 What is the source of magnetic fields in nature? The B.O. Savard law gives us one insight into answering this question, that it is the movement of electric charge that creates magnetic field. Let's begin to explore the implications of this fundamental law of nature and how it can relate moving charge to the magnetic fields that are generated by those charges. Okay, so let me demonstrate something here. Okay. I'm going to just take this camera, and the camera is primarily made of plastic, non-conductive, non-magnetizable material. All right, so I have here a compass, all right, and we'll try moving it here. All right, so if I let that settle for a second, you'll see that it points in the direction that north is supposed to be. So let me line up N on the compass roughly if I can. Okay, so now we know which way is east and south and west and so forth, okay? So there we go, and get the reflection off this. I'm going to kill the lights, so brace yourselves, okay, a little easier to see now. Okay, so I have this assembly here now with a coil of wire, that's the orange stuff, so it's just wire that's been looped many times around a piece of plastic, and it's coated with a thin plastic coating so that the wire doesn't actually come in electrical contact with itself. So those loops are actually electrically isolated from one another, but just a thin little clear plastic coating. This is a very common way of making coils of wire like this. If you didn't do this, then you short the loop. So the current would not flow through each cycle of the loop, it would just run the shortest path to the other end and come back to the power source. Okay, so I have a switch here, all right, and I have power coming in from this low voltage power supply over here on the right. So it's plugged in, and at this point I can switch this on, you should see a green light come on. Now I'm only going to get to do this experiment a couple of times, so I need you to watch very carefully, all right? So we have here, again, the compass still pointing north, loop of wire over here, and I'm going to throw the switch, watch the compass. All I'm doing is running an electric current through the loop of wire, there are no magnets anywhere else near this, there's no magnets in my hand, okay? All I have to do is close the switch and make a current flow, and I can change the direction of north locally. This in and of itself was an astounding discovery. For almost 2,000 years, human beings have been playing around with magnets, not at all understanding where magnetic fields came from. They seem to be something apart as chemists and physicists and mathematicians started studying the electric phenomenon. The magnetic phenomenon seemed eerily similar, in fact there were scientists that suspected a connection between the electric phenomenon and the magnetic phenomenon, but no one could quite prove it until one day, quite fortuitously, while doing a lecture demonstration for his students in a physics class, Hans-Christian Ursted closed a switch on a battery and noticed that a very thin metal compass needle deflected nearby, and he did it again, and the compass needle deflected again, and it was a eureka moment for him, because he realized it's not static electric charge, I mean I could put for instance that device I showed you at the beginning of class that can hold charge, for instance I could rub PVC pipe and hold it next to this very still, and this compass won't respond, but if you move charge it creates a magnetic field that affects permanent magnets, in this case it made the compass needle deflect, terrifying, let me get rid of that, so let me reinforce this a bit, so Hans-Christian Ursted, and as I said he observed in 1820 that electric current caused a compass needle to deflect, that was his basic simple observation of the natural world, and it caused quite a stir when he published his results, I mean anybody could reproduce it if they had an electrochemical cell, what we call a bowery now, and a compass needle, some wire and a compass needle, now the reason I only get to do this a few times is this is copper wire, this is copper wire, there's not a whole lot of resistance in this circuit, I didn't put a light bulb in here, so there's a big current of order in amp or two that's flowing through this copper wire straight back into the power supply, I'm effectively shorting the power supply through the circuit, so eventually if I leave this on long enough it's actually warm to the touch now, there's a safety mechanism that when it gets too hot it pops a little switch and the power supply doesn't work anymore, if I do that four or five times in a row quickly I can short the power supply and it will save itself by breaking the circuit, when it cools down again I can pop the switch back in, so I didn't actually get it to flip this time but I did this while I was practicing the demonstration, it's actually quite difficult to get this to work, you need a lot of current and so you know, Ersted needed really strong chemical cells and he wasn't the one that developed them, other people developed them and he was using them in experiments, so as I said he actually first observed this amazing phenomenon while doing a lecture demonstration of electricity in front of his students in his class, so this marks the discovery of what we now call electromagnetism, which is a relationship between electricity, the electric force, and magnetism, the magnetic force that's intimate and related through the motion of charge, so what we learn at this point is magnets, okay, like these permanent magnets will respond to other permanent magnets like the earth, so the north will line up with the self-magnetic pole of the earth, which is the geographic north pole, okay, we also know that electric charge, okay, we can take a magnet, we can put it in the path of electrons and they will deflect if their velocity vector is at any angle to the magnetic field from the permanent magnet, magnets affect electric charge, but we also see a beautiful symmetry here, electric charge in motion affects magnets, they generate magnetic fields, and