that you observe due to that moving charged particle we take that V cross R where R is a vector that goes from the point charge to you our points to me right and so that told us the direction we generalize that to a little segment of current wire okay current carrying wire so this is a little segment Delta L of a current carrying wire and so the biot-savart law and that in that case became Delta B equals mu naught over 4 pi I Delta L cross R hat over R squared and so the only difference between these two is that the QV of the point charge becomes I Delta L in the case of the current carrying wire do you have any questions about that okay all right so today we will build that up then to not just the current and a little piece of the wire but we're going to integrate all along a long straight wire to find the total magnetic field that that wire produces and then we'll apply the same method to a loop of current so if I take current and run it in a circle what kind of magnetic field does that produce I've got parentheses here because having seen the guts of this calculation for the straight wire I'm not going to show you all the guts of the calculation for the current loop it's the same method so I'll show you the steps but what you should focus on there is just the result and then we'll talk about a magnetic dipole moment and what that means compare that to a bar magnet and we'll see what it is inside of materials that causes magnetism so first let's think about the magnetic field of a straight wire let's say that I have a wire carrying current and the currents going to go up in this case I have x-axis this way y-axis goes to the back and Z is straight up so let me have a current carrying wire pointing straight up due to the little segment here I can calculate the magnetic field over here I'm going to calculate the magnetic field in the bisecting plane of that wire okay does this look a little bit like something we did in electrostatics okay we did a long rod in electrostatics right so let me remind you first how that went so in the case of the the rod of charge okay this is straight out of lecture five and we'll use this analogy to feed into how to calculate the magnetic field in the current carrying case so if I just had the rod of charge the way this calculation went was we calculated the electric field in the bisecting plane and the reason we did the bisecting plane is because that was a high symmetry point it was easier to calculate in the bisecting plane the method works anywhere in space it's just that anywhere in space gives you a really yucky looking integral if you go down to the bisecting plane it simplifies okay so we went to the bisecting plane and we saw that what we needed to do was stand at the observation point calculate the electric field at the observation point due to a chunk wire okay so due to a chunk of the rod and we wrote that down explicitly and then we integrated over all the little pieces of the rod that was this some of our Delta Q's adds up all the charge and then we converted that into an integral over the length of the wire that's straight out Electrify what we're gonna do today is very similar so same methodology it's just that now we're calculating a magnetic field so I'll again take the bisecting plane of the long wire okay so get in that bisecting plane put yourself in the observation point right here and I'll calculate the magnetic field contribution at this point due just to this little piece of the wire so step one draw a really careful diagram where you know right where the rod is we've put the rod on the on the z-axis I'm gonna have an observation point on the x-axis in the bisecting plane and I want to calculate onna write down what's the contribution of the magnetic field right there due to a piece of that current carrying wire so that's step one so in doing that what I like to do first in all these problems right first draw the really good diagram and then after that write down our remember that our points to me so you imagine yourself at the observation point and our is a vector that goes from the current element you're considering from this little chunk up here it points from that to you that's where our is so and my diagram here R goes from the current element to me at the observation point our points to me so now I write down the vector R it is according to the diagram it's got a positive x component so I write that down it's got nothing along the y axis and and going from here to here I go negative amount along the z axis so there's a minus Z right there so I wrote that vector down after having drawn the really careful diagram now from our I calculate its magnitude and its direction so the magnitude of R squared is going to come directly into our equation so I need R squared and that's x squared plus Z squared to get our hat I take R divided by its magnet so R is X comma 0 comma minus Z and it's magnitude is square root of x squared plus Z squared I like to just write all that stuff out so that then when I go putting it in the equation it's crystal clear so now I'm ready to put these things in the equation the first thing I need in calculating this little piece of magnetic field as I need something called Delta L cross R hat in our case Delta L okay so Delta L was always parallel to your wire right so it's wherever wherever you are in that little wire look at the direction of the wire and then Delta L points in the direction of the current so here I've got