 Magnets. How do those freaking things work? We have written records going back to 400 BC or so from cultures such as the Chinese or the Greeks in the ancient world that describe what we now know as the magnetic phenomenon. This is simply an attractive or repulsive force between two seemingly inanimate objects that have no physical contact with one another. It's very easily demonstrated. Many of you have definitely seen this before, you probably rely on magnets all the time in your day-to-day life. These are just craft magnets purchased for cheap from a craft store and depending on which end of the magnet I actually pointed another stack of magnets, I can get an attractive or repulsive force. So for instance, there is attraction and I can get repulsion by aiming just the right end at the magnets themselves. So we see that this phenomenon has an attractive and a repulsive aspect. We've seen this before. Electric charge also has an attractive and a repulsive aspect we've described these aspects using electric fields, electric charges. So is there a magnetic charge? What is the source of magnetism? Well in this lecture video, we're going to explore that and our key thing that we're going to want to figure out is just Magnets. How do they work? And I don't want to talk to a scientist You're my f***ing clients and getting me pissed Solar eclipse and vicious weather 15,000 juggalos together Anyone who tells you these days that magnetism is not understood, that it's a complete mystery to our species is either lying to you or very ignorant. I mean the fact is that we've actually had an excellent working understanding of magnetism going back to well after the Greeks or the Chinese were playing with signals like this that they found in the earth. People were understanding compasses and the motion of the compass needle. They were understanding the behaviors of man-made magnets engineered more like this one well through the 1800s until the very end of the 1800s when thanks to the tireless efforts of people like Carl Friedrich Gauss, Christian Ursted, Bio, Savar, Faraday, famous names now in the physics community, a great deal was not only discovered about the magnetic phenomenon but explained. And in this lecture what I'm going to do is talk a little bit about magnetic fields and what magnets do under the influence of magnetic fields but also what electric charge does in response to the presence of a magnetic field. It's exploring charge and its relationship to magnetic fields and magnetic fields and their relationships to electric charge something we've been looking at so far in the context of the electric field tells us the most about magnetism and its deeper, more fundamental nature, not just arising seemingly out of little inanimate blocks like this but actually having a fundamental connection to subatomic particles like the electron. We're usually exposed to magnetism early on in childhood. For instance, if we go camping or go wandering in the woods behind our house, if we had woods behind our house, we might take a compass with us. A compass is nothing more than a magnet that can respond to the much larger magnetic field of the earth, which emanates from its magnetic poles north and south. And we can use that to move other magnets in our hand and from that figure out the direction of north. Or perhaps we played with things like refrigerator magnets. These for instance will set to surfaces and they can hold notes. They can do all kinds of useful work for us. These days magnets are used in things like the Apple computer power adapter so that you no longer have a mechanical connection of power into your laptop. You have a magnetic connection and that can be easily broken without doing mechanical damage to the plug. It's an attractive feature of magnets that they can connect without physical mechanical connections that have to be made that uses simply the magnetic force. So first what I'd like to do is I'd like to explore the description of the magnetic force, the attraction and the repulsion through the application of the field concept. That is that from magnets there arises magnetic field, so for instance often on a magnet you'll see an N marking a north pole, something like that, and an S marking a south pole. What does that mean? Well in the magnetic analogy to the electric charge, north poles are the source of the magnetic field. It would perhaps emanate out in a loop like this and it would end on a south pole. Much in the same way that in the electric charge you have positive charges that are the sources of electric field, negative charges that are the recipients, the sinks of magnetic field. So north pole, south pole, north is the source of magnetic field, south is the recipient or the sink of magnetic field. Things to keep in mind, definitions that we'll need going forward. Now magnets are actually quite a lot of fun and we can begin to explore the field a little bit and the structure of the field just by taking some of these little craft magnets and playing with them. I've made a stack out of them, you can see that this is just made from a bunch of little magnets all stacked together. And what I can try to do is I can try to bring for instance two south poles next to each other. This would be equivalent to bringing for instance two negative charges next to one another. And we kind of know what happens when you bring two like sign charges near each other, they repel. And very similarly you see that it's very hard for me to bring these together. I'm struggling actually to get them to even physically contact each other and it's almost impossible. I actually can't even get the materials to press against one another. It's a very effective force. Now if I were to flip this around on the other hand, keep one of the poles south for instance and now expose the