this was a stunning revelation at the time, and we have benefited immensely from this very basic observation about the natural world, so let's watch this, this is a magnetic fluid by the way, has anyone ever seen this before, let me go back here, so magnetic fluids are just like solid permanent magnets except liquid form, and they're neat because if you spritz them onto a permanent solid magnet you can see the magnetic field lines, see the dipole field lines shooting out of the top, coming out of the sides, you can suck the fluid back into the plunger, it was a famous car commercial a couple of years ago and I first started teaching the course, and it was a car commercial about the fact that this car company uses a magnetic fluid as part of their brake system to create a more responsive brake system, so that fluid is actually used by car manufacturers to improve the quality of safety systems on a vehicle like brakes, he's going to do another demonstration in a moment, and it's similar to what I just did, I used to have to only show the video because I couldn't get enough current to do this but this year I got it to work, but the standard judicial way of showing magnetic fields is with a compass, I've got a wire, big-ass wires, got it surrounded by compasses, compass points towards a magnetic field, if you were to bring a strong magnet near it they would point towards it, it's like a little strong magnet, my hand eats it, they point towards it, and an interesting phenomena takes place, we get it lined up, when we put electricity, so they're all pointing north, and then they all start to point around the wire, and if you wrap the fingers of your hand they go in the direction of the magnetic field, you curl your hand around the wire, that shows you the magnetic field, like this, curl the fingers around your hand around it, the thumb indicates the direction of current, it's got this picture upside down, I don't know why you did that, but okay so we'll come back to that in a minute, but let's just sort of take a look at a photograph of that experiment, not using compass needles which are big, but using little iron filings, and each piece of the iron filing, each iron filing is a little north pole and a little south pole, a little bar magnet, and if you take a wire and run a couple of amps through it, and in this case they cut a hole in a piece of paper, so non-conductive paper, they ran a conductive wire up perpendicular through the paper, okay, so out of the plane of the paper, they ran an electric current through the wire, and the iron filings on the paper, which were originally just shaken randomly and uniformly on the paper, they line up, and isn't it, this is gorgeous, right, you can see the current is going up the wire, and the magnetic field is circulating around that wire, and it was this observation that two physicists used to now write down a law that we understand deeply predicts the magnetic fields caused by electric currents. Katherine, did you have a question? Yeah, I was wondering why they wouldn't all line up, like, on top of each other, because wouldn't they want to get as close as possible to the current? Absolutely, and the paper has friction, so their rotating is pretty easy, you know, but sliding that whole little bit of... So they're just like placed all on there? Well, they're just shaken on the paper, and then you kind of just kind of tap the paper to get them to settle uniformly, and then you turn on the current and wait, and the little magnets will all line up, and they'll line up along magnetic field lines, so what you're seeing here is the little bar magnets are, you know, each of them is sort of like a little, it's like a little bit of, you're kind of looking at calculus here, but done physically. Each little bar magnet is like a piece of a much larger magnet, and each little piece is lining up along the magnetic field lines, which are big swooping circles that it go around the wire. So the magnetic fields appear to be in the plane of the paper, and they are circulating perpendicular to the current flow. Again, you can't get away from the cross product with magnetic fields. The current goes this way, the magnetic field goes at an angle to that that's perpendicular. We'll see what Biot and Savart did in a moment. Used a lot of experiments using compass needles to figure out the form of the rule, okay? But yeah, so I mean, if you waited long enough and there was a less, you know, low enough friction that would allow motion in the plane as well, yeah, eventually those little ferromagnets, the little iron filings would clump closer and closer to the wire. That would be dangerous because you start making electrical connections between the wire and iron, which is conductive, and well, eventually you could cause shorts or sparks or things like that, because that's, again, that's like probably two or three amps of current going through that. That's a lot of current. You know, you don't want to short that or touch that with your finger, because we know what just a few milliamps will do to a human being. So these are they. Actually, I thought it was kind of interesting as I was doing a little bit of historical homework for this slide, that there's a photograph of Jean-Baptiste Biot from 1851 or so, so maybe about a decade before he died. Photography was only commercially available as of about 1839 or so. It'd been invented in the decades prior to that, but in terms of commercial photography, that wasn't really something you could get. You couldn't get a commercial photograph until almost 1840. So, you know, within a decade after commercial photography first appeared, people were photographing other people, and there's a picture of a very old Jean-Baptiste Biot, and this is probably closer to when he was doing his work in the, what, 18, well, 1820, actually, the year that Ursted published his results was the year that Savard and Biot did their research with compass needles and currents to figure out what the rule was for having a current in some length of wire and then getting some resulting magnetic field from that. They figured