current pointed up so Delta L is pointed up right there which means I can write it as Delta Z okay so in this case Delta L is parallel to Delta Z so I'm gonna write this as Delta Z cross our hat sorry our Delta L is Delta Z which in vector notation is 0 comma 0 comma Delta Z so now any Delta Z cross our hat and that's looking up here at what our hat is I'll be taking Delta Z cross R over its magnitude and here's how to calculate that I use determinants to calculate these cross products so there's the 1 over magnitude of R right here and to calculate them the cross product of Delta Z crossed into R I write in the first row I write X hat Y hat and z hat in the next row I write the components of Delta Z that 0 comma 0 comma Delta Z and in the next row I write the components of R which is X 0 and minus C and then I take the determinant of that now the way that I think about taking determinants is I think of drawing diagonal lines on this matrix so if I draw a line diagonally down to the right then anything I pick up along the way gets multiplied together with a plus sign so X at time 0 times minus Z gives me 0 so I got nothing there and then I go to the next spot I get Y hat times Delta Z times X ok that gave me this component here x times Delta Z in the white hat direction and then take going diagonally down to the right starting from Z hat I get Z hat times 0 ok so that one went away and then the other thing I have to do is go diagonally up into the right and those things contribute with a minus sign so going diagonally up to the right I'll get it x times a 0 ok that turn went away I'll get a 0 times a delta Z well that term went away and I get a minus Z times a 0 that term went away ok so the only term that contributed was X Delta Z in the Y hat direction all right so that's all together Delta Z cross R I'm sorry cross our hat now I should go back and check it right so I need to use the right that was math then we should always do the Baths extremely carefully so we always get the right answer and then do the double check so you know you got the right answer do the right hand rule real quick to make sure that this is in the right direction so Delta Z crossed into our Hat according to my diagram Delta Z is up our hat is pointing a little bit down like that towards the x-axis so by thumb points backwards which is in the Y direction here so I have Delta Z cross our hat and then it's pointing in the positive Y axis so that worked so that gives me a double-check helps me know well if I had made a mistake it probably would have shown up there all right so now then this is X Delta Z over square root of x squared plus Z squared in place of the magnitude of R I wrote square root of x squared plus a Z squared all of this is in the Y hat direction as we said and that is Delta Z cross R hat so I've one step along the way now to calculating this guy all I've done so far is the Delta L cross R hat term in the biot-savart law do you have any questions so far ok and and really to to do physics really well you want to do both of those in parallel you want to do your diagram really carefully right do the math really carefully and go back and check it physically with the right hand rule to make sure you got the right answer ok all right and so here's what we had on the last slide Delta B is mu naught over 4 pi I Delta Z crossed our hat over x squared plus a Z squared we also calculated Delta Z cross our hat so now I'm going to plug that in ok and all together I'll get mu naught over 4 pi there's a y hat here there's an X Delta Z that's all here and now I need to combine the denominator I'm gonna have x squared plus Z squared times square root of x squared plus Z squared so that's a power of 1 here and a power of 1/2 1 plus 1/2 gives me all together 3 halves power down there in the denominator you okay with that alright okay that's just one piece right now I need to take an integration right so this was just the contribution to the magnetic field due to that little bit of wire I need to sum up the whole length of wire I'm gonna assume now that I have a length of current-carrying wire that's that's L okay of length L and that I'm observing at the midpoint so I'm gonna set the origin at the midpoint of that current-carrying wire so I'm gonna take an integral that runs lengthwise it's gonna run from minus L over 2 to positive L over 2 what I'm really doing is I'm integrating the little Delta B's that contribute right so there'll be a delta B from one piece and a delta B from another piece and so forth so I'm an integral Delta B but in order to take that mathematically I need to convert it into our x and y coordinates here and so that becomes integral right this stuff appears now going to become an integral the Delta Z is the thing that turns into DZ in the integration so now I have that whole thing integrated DZ from minus L over 2 to L over 2 questions so far ok all right and you all remember how I take integrals right at this point I look it up in Wolfram Alpha okay but you have to know what's constant and what's changing so tell me in this integral does exchange during the integration know all right how do you know that are you sure yeah pardon oh yeah okay all right so the integral is in relation to Z all right so X is another independent coordinate so I'm integrating over Z at the same time while I'm doing this integration the observation point is not moving in fact that's pretty much always true right your observation