north pole to the other side, no problem. So you have that very nice crisp attractive force. You can get a very satisfying click out of these little craft magnets. So we clearly can demonstrate the attractive and repulsive aspects of the magnetic phenomenon. And it seems to lend itself very well to a field description. After all we have action at a distance with no physical contact whatsoever. In fact bringing two same poles together will result in zero physical contact in this case. They really don't want to be with one another. So we won't make them go that way. But we can take a look at this a little bit and describe this using the field concept. We can explore the same phenomena that I've been showing you in person using a computer simulation. Now this is one of the so-called FET demonstrators. It can demonstrate physical phenomena, physics principles using a computer simulation. So you see here I have a bar magnet, which is a typical magnet. It's got a north end, which emits magnetic field. It's got a south end that absorbs magnetic field. And this simulation is designed to mimic what we see in the world around us. Whenever we have a magnetic material, it always seems to have a north end and a south end. That is if you bring another magnet near it, there will always be two ends that repel when put together. Flip them around and they'll still repel. Flip only one of them and you'll get an attraction between them. We've never seen in nature what's called a magnetic monopole. A monopole is just a north or a south pole in isolation all by itself with no accompanying other. So a north with no south, a south with no north. We always see things like this in nature. And so the compass is just a little bar magnet and here we have a little bar magnet. As far as we can tell we have bar magnets or bar magnets. And we'll explore a little bit about why this is as we get into something a little bit later in the course. But as far as we can tell, north ends always come with south ends and south ends always come with north ends. And it's unavoidable. So magnetism is already a little bit different than what we are used to thinking about from electricity, where you have positive charge and negative charge and you can separate them and isolate them. One of them we treat as the source of electric field one of them we treat as the sink of electric field. We can visualize the electric field using the show field option here. And so all they've done is at these points in space they've placed teensy weensy little bar magnets. And little bar magnets spin and they respond to the presence of the bar magnet. So you see that the compass end, the red, is the north pole that's attracted to the south pole. And these little magnets out here that are mapping out our magnetic field for us, the little red ends point toward the south and the little white ends point toward the north. So the opposites attract each other. But every one of these little magnets has a north and south end. Because again in nature we just haven't seen mono poles. We haven't seen singular north and south poles. They may be out there but no one's ever managed to observe one. And our understanding of where magnetism arises from doesn't necessarily need magnetic mono poles in order to make sense of the field around us. But it would be actually kind of neat if they existed for reasons I won't get into. But if you're curious about it you can ask me. Now what's interesting about this, the fact that we've never seen a separated north pole from a south pole, is that it means that the minimum magnetic field that you will ever find in nature is our friend the dipole field. Remember the dipole from electricity, a positive end and an equal magnitude but opposite sign, negative end, bound together, emitting the sort of funky field which radiates outward on the ends and circles in the middle back on itself. Well, there it is. There's the dipole field. And that's because the minimum configuration of magnetic fields that we've ever observed in nature are dipole fields. There's always a north end. There's always a south end. So you always have magnetic field coming out on the north cycling back to the south. Now some of it will shoot all the way out the ends and find a south pole on another magnet someplace. Maybe over here you've got this north pole which is now a source of magnetic fields that can point into this south pole over here. But basically that's it dipole fields. And so all the behavior of magnets that we know about are at minimum dipole behaviors. Magnets will orient themselves in the magnetic fields of another magnet. They'll spin in response to the presence of the field. Any attraction that occurs occurs very similar to the way that dipole attraction works between dipoles. It's a non-uniform field. So you have different forces on the ends of the poles and you get attraction as a result, or appulsion as a result. You get acceleration basically. If I flip the polarity around here I can put the north pole closer to the compass over on the left and you see that the south end flips around to meet this. So I can keep doing this and this, flipping it back and forth. This is actually kind of interesting because you've already seen something kind of cool that can happen. You might imagine that if we could actually get this to happen in real life we could get something to spin mechanically in response to a changing flipping of polarity, a switching of the poles north and south. If we're able to do this in the right rhythm I could keep something spinning for a really long time. Look at that. And believe it or not that principle is really super important in the world around us. We'll come back to it later and explore that in more depth. I can do a real demonstration of this magnet and compass thing. So if I take a magnet and I