out the mathematical relationship for that. So they worked together, you know, about 1820 and came up with this thing we now call the Biot-Savart Law. So let me show this to you, and I'll spend the last five minutes kind of talking a little bit about this, all right? It looks nasty, all right? So dust off your calculus, because here we go. What I'll do in the next lecture is I will show you how to calculate the magnetic field from a straight wire, and if there's time, I'll also do a circle. And then we're going to do an exercise next week that synthesizes everything that we've been learning about magnetism so far. And those things are magnets cause other magnets to repel or attract, okay? We know that from just playing with magnets. We know from experimenting, I showed you the cathode ray tube, we know that electric charge can be deflected by magnetic field. So magnetic fields affect moving charge. Moving charge generates magnetic fields. That's what the Biot-Savart Law encapsulates. We observe the phenomenon in nature, we describe it mathematically. And it requires calculus. So just like Coulomb's law, this is the Coulomb's law of magnetism. This law tells you that thanks to moving charge, I, Coulomb's per second, you get a magnetic field B resulting from that. And in order to figure out the magnitude and the direction of the field given the current and the length of the conductor, there's a relationship, okay? So the magnetic field, so let's take a look at this bendy wire I have over there. Let's focus on just a little piece of the wire, which I'll call dL vector. So dL vector is just a little piece of the length of the wire. And the direction that that little vector points is the direction that current I is flowing, okay? So this is current I. Comes in the wire, snakes around, goes out that end over there. I'm only showing you a little piece of it right now. And if we were over at some point P, which I've indicated with the black dot, we would experience a little bit of magnetic field due to that little bit of conductor. Now, of course, the total magnetic field is the sum of the effects of the current here and here, here, here, here, all the way up there. But if we can calculate using coordinate systems and vectors, one piece of the wire, and then we can integrate over the whole wire using some trick, we can get the total magnetic field. We can sum all the little dVs that result from these little IDLs, these little current times these little length vectors in the conductor. Now, we know from experiment, so like the demonstration I did here, it's easier to see in the demonstration in the video, or the little paper with the wire going through it, and you can see the circles of magnetic field around the wire. We know from that that the direction that the current flows is perpendicular to the direction that the magnetic field points. So if the current points this way, up here the magnetic field might point that way, over here it might point down. It's always perpendicular to the direction of the current flow, and that's what gives you a circulating magnetic field around the wire. So that's an empirical observation. When you do the experiment, that's what nature does, and so mathematics you use to describe that phenomenon and try to figure out the exact precise relationship. So beyond Sivar worked really hard with little compass needles to do this. They mapped out magnetic fields from wires, and they came up with the rule that the magnetic field direction is at right angles to the current flow, which we already knew, but the trick is if you draw a vector from the little piece of wire out to where you're observing the field, and you use a right hand rule, OK, dL points that way, r points that way, the magnetic field at point P will point out of the page according to the direction your thumb is pointing. So again, dL, r, dB. That's what they figured out, and the only way you can describe that is with a cross product. You have to know the dL, you have to write down the r, and you have to then from that compute the direction of B and the magnitude of B. Now the magnitude of B comes from multiplying idL and then the unit vector r hat and dividing that whole thing by the distance squared, they found that much like with Coulomb's law, the magnetic field strength seems to decrease as a function of distance, and the rule that they worked out is just like Coulomb's law, it's an inverse square law, which is already in and of itself fascinating that that happens to be the same. And then it turns out you have to multiply the number you get from this by a constant. OK, so this is another number that you can commit to memory or I'll write on the formula sheet. It has a name, let me just skip to the next slide here, it has a name. Epsilon knot, which we've seen already in electricity, OK, so that's related to K, 1 over 4 pi epsilon knot is equal to K, that constant we encountered at the beginning of the course. Epsilon knot is actually considered to be the fundamental number, it's the number that physicists talk about when they talk about electricity. And it has a name, it's called the permittivity of free space, I think I've mentioned this before, but I'll formally mention it now. It's given its own name, it's so important in electricity that we named it. And from a capacitor, if you remember the field inside of a capacitor, an ideal parallel plate capacitor, it was proportional to epsilon knot, right, it was the area of the plates divided by the separation of the plates times epsilon knot. And so epsilon knot is meant to describe somehow, if you have nothing in between the plates, it sort of describes the freedom with which electric fields have to propagate through empty space. And if you add any material into that empty space, if you put a dielectric inside the capacitor, you will weaken the electric field. So you'll change the permittivity of free space to something else by multiplying it by a number of kappa that's 1 or greater, OK. So this number physically represents what is known as the dielectric properties of empty space. And similarly, mu knot