point should not be moving during these integrations and the observation point is here it's a distance X from the rod so yes as he's pointing out the Z the X doesn't change during the integration just the Z so observation point X doesn't change during the integration I look up the integral in order to take it and I get that the magnetic field there is mu knot over 4 pi il over X square root of x squared plus L over 2 squared that's the magnetic field of a long straight wire and I've put magnitude B Y here to remind us that it sits in that Y hat direction at that spot in space so right here at this spot a distance you know X along the x axis it points back in the Y hat direction do you have any questions about that ok all right let's think about what if I rotate this wire now leave the wire right where it is I'm just gonna take the wire and spin it around the z axis okay so if this I clicker represents the wire I'm just gonna turn it around does my answer change no all right what if I leave the wire where it is and I rotate my coordinate system around it I should still get the same physics right so that means that anywhere I am that's a particular distance X from the wire okay from the mid point of the wire I could walk on around in a circle on that plane and I should get the exact same answer I should always get that if I walked around fight took a circle in that plane a distance R let's say where distance R now for radius away then I would find that the field if I walked counterclockwise the field was always parallel to the direction I was facing ok because I could rotate the coordinate system where I could rotate the wire and I'd get the same thing that's actually equivalent to what's happening up here what I have here is a bunch of little compasses okay and then I have wire here and then when I turn on the current for the wire all right now the compasses all choose which direction they should point based on the magnetic field that they feel so they're locally feeling a particular magnetic field all right and the magnetic field pattern here corresponds to what we just calculated okay do you see how that's how that's working if I stand let me focus that a little better for you all right so there's teeny tiny compasses all over the place and if I look at this diagram on the board and this situation here I can see that basically we were a distant distance X out and the magnetic field is pointing back along the y-axis okay so the current in this wire must be coming out towards us so right here the current is coming up just like in the in the diagram on the board do you have any questions about that yeah ah oh I'm glad you asked that because we're gonna calculate that in like two slides sound good okay all right now I want you to start thinking about space as filled with teeny tiny compasses this helps you visualize the magnetic field so think about anywhere in space there's a teeny tiny compass like that and if you zoom into any spot in space with your mind's eye you should be able to read that compass and the compass will point in the direction of the total magnetic field there so if I think then of how this pattern looks all throughout space it's basically concentric circles okay so B is always pointing along I'm gonna turn this off B is always pointing along circles in this case so I've current going up and this is the pattern we see on the board there and this is the pattern that we calculated a better way to express this physics is in cylindrical coordinates I derived it in X Y Z in Cartesian coordinates but the problem really is better suited for cylindrical coordinates so cylindrical coordinates say okay there's an axis as the axis here and then anywhere in space I'll take the shortest distance to the z-axis and call that coordinate R so R is now just a distance shortest distance to the z axis and then there's this coordinate theta which tells you where you are around the circle okay that's cylindrical coordinates and this problem is lends itself to cylindrical coordinates so as we said I could walk in a circle of radius R around this thing and if I walk in that that direction of the magnetic field then this math here in cylindrical coordinates represents that idea I've got to say to hat there okay so theta hat just says well what's the direction that the angle will go around all right that's the direction of the magnetic field so the direction of the magnetic field lines here are always pointing in the theta hat direction I've taken our result from Cartesian coordinates I've switched out X for R where R now represents the shortest distance to the z axis do you have any questions about how that happened okay so this is like this is like another right hand rule basically so I have here did you guys see my video last night to help you do the homework okay I thought I thought we came up with that in office hours how to make your own little magnetic field visualizer so here I have a bigger version okay and I will point out that it's the end of the summer so pool noodles are on sale right now if you want your own for your dorm room get your own little magnetic field big magnetic field visualizer so have your current okay remember current means conventional current so I wrote that too just in case I get confused alright so this is current and here I have a shape of a magnetic field it could either be going clockwise or counter clockwise so according to that diagram there how should I poke this current