put it next to a compass you see that it flips, the compass needle flips in response to which pole of the magnet I aim at it. And I can keep flipping the magnet and getting the compass needle to spin around as a result. So you can really do this on your own. You just get some refrigerator magnets, just get a compass. It's cheap and you can play around with this a little bit. You see the compass needle just keeps moving wherever I move one pole and then I can keep reorienting the magnet, flip it around, flip the compass needle around as a result. Very easy to do and is the principle of a very important device that we use all the time in the modern world. So I have here just a vessel that's filled with water and ordinary iron filings. So you can just kind of tip it around, see the little iron filings move inside. And if I shake this up to really uniformly distribute the iron filings so that they're nice and uniform inside of the water in the vessel. There we go. Alright now it's all pretty much homogenous inside there. And I can take a little bar magnet and I see it here. It's got a north end, it's got a south end. You'll see it in a minute. Drop it into this vessel and watch. Watch very carefully. As the magnet sits there for a while all the little iron filings which are themselves tiny little bar magnets, each with a north end and a south end, they'll begin to line up just like those compasses in the computer simulation and map out the magnetic field of the bar magnet. You can see that there's one pole and then at the top end you've got the other pole. You can see all the little filings clumping around the poles. The field bends out from the, bows out from the center connecting the north pole and the south pole and then field lines radiate out from the north poles and south poles out into space as well. It's really quite a cool little demonstration easy to do with some iron filings, water and a little bar magnet. You can map out the magnetic field and see that it's a dipole field. Looks exactly like the dipole field we saw from electric charge. But this is what the minimum field of magnetism looks like. This is a device called a cathode ray tube. It's actually the device that a gentleman named J.J. Thompson used to discover the electron, the first subatomic particle ever discovered by a human being. This is a sealed tube it's filled with gas. It's got a metal plate on one end and a metal plate on the other. On the right side the metal plate is known as the cathode. It's the emitter of cathode rays which is what people were studying before they discovered the electron, which is what cathode rays actually are. The other end, the other metal electrode is known as the anode. Now I have this wired up so that the anode connects to ground. That is any charge that is present in the system. If I put an electric potential difference on it the electric potential at the anode end will be zero by definition and I can put a really high potential on the cathode end using something called a tesla coil. This is a handheld device it's kind of nasty as I operate this and you can hear the buzzing sound here in a moment. As I operate this you create ozone and you can smell that sort of sickly sweet ozone smell in the air as I run this. Let's take a listen and hear this thing because it's quite a brute when it's operating. Alright, so I've used the handheld tesla coil to establish a strong electric potential on one end. I've got a zero electric potential on the other so I have a nice electric potential difference and therefore a big electric field inside of this tube. What the cathode ray tube does is the potential is so large it actually strips electrons off of atoms on the cathode and they fly toward the anode under the influence of the electric field. The blue light that you're seeing is actually not the electrons themselves but it's the light given off as the electrons interact with the metal white plate at the back which is just a screen that the light can shine on and the gas molecules more importantly that are present inside of the tube. That blue light traces out the path that the electron beam is taking from the cathode to the anode. If you've never seen electrons before, this is about the closest you're going to get to it. That bright blue light is the result of a beam of electrons streaming from the cathode to the anode and crashing into atoms as they go emitting light after the collisions occur. So let's analyze this. We have electrons leaving the cathode traveling to the anode. They're moving from the right to the left so we could even draw a little velocity vector that represents this. We have a magnet that we can now use to expose these electrically charged particles to a magnetic field and the question is what will happen? When we know what happens if you were to expose these particles to an electric field they would accelerate against the direction of the electric field lines. But what happens when you expose electrically charged particles like electrons to a magnetic field? Let's fire up the cathode ray tube. There we go. There's our nice bright blue light. Here's our magnet and I'm going to take one pole of the magnet and I'm going to aim it perpendicular to the beam. So now the magnetic field lines are passing through the beam perpendicular to the flight direction. What do we observe? I'll flip it around and do it again. The beam doesn't appear to continue on a straight line. It bends. If I flip the magnet around again you'll see here it bends upward and I can hold it there. You see how the beam is bending? It's perpendicular to both the original velocity direction and the magnetic field line sticking out of the end of my magnet. Flip it around again and I can bend it the other way. So electric charge does respond to magnetic fields but not in the way that