represents the, in a magnetic sense, the dielectric properties of empty space. So mu knot is the equivalent of epsilon knot, but for magnetic fields. And it tells us how permeable free space is to magnetic field propagation, in the same way that epsilon knot tells us how much empty space permits electric fields to propagate through it. Now this might seem kind of random. What the heck is with these numbers? I mean, they're just numbers that some French guys figured out in a lab. So be it. But it turns out that they actually have a deep connection to a much more important number. And I won't talk about it today, but in one of the lectures that you're going to get in the next couple of weeks, there is going to be a lecture that talks about the nature of light. And what is light exactly before we get into optics? And there isn't going to be any reading for that lecture. It's going to be just kind of a nice, fun lecture to break up the course a little bit, OK. And it will unite absolutely everything that we'll have talked about up to that point in the course before we get into optics. And optics is just the transportation of light. So going back to biosovar, let's do an exercise with biosovar. And the exercise that I'm interested in doing is the magnetic field from a long straight wire. Let's imagine we have a long straight current carrying wire. So I'm going to just draw a little section of it, but I want you to imagine that it continues on to infinity in both directions. So let me just say that that goes up to plus infinity and that goes up to minus infinity. I'm really under-representing this wire, OK. So really long straight wire. So we don't have to worry about what's going on at the end. They're very far away from us. And we're up here at this point, P, and we're going to try to calculate the magnetic field at that point. So far this is looking a whole lot like Coulomb's law trying to figure out the electric field from a long line of charge with some charge density lambda. Well, there's no lambda anymore for this. There's just current, I, constant current. So in place of linear charge density for this problem, we'll have I to deal with, OK. In place of DQ, OK, in place of DQ, we have to deal with DL vector, the little piece of the wire where the direction of the L vector points in the direction of the current. All right, so let's go ahead and pick a little piece of the wire. I'll shade it in here, OK. So this right here is our little so-called current element, current I DL vector. I haven't done anything with coordinate systems yet. I'm just labeling pieces using Biosovar. Well, that little piece of current, that little current element, generates a little magnetic field up here, OK. And we have one more vector to write. So actually, let's write the vector here. So as I said, we have to say where the current is going. So let's say current is going to the right. And actually, in anticipation of using Cartesian coordinates, let me make a quick little notation change here. And let me switch this to big I. So we have a current big I going to the right, because I'm going to need I hats and J hats and J hats. OK. And then finally, just like with Coulomb's law problems, the R vector, where does that go from and to? Well, it goes from the thing creating the magnetic field to the place where we're measuring the magnetic field. So it goes from the current element to the point P. But that convention remains exactly the same as before. OK. We're missing a coordinate system here. We need to start writing things down in terms of stuff we can actually maybe integrate, for instance, OK. So we need some coordinates. And I'm going to choose to put my observation point right on the Y axis. And I'm going to choose my wire to Y along the X axis. So positive I hat is to the right. Positive J hat is up. And again, using this little right hand rule, X, Y, Z, Z points out of the board. So let me just go ahead and draw. There's the Z, the Z axis points out of the board like that at a 90 degree angle to both X and Y. So classic Cartesian coordinate system. All right, angles. OK. Well, we can already basically use the right hand rule for a cross product to anticipate where this magnetic field VB vector is going to have to point from this current element. So, for instance, we can take our index finger. We can point it in the direction of DL vector, which points in the direction of the current, OK. I can then point my index finger in the direction of our hat, which is the direction that our vector points in. So I'm actually already doing that. And then my thumb indicates the direction of the magnetic field. The little piece of magnetic field VB vector will point in this picture. So without doing any calculations, I can already create a situation that I can use to cross check my answer later. I expect that little piece of magnetic field that when I write it down, I expect it to point out in the positive Z direction, the K hat direction. So let's just mark that for later. It appears that DB vector will be written as something like a magnitude times K hat. We don't know what DB is yet. We've got to write that in terms of I and L and R and hopefully some Cartesian coordinates in there someplace, OK. We'll try to rewrite this in terms of our coordinate system. But we can anticipate that that's what we expect to see, all right. Something that points in the K hat direction. Something that points along the Z axis. And that makes sense because the Beaus of Art law says that the magnetic field element from that current element had better point perpendicular to both DL and R hat. And since DL is in the plane of the blackboard and R is in the plane of the blackboard, the only place where you can be at right angles to both of those is either out of the plane or into the plane. So perpendicular to the plane like this, OK. So let's just, let's file that away for a second and let's do some math. Just like Coulomb's law, we now seek to represent R, R hat, R vector, DL in terms of coordinates. So let's do IDL first, OK. IDL vector. Well, a couple of things we notice. So first of all, this is