into the magnetic field should it be this way or should it be the red side red side okay so all right so just to double-check this is if I have current pointing up like the diagram then this is going to have magnetic field going in that direction which is the same as our diagram up there okay so there is another right hand rule going on here which is that if current is going this direction I can point my thumb along the direction of current and then my fingers curl in the direction of the magnetic field okay and it works up here too even though that's the black side the current is pointing this direction and I curl my fingers in the direction of the magnetic field okay and you'll notice that in going from the the red side here has magnetic field this way the black side has the magnetic field in the exact same direction right it's this direction on both sides okay so when the currents facing away from you magnetic field goes clockwise or counterclockwise yep and when the currents facing towards you counterclockwise alright now that's so important that I don't want you to forget it hmm don't fall off okay so have a little a little story to tell you while I draw something here is anybody here a morning person anybody more yeah you're not going to get hurt by the way I'm not asking for volunteers to come up or anything like that just curious if anybody's a morning person I am NOT a morning person so I don't like my alarm clock I'm drawing a clock face by the way I don't like my alarm clock in the morning the alarm clock goes off and although I'm really excited to get up and come teach you physics at the time I'm not happy to hear the alarm clock right so there's my alarm clock in the morning mm-hmm and not being a morning person I kind of don't like my alarm clock okay so this is kind of how I feel in the morning anybody else feel like this when they hear me alarm go off okay all right I want you to take all that rage from the morning I'm gonna step back from the front row they're getting nervous okay take all that rage from the morning when your alarm clock is going beep beep beep beep beep Wawa or whatever does pedo pedo pedo if you like you know minions okay and just do this to it all right pretend that was in the center okay mm-hmm this now did that see that rage is gonna help you remember the right hand rule right because the knife represents the direction of current and then the magnetic field goes clockwise around that all right so just imagine yourself stabbing your alarm clock in the morning Couric you're never going to forget this now all right okay card goes this way magnetic field goes that way okay and no one's gonna sit in the front row again I bet okay mm-hmm no students were harmed in the making of this lecture all right good any questions about that okay so this idea here tells you what that theta hat means all right now you asked the great question of what happens if we have an infinitely long wire and there's two ways to think about that you can either think about the wire getting infinitely long or we can think about the observation point getting infinitely close to the wire so either way and so here's the equation we had we had that the magnetic field coming off of a long straight wire is mu naught over 4 pi I L over R square root of R squared plus l squared no plus L over 2 squared and all that is in the theta hat direction now very close to the wire all right the cylindrical coordinates are about the cylindrical coordinates are about the shortest distance to the z axis so align your current width with the z axis and an observation point here the shortest distance to the wire is this that's R if could be down here the shortest distance to the wires are okay now so that's the R I'm talking about very close to the wire R becomes very small compared to the length of the wire all right so remember how to do approximations doing approximations you want to find two length scales so here I have are it's gonna be small and I have L it's going to be large right so you find your two length scales one big one small pretend they're money right pretend the big length scale is billions of dollars and the small length scale is like pennies and pretend you're the billionaire right do you care about the pennies if you're a billionaire probably not right you get you know you like walking along you're like I'm a billionaire you see pennies on the ground you don't pick them up you don't care you're billionaire right so then you look in the equation for places where pennies are added to billions of dollars if they're multiplied you can't make an approximation right because a billion dollars times a very small number matters to you right but a billion dollars plus pennies doesn't matter to you so here's the billion dollars L over two and here's the pennies which is the R term so I neglect the R term and say that all together square root of R squared plus L over two squared is approximately equal to L over two if L is really large okay and then very close to the wire this is what the equation looks like so then that becomes there the magnetic field is mu naught over 4 pi IL over R times L over 2 in the theta hat direction which is all together Mew naught over 4 pi 2i over R in the theta hat direction close to the wire or for an infinitely long wire do you have any questions about that and here's the question I have to current-carrying wires this one is current carrying current so the left end this one is