it responds to electric fields. The force due to magnetic fields on charged particles appears to be perpendicular to both the magnetic field and the velocity of the charge. And we'll explore this now a bit in mathematics. So using this field concept that we've developed that north poles are the source of magnetic fields and south poles are the ends or recipients, the sinks of magnetic fields, we can actually describe the phenomenon we've just observed in the cathode ray tube. And that was a very strange phenomenon. That wasn't quite perhaps what we expected. In electric fields, if you're in electric charge and I drop you into a region of electric field so let's say I put a positive charge in a region of electric field, we kind of know what's going to happen based on our experience, all the exercise and practice we've been doing. We would expect this to feel a force that causes an acceleration for the positive charge in the direction of the electric field lines. But that's not at all what we observed in the case of charge and magnetic field. If we have a magnetic field now. So for instance emanating from a north pole and going to a south pole like this and we put an electric charge with some velocity into this field. So for instance if we were to put the charge going into the board so point a beam of charge moving into the board which I'll denote like this. So this is a velocity vector for for instance for some positive charge that points into the board. Well if I want to figure out the direction that the magnetic force acts on this velocity, I don't have to look at the direction the field is pointing according to the cathode ray tube experiment. We just have to look at right angles to both the velocity and the magnetic field. Well if the velocity is pointing into the board and the magnetic field is pointing in the plane of the board along that plane. The only direction is down, up or down in this plane. That's the only direction that's perpendicular to both of these. And there's a rule. There's a simple rule that you can use to remember which way positive charge will be pushed by a magnetic field when it enters a region with magnetic field. And that's the so called right hand rule. All you have to do is take your fingers in your right hand and point them in the direction that the velocity vector points. Alright so I can take my arm and I can point it into the board like this because that's the direction my beam of particles is going, my positive charge. Then you have to curl the fingers on your right hand curl the fingers on your right hand in the direction of the magnetic field. Okay so I'm going to point my hand into the board and I'm going to curl my fingers in the direction that the magnetic field points which is this way. My thumb, I'm giving it a thumbs up here, my thumb now indicates the direction of the force, the acceleration on the particle. And again it's perpendicular as we saw from the cathode ray tube experiment to both the direction of the velocity vector and the magnetic field being. Alright so we'll just do that. So again we take our hand and we point it into the board. We curl our fingers in the direction that the magnetic field points and our thumb will indicate the direction of acceleration of positive charge under the influence of this magnetic field. And so what we see is that the acceleration okay the force and the acceleration both would point for a positive charge down. We would expect this beam of particles coming in entering the region of magnetic field and we would begin to bend downward. Okay and at every point, at every point as it bends in the magnetic field the force will remain perpendicular to the magnetic field and the velocity. And so this is very similar to centripetal motion, orbital motion. You have a velocity of a satellite, you have a magnetic you have a gravitational field that's acting on the satellite and it's pulling the satellite, it's accelerating it down toward for instance the center of the earth. And so the acceleration of the satellite is not along its direction of motion it's perpendicular to its direction of motion and that's what holds it in orbit around the gravitational, the planetary body. Alright this is a little bit weirder because here you have a force that points at right angles to both the magnetic field okay and the velocity very different from gravity, very different from electricity. And this is something where a lot of people can get hung up alright. So you have to practice this a little bit. Now we have a mathematical description that we use for this and I'll give you the equation and I'll talk a little bit about it. We say that the force due to some magnetic field which we denote B is equal to the charge times the velocity vector and then this is the cross product. Now at this point you should go to Khan Academy, you should go to the video that I have linked from the Physics 1308 homepage. You should look at how to calculate a cross product, they're going to need it. Okay but there are some gross features of the cross product that I won't review here where they come from but I will simply write them down. The magnitude of any cross product between two vectors A and B is simply the magnitudes of the vectors and the sign of the angle between them, theta. Okay so if I have two vectors A and B and they have some angle theta between them, the magnitude of the cross product is the magnitude of A times the magnitude of B times the sign of the angle theta between them. Now if I want to know the direction of A and B, so let's say A cross B equals some new vector C, that's the beauty of the cross product. The dot product gives you a number back. The dot product of two vectors gives you a number. The cross product gives you a vector. The force is a vector so we have two vectors and we need a third