supposed to point in the direction that the current is pointing. And what direction is the current pointing by construction in my coordinate system? S, yeah. So it's pointing in the positive I hat direction. In fact, any current element I look at in here, the current element vector will always point along I hat. So I can write this one, for instance, as IDL I hat. So I've got the direction I'm taking care of in terms of my coordinate system. That's done. DL, all right, well what is this? This is a little piece of the length of this wire which goes out to positive infinity that way and negative infinity that way. It's just a little piece of the x-axis. So DL can be identified merely as DX. And if I were to look at another piece of this, it would be a DX over here or maybe a DX over here or over here. But they're all little DXs. They're all pieces of the x-axis for this conductor. So that's about as far as I can go. I mean, I would assume in a problem like this that you'd be given the current, OK, so you'd be given maybe a number for I. Now DX is something we can integrate. You can imagine that our integral, when we sum up all the little current elements to get the magnetic field, we're going to be summing over DXs. And DXs are going to go from negative infinity to positive infinity. So we already have our limits of integration figured up too. Let's move on to R. OK, so R vector, well, R vector is the vector that we have to take. We're going to break it into Cartesian components, X's and Y's and Z's as needed. It's the path that we want to take to go from the current element to P, all right, along our axes. So to go from our current element at some distance X out from the origin of the coordinate system back to the origin and then from the origin up to P, OK, which we'll write as being some distance D above the line. So we're some distance D above the line in this. It's actually not too bad. We have to go a distance X in the negative I hat direction and then we have to go a distance D in the positive J hat direction. And that's our R vector in terms of our coordinate systems. OK, and X, well, X is the thing we're going to integrate over later, so that's good. We've got X in here. D is just a constant. We're just at a fixed height above the wire. In fact, no matter which current element you pick anywhere on this wire, it'll always be this height D above the line. OK, that's shortest distance from the wire to the point P is always D. So D never changes for any of the current elements. X changes, but D doesn't. And we're going to integrate over X, so that's good news. And then, great, so magnitude of R is just going to be, well, R squared is going to be X squared plus D squared. OK, that's easy enough to do from Pythagorean theorem, or you can square, you can take the dot product of R vector with itself to get R squared, whatever you like. But Pythagorean theorem works just fine for this. It's a right triangle. So you want to get the length of the hypotenuse, you take X squared plus D squared, that's the hypotenuse length squared. Done. That's pretty cool. Yeah? Just to clarify, I guess we're writing terms. We're not using dot, I mean, formula terms. I haven't played getting the term thing. I'm just getting things into the coordinates. Yeah, using my coordinate system, which is usually the longest step. Getting things into the coordinates. Outline and then we'll use the formula term. And then we'll get this into an integral. OK. And then we'll integrate it. All right, so finally we have R hat. Well, that's just R vector divided by its length. And we already have R vector, and we already have the length. So we just have to write an ugly looking ratio here. So this is going to be negative X i hat plus D j hat all over the, I'll write it this way, X squared plus D squared to the one half power, or the square root of X squared plus D squared. That's what R is. It's the square root of this equation. OK. I'm going to write that as just to the one half, because in a moment it will be convenient to combine exponents with other parts of the equation. OK. Well, let's see. We've got I D L vector all sorted out. R hat all sorted out. R squared all sorted out. Now we can plug this into the B O sub R bar. And I'm going to assume that you're at this point, if you weren't at the beginning of the course, you certainly are now that you're capable of popping up in a book and figuring out what an integral is supposed to be. Or flipping through the formula sheet on the exam and reading off the result of an integral. Or asking Wolfram Alphra, how do I integrate this? And then getting the answer from it. OK. We're going to kind of skip steps for the integrals and then a miracle is going to occur and we're just going to move on with our lives. OK. So we need the little db vector due to this little current element I've written here. Well, the easy part of the constant's out in front. U not over 4 pi. They never change. Now let's deal with the numerator up here. So we've got I D L vector. Well, that's I DX I hat cross product with R hat. Well, R hat's a little more complicated. So let's see here. So this is going to be, let's put the numerator of R hat up here. It's this thing negative X I hat plus D J hat. OK. And then down in the denominator we've got R squared in the denominator of R hat, which is X squared plus D squared to the one half. So I can already simplify this by multiplying these two denominators together. OK. So let me rewrite this. U not over 4 pi. In fact, let me just group U not and I together up here. U not over, U not I over 4 pi. Put the DX here on its own. And then we have D J hat all over X squared plus D squared. And in fact, the integrals that we have to do here are eerily similar to the integrals that one has to do for the electric field due to a line of static charge or any line of charge. OK. So if you dig way back in your notes, you'll see that we had to integrate something just like this. Alright, so let me go ahead and now, let's see, I think I've got all the pieces. So let's look at just, let me just look at, so look at, isolate that little piece up