carrying current to the left at an OPS and assume they're carrying the same current and add an observation point directly in the middle of those two wires what kind of magnetic fields you observe ok what's the direction of the magnetic field at that location okay what do you thinking what do we need to take into account here what so what's a line of reasoning that we need to consider for this problem and it's always okay to tell me your neighbors line of reasoning that's okay oh yeah okay so he's arguing that I'll repeat this for the people in the back tell me if I get it right so he's saying the upper wire actually I see I could use the pool noodle all right so he's saying the upper wire has the current going this way therefore the magnetic field is coming out towards you and then the lower wire has the current going that way but but at the observation point the magnetic fields going back and those are two different directions so the one up top has filled towards you the one on the bottom has filled away from you and those contributions tend to cancel and in the case where the currents are exactly the same and we're measuring a midpoint they exactly cancel now I want you to do it for the other case all right so now I have conventional current going to the left up top and going to the right on the bottom what's up what's a line of reasoning we need to consider again we're much more interested in lines of reasoning than n the end answer right because the thing you need to get to the answer is actually the line of reasoning so yeah right here okay okay so you said mm-hmm by symmetry whatever fields they're producing are going to be in the same direction at the X point so I you just calculate the top one and the top one has current going this way and feel pointing back towards you all right other other lines of reasoning we need to tangle out yeah you good okay all right and of course you should make your own magnetic field visualizer a Tron thingamajig to do your homework that's its official name we're gonna patent that right so the top guy has current going this direction and coming out towards you is this magnetic field right and then the bottom guy has the current going the other direction and it still has current coming out I'm sorry field coming out towards you so in this case the fields add and they're all coming out towards you which in the diagram is the positive z-axis all right now now that we've got that in our heads that when I think about the magnetic field that a current carrying wire makes I want to put my thumb along the direction of the current my fingers curl in the direction of the magnetic field okay or you could always think of stabbing your alarm clock okay when I stab my alarm clock in the morning that's like current going into my alarm clock and then the magnetic field is like the clockwise direction on the clock mm-hmm now we're ready to think about another geometry so let's have a current loop let's have a loop of current so I have a loop of wire here I'm gonna run current this direction and I want to think about well what kind of magnetic field pattern does this make all right and what we're going to make by the way is an electromagnet when I turn the the current on I'm gonna get a magnetic field and when I turn it off I won't have the magnetic field so I can use this biot-savart law for currents and a wire in order to calculate the magnetic field of the the loop of wire here and again I'm going to stick to a high symmetry point we stick to the high symmetry points to do the actual calculations because they're easier to calculate right the same principles will apply everywhere in space and you'll get a complicated formula in the rest of space so I'd like to orient this loop of wire along the z-axis I'll take an observation point on the z so let's sit right here on the z-axis have the current going this direction and I need to think about what direction does the magnetic field go in right there on the z axis so the thing I'm going to do is do the right-hand rule right so I have Delta Delta L up here at the top Delta L and then I need to cross that into R which comes now pointing along towards the z axis and then that used me can't quite get it right can I so Delta L crossed in the are points up in this direction there's the Delta B pointing up now I need to think about that's just the contribution from that top piece of the wire I need to think about what happens when I sum up all the contributions all along the entire loop all right and as I do that all right this piece from the top of the wire gives me something that's pointing a little bit up as I sum around the wire I'm going to get contributions that go like this around the z axis all the components that are off the z axis will cancel each other as I go around and so I just get something that's net in the Z direction all right mm-hmm so the net magnetic field along the axis is just on the z axis all right and and this procedure that we always use we write down now the magnetic field due to one piece and in this case the magnetic field then due to one piece is this now that's lots of math okay I this this is a step I this is tons of math go in between this right Delta BZ due to one piece and this answer here okay it's just that it's the same method that I've showed you a few times so I'm not going to show you all the little steps this time in fact we're doing something well like this I think you should be more specific here