here. OK. So we have I hat cross negative X plus, I'm just distributing the cross product. I hat, all I did was I took I hat cross that term plus I hat cross that term. There's a gift. I hat is zero. Awesome. Don't even have to worry about that stupid term. OK. Well, we have this one left now to deal with. So that thing's zero. So we have zero plus whatever this thing is. So I can pull the D out in front. That's just a number that multiplies either the I hat or the J hat. Again, a cross product is just like a multiplication sign. And the same algebraic rules that apply to algebra, you know, distribution and so forth, association, that's all that holds here. And this is D is just a number. It's not a vector. It has no direction. So you can just pull out in front and just leave behind this cross product. And what's I hat cross J hat? K hat. OK. Well, let me go ahead and rewrite this thing now. DB vector using what I've just learned. So I've got U naught I over 4 pi. I got DX. And now I have just the distance D K hat up in the numerator. X squared plus D squared for the three halves. And that's about as far as I can take this thing without integrating it. So let's take a look at this. This has got all the thing, well, it's got the key thing that we kind of anticipated at the beginning. OK. So right now we've got U naught positive number. I positive number. 4 pi positive number. DX, we're going to integrate that away in a second. We got X squared and D squared. Even if those are negative, they become positive by squaring. So this whole thing in the denominator is a positive number. And D by construction is a positive number. So we have a positive number times K hat. Number done a good job. We can use the right hand rule to anticipate what the answer should look like. We can check to make sure the answer looks like that when we get through all the math. That's the nice thing about the right hand rule. It gives you a chance to think about things slowly, write a few notes down, and then proceed blindly into a blazing calculation of mathematical stupendous tricks, or whatever this is, and then hopefully check your answer at the end, which is more valuable than anything else. OK. Awesome. So now we have to do an integral. So let's do an integral. Is it OK if I clear this out? Yeah. Not a whole lot going on here anymore. OK. So now to get the total magnetic field due to every current element in that liar, I have to do some integral over the little db values. OK. So that's going to be an integral over mu naught i over 4 pi dx d over x squared plus d squared to the 3 halves. Well, I only need to integrate over things that have a dependence on x. And so I can take mu naught and put it out in front. I can take i put out in front. No dependence on x. 4 pi out in front. That d, that d can go out in front because it doesn't depend on x. Now this nasty thing in the denominator depends on x, so I got to leave it in the integral. And k hat doesn't depend on x, so I can move it out. So this winds up being an integral of mu naught i over 4 pi k hat integral of dx all over x squared plus d squared to the 3 halves. And remember I have my limits of integration from before. I'm going from x equals negative infinity to x equals positive infinity, infinitely long wire. Now that's an integral. You can look up a book, ask for from Alpha for the answer, whatever you feel like you need to do. OK. So I'll just write down the answer. It turns out that the answer to this is just 2 over d squared. That's it. Not a bad integral to have to write down the answer to at the end. And so we can just plug that in for that integral and we're done. We've got the magnetic field around the wire at a distance d from the wire is just mu naught i all over. We've got a 2 in the numerator and a 4 in the denominator. So we're left with a 2 in the denominator. Pi is not quite it. That is the direction and magnitude of the magnetic field around a long straight wire, which essentially is what B Owens-Savart used to understand the law they wrote down later in the beginning. Let's look at the elements of this. If I double my distance away from the wire, if I go from d to 2d, I cut the strength of the magnetic field in half. So it has a linear dependence on the distance you are from the wire. So it goes as 1 over the distance from the wire. Similarly, if you remember for the electric field from a line of charge, that also fell off linearly with distance. So if we were to draw what these magnetic fields and electric fields look like from a current. So let's say I have a current coming out of the board. So it's coming at you out of the board. Let's just think about the electric field lines from the charges that make up this current. They're going to radiate outward from the wire. So the electric field lines, so they're at ease. And then through the number of technology, the magnetic field lines on there, the hand, they go in circles around the wire. We saw this from an experiment. Let's go back here. Here. So that's what the magnetic field lines look like. And let's do far. There we go. So again, let's kind of look back at this wire picture for a second here. If I'm up here at this distance d, viewing the magnetic field, and I use the right hand rule to figure out what direction the magnetic field points. Remember you do current, that index finger, middle finger points along our vector. Thumb indicates the direction of the magnetic field. If I go out here and I observe the magnetic field from, again, a distance d, but now along the positive z-axis. So now what I have to do to get out to that point is I have to point my finger along i. I have to point my middle finger out toward that point p. And my thumb indicates the direction of the magnetic field. So up here, the magnetic field points out. Over here, it points down. If you go below the wire, it points into the board. If you go behind the wire, it points up. It's circulating. And so the magnetic field around a current carrying wire looks like this. This is a rich set of phenomena. Every time you elect chord on the floor, that thing in principle is emitting electric field lines radially outward. So charges could be accelerated by it along the radii. And it's got a circulating magnetic field around it, such that if a charge was up here and started moving toward the wire, its velocity would be at right angles to the magnetic field, and it would start to bend in the magnetic field. So apart from the various reasons you'd want to shield a lamp cord from being touched. You don't want to electrocute yourself. Shielding like putting plastic around this is also useful for basically dimming the electric field lines that come from a wire and the magnetic field lines that come from a wire. So that there aren't stray electric and magnetic fields all over the room. Yeah? Can you do the right hand rule for the two points? The two points? Yeah. I'm just telling you. Okay, so what I want to do is let me actually, or it doesn't matter where I do this, as long as I get my directions right. So if I put my hand up at this point p and I point my index finger in the direction of I, which is along the positive x axis. How does that always? You point your index finger in the direction? Well, you always point your index finger in the direction of the first vector, and then you point your middle finger in the direction of the second vector, so A cross B equals C. Always. Yeah, so A, B, C. Or in this case, D, L, R hat, B. Okay? So up here, that's where the magnetic field points. Out here, you have to rotate your hand, and this is really painful, I promise you. You still have your index finger pointing in the positive x direction, but now to get from the current element out to this point, you have to follow a line indicated by my middle finger, and so your thumb points down, and that's about as far as I can, as long as I can. All right, so let me construct this. All right, so we're going to do, like, this is where having one of these all pre-built would be really handy. All right, so there's my little coordinate system. Okay? So, D, L vector point, this in the direction of R hat, that indicates the direction of the magnetic field. So if I apply that here, okay, I just have to adjust that back so that it points along R, and then the magnetic field is at right angles to both of them, okay? So I can translate that up here. That's what that vector would look like. And if I go out here, I have to rotate this so the green still indicates the direction of the current, the R vector points out this way, okay, and magnetic field now points down, okay? And then if I rotate that around one more time, I keep going, and this is a circulating magnetic field line, okay? There's actually an easy way to remember. There's another right-hand roll you can bust out. There's always another right-hand roll. You can remember which way the magnetic field lines are circulating around a segment of a wire, so even if you have a really complicated wire, like the one I showed you here, if you just isolate a tiny little piece of it and you want to know where its little magnetic field lines are circulating before they're summed up with all the others, you just take your thumb, you point it in the direction of the current, and the way your fingers bend, they tell you the direction of the magnetic field. So thumb out, current, and look at that, they curl in the direction of P. If the current pointed in, the magnetic field lines would circulate in the opposite direction. So this is another way you can remember current, sense of rotation of the magnetic field. All right, so I already went through that. All right, so we went through this. So the, well, okay, so this is just the wire again. I already went through the right-hand roll. Okay, so something I want to talk about now, what are the consequences of the B.O. Savart Law? Let's go back to the wire again, all right? So the magnetic field from a wire, if, let's focus on the wire A up here. So I have a long wire labeled A. It's carrying a current that way, so I'll point my thumb in it. And if I curl my fingers around my thumb, that tells me the direction that magnetic fields are circulating around this wire due to the moving charge. So the magnetic field lines are circulating out of the, this is sort of a funny angle, around at right angles to the wire. What are those magnetic field lines doing? There's another wire over here, and it too is carrying your current, IB, let's say. So wire B points in that direction, and suddenly it encounters a magnetic field due to the first wire. What happens when you have a current traveling through an external magnetic field? What does the current do? I failed as a teacher. I failed. That's it, they won't pay me this year. External magnetic field, B, caused by something else, we don't care what. I, traveling through the magnetic field along some line L. What's that? Yeah, there's a force due to that external magnetic field on the other current. So if the source of the magnetic field is another wire, which we know from B. O. Savard is absolutely possible, in fact very likely, this wire will feel a force due to the magnetic field from wire A penetrating the current on wire B. All right, so where does that magnetic field point? Well, it's circulating around wire A. So over here, it points down, and that's what's indicated here. B vector magnetic field due to wire A points down. And so if we think about the forces involved, we have a current, a magnetic field I L cross B. And then the resulting, well, I'm trying to do this way, the resulting force points toward the other wire. This wire exerts a force due to its magnetic field on this wire that pulls them together. And if you look at the problem the other way around, well, what about B? Doesn't B cause a force on A? I mean B carries a current. It's got a magnetic field that circulates around it. That's that magnetic field penetrates the conductor A, which is carrying a current I A. It too feels a force, and that force points from A to B. So they pull each other together. When you have currents pointing in the same direction by two neighboring wires, they exert mutual forces on each other that cause the wires to move together. If