in step two I just want you to remember the method right how we get it and then I'll take this guy and integrate to find the total B so when I integrate around that guy lots of math happens down here and here's the answer that I want you to take home okay so is the method clear draw this careful diagram look at the observation point use the biot-savart law to calculate the magnetic field due to one piece integrate all around the the the current and then that gets you the total magnetic field that method gets you this answer so it's the same method we used before for the long straight wire and now what I want you to focus on is the answer itself so here's the magnetic field then due to a loop of wire on the axis it has a Mew naught over 4 pi in front to get the unit's right there's AI which is the currents and the wire R squared 2 pi over R squared plus Z squared to the three-halves r is talking about the radius of the loop of wire so that's how big this wire is z is how far away I am along the axis of the loop of wire so if I Center the loop of wire on the origin and have its axis coincide with the z axis this Z is how far away I am from the loop of wire ok so that's the answer and it's it's a lot of math to get from here to there which we're skipping now I'm gonna double check real quick and make sure the units are right the unit's here mu naught over 4 pi has units of Tesla meters per amps this stuff on the right has amps meters squared that's amps meter squared and in the denominator here I have meters squared to the three-halves okay so does all that work out all right I can take this 2 times the 3 halves gives me altogether a 3 so have three powers of meters that cancel these two plus that power of meters so that works amps cancels amps and I'm left with Tesla so the units work check alright and then I should check and make sure it fits the right hand rule right so have a loop of wire with current going in that direction and let me orient my thumb along the direction of the current okay when I orient my thumb along the direction of the current so let's be at the top think from the top wire top of the wire here the currents coming towards you so field should be poking this direction along the z-axis right and I can orient my thumb over here on this side of the wire currents pointing down but magnetic fields still poking in the center and so on all the way around I'll get to the magnetic field points along the z axis the whole time do you have any questions about that okay so the direction is working out yeah a question here Z Z is distance along the z axis this should have been labeled so this is this diagram here the this axis coming along this direction is the z axis I'm sorry I should have labeled that so I'll post that online I'll post that correction this is the z axis here does that help yes so if I take the current loop and put the current loop so that its center is at the origin of the coordinate system and then for any loop there's an axis for it so that's I'm putting that axis along the z axis so to show you in terms of let's see there's another paper plate so if I say that current is going along the rim of this guy right okay pretend I'm a better artist than that so to match the diagram right currents is going along the rim of this guy then the z axis points straight out of the middle like that okay so Z is Z is going to be this equation is 4 on that z axis and how far am i from the ring okay and all that direction the the magnetic field points this way and then appoints that way as well thanks for asking the diagram should have been clearer I'll post a correction do you have any more questions about that ok all right so here this might make it clearer to if we think in terms of that loop of wire anywhere along the loop of wire you can orient your thumb on the direction of current and then your fingers poke in the direction of the magnetic field so in the interior of the series circle here I've got magnetic fields coming out towards you all right if you like you can turn that into yet another right-hand rule which is over here which is where I take the loop of current and now it's a different right-hand rule if I put my fingers in the direction of the current in the middle of the of a circle then my thumb points in the direction of magnetic field in the middle okay and then outside of that it's pointing in the other direction do you have any questions about how that guy went okay all right the overall shape of this thing so here's here's a current carrying wire all right the overall shape of it is that there's a magnetic field that on the z axis points all along the z axis and then loops back around in space right ultimately we have current coming out here and I need to think about the right-hand rule up there think about current going into the board of this spot and then the magnetic field lines wrap around that and now fill all of space with those patterns and I get this where the magnetic field comes out here comes back around and back in the other side so this shape actually is very similar to the shape we had for an electric dipole okay when we talked about an electric dipole that's a positive charge here and a negative charge here and the electric field comes out of the positive side wraps back around into the negative side at least far away from it not in the middle in the middle it's different but far away from it the fields look very similar now if I think of the current loop