the currents point in opposite directions, they repel. So this is a bit like, you know, to a north and a south pole next to each other, a north and a north pole next to each other. Has anyone ever been under high voltage power lines before? Anyone ever gone running or hiking in the wilderness around the city, just walked, like maybe by the Cady trail, on the Cady trail, I think there's some high voltage lines that go over that. Has anyone ever done that before? Morgan? Yeah? Or is that just the pen in the air? Okay. Lynn? Yeah? What have you noticed? What have your senses picked up when you're around high voltage power wires? Something really obvious. Yeah, loud crackling sound. Actually, it's a buzzing sound. It sounds some, well, okay. So that's what a high voltage, a high tension wire looks like. All of these conductors are stuffed together in a single bundle, and they're all carrying current in the same direction. And actually, they carry alternating current. So we're not going to cover this in the course, but alternating current is sort of just what it sounds like. For one moment in time, the current points one way, and then another moment in time, the current goes to zero, and then it points the other way. And then it goes back to zero and points the first way. This was, this way, you get a very good transmission of electricity at low loss over long distances, which is important in an electric power transmission system like in the U.S. or any country, okay? So alternating current is just the current goes one way in, let's say, one sixtieth of a second, and it goes the other way, the next sixtieth of the second, and then back and forth, and back and forth, and back and forth, and you still do work with that current. And after all, that's what we want. Alternating current is what powers the lights in here. Alternating current is what comes out of this wall socket here. And you can analyze it. It's a whole chapter in the book on it. We're not doing it, okay? But it is consequences. And that means that for one sixtieth of a second, all the currents in these wires point one way. So what do the wires do when they're at max current going one way? What do they do? They're tracked. Then the current goes to zero. What happens to the wires? They pull apart. Now the current flips the other way. Again, all going in the same direction. What do the wires do? They're tracked. Then they relax. Then they're tracked. They relax. And they're tracked. And they do that about 120 times a second, okay? So if I could do that 120 times a second, this is what you'd hear. Coming out of this little section of trail, come out to this power line that's making like a buzzing noise. Hope you can hear it. It's not the happiest noise I've ever heard a power line make. All right, so the power lines make a little buzzing sound. Like that. That buzzing sound is from the mechanical force of the wires actually colliding with one another in the bundle. Okay, so this is why it's important to, when you design systems like this or if you're working with systems like this, you need to be aware that any high current carrying wire with other wires near it will cause a deflection of the neighboring wires toward the high current carrying one because of its strong magnetic field, okay? Let me bring these slides back up. All right, so there's one more thing I want to show you and this is something I hinted at before but now you can really see it, okay? And that is an MRI machine and what happens if there are strong magnetic fields and high strength current carrying wires. So I'll show you this image first. Okay, give it a second. There we go. So this is, I'll go into this thing next lecture. All right, this thing is called a solenoid and it is nothing more than a bunch of loops of wire placed right next to one another. I'll show you one in class next time. They all carry current I and together they make a big magnetic field in the center of the loops. It's very uniform. In fact, it's almost perfectly uniform if you design your solenoid correctly. Okay, I've got two minutes left here, folks. And this is great for human beings. Once we learned that there's something called nuclear magnetic resonance, that is that you can expose and identify to alternating magnetic fields and get them to sympathetically flip back and forth in the magnetic field. And by doing that, looking at the frequency, you can figure out, oh, that's calcium. That's fluorine. That's carbon. So you can actually identify atoms in a substance, in a material. You can then put a human being inside of a very long, straight, powerful magnetic field, get all of the little electronic and nuclear, most importantly, the nuclear spins all point in the same direction, because they're all little dipole bar magnets. And then you can blast the body with radio frequency waves, which causes the bar magnets to flip in the magnetic field. And the frequency of the flip can be used to image the body. So that's what a computer-aided imaging of the magnetic field of one of these MRIs look like. Note the scale. The red is one Tesla. These are really big magnetic fields. They go up to two Tesla. They blast the magnetic fields in my experiment, the Large Hadron Collider, which we use to steer protons. They go up to like 6, 7, 8 Tesla. They're really powerful magnets. When they explode, it's catastrophic. And that happened in 2008. So here's why you should secure your high-current carrying cables going into your MRI magnet, because there's leaking fields outside of this really powerful solenoid, this really powerful big loop of current. So there's the MRI magnet. This is one of those portable MRIs. And someone's done a crap job of securing the high-current cables that feed into the wires that create the magnetic field in that device. And so it looks like, let me go back and do that again, it looks like they're flapping in a breeze, but they're actually whipping back and forth. I mean, those are really thick cables whipping back and forth almost effortlessly in the stray magnetic fields of the solenoid. All right? Okay, we'll pick this up next time.