and now let me take an observation point far away from the current loop so here's our equation for on the axis of the current loop its mu naught over 4 pi times 2 PI R squared I over R squared plus Z squared to the three-halves by the way what happens at that top equation so let me take that top equation let me put in a negative Z what happens to the negative Z in there yeah the minus sign goes away right because I have a minus C times a minus C so in fact the arrows are telling you something about that as well right the arrows are showing you that the magnetic field is always pointing in the positive z direction and the equation represents that as well all right now far from the loop if I take Z and much much greater than our okay remember how approximations go you look for two different lengths of different sizes one length is large the other one's small I call the large length like billions of dollars I call the small length like pennies and then I look for a place where they're added together here's where they're added together I have R squared plus Z squared to the three-halves so in that case if if the Z is billions of dollars in the r is pennies I'm a billionaire so I don't care about the pennies I've cross them off and I get all together Z squared to the three-halves which is all together Z cubed okay so that gives us the magnetic field along the axis far from the loop here I've taken the Z and just measured it in terms of little R as distance from the loop itself do you have any questions about that okay all right so basically far from any dipole whether it's a magnetic dipole or whether it's an electric dipole when you're far from the dipole the strength of the field the magnitude of the field falls off like 1 over R cubed whether it was an electric dipole or whether it was this thing here which is actually a magnetic dipole this is the same field configuration that you actually have for a bar magnet as well so if I have a bar magnet which is a magnetic dipole this is not labeled so that you can see it very well but this is north and this is south the field configuration coming out of a bar magnet does the exact same thing as this current loop alright when you look far enough away so if I have a bar magnet here and I measure its field far away it's gonna look like that if I replace this with a current loop and I went far away and measured the field it would look exactly the same all right so they both fall off like 1 over R cubed now one of the things I'd like to know though is why does this bar magnet act like a current loop alright it probably means that inside the bar magnet there's little current loops all right and in fact that's the case there are little current loops inside here so let me think about where they might be a current loop makes an electromagnet right so I turn on the current and I get an electromagnet turn off the current and the magnetic field is gone all right so in fact it turns out that electrons are doing something very similar inside this piece of material okay so if I think deep inside the material okay put yourself deep inside think in terms of what the electrons are doing inside electrons orbit their atoms all right as they orbit their atoms they sometimes make motions that are very much like a little current loop and when they do that they're making a little electromagnet now in most materials those little electromagnets all cancel out in the material and a permanent magnet that what's going on is that every single little electric current loop has a line and that's what made this thing a permanent magnet all the little current loops lined up in the same direction and that all their magnetic fields add up and add up and add up right and then that gives me this total permanent magnet all right now there's one other thing that's going on inside of a bar magnet which is that the electrons actually spin on their axis and that's kind of weird so think in this case think in terms of an analogy with what the earth does in the solar system right so the earth goes around the Sun all right and that's like the electron orbiting its atom and making that big current loop that's like the big current loop but the earth also spins on its axis all right now if I think of the earth as being a big ball of charge then if it were spinning on its axis and it's a big ball of charge then that's also creating a little current loop there as it spins on its axis would you believe that electrons actually act like that okay so electrons have a charge and it looks as though every single electron in the universe is spinning on its axis the weird weird weird thing about that is that as far as we can tell electrons don't have any size they have zero radius so that'll Wiggy out but it's quantum mechanics okay all right so there's two reasons that electrons act like little electromagnets when they're all aligned in a material that gives you a permanent magnet okay all right we're done for today I'll see you guys next week all right I want you to take all that rage from the morning I'm gonna step back from the front row they're getting nervous okay and just do this to it no students were harmed in the making of this lecture

Channel: Prof. Carlson

Views: 1,322

Rating: 5 out of 5

Keywords: iMovie, Physics, Electricity and Magnetism, Right Hand Rule, Matter and Interactions, Electricity, Magnetism, Current, Current Loop

Id: JnhxIY0mHuw

Length: 42min 59sec (2579 seconds)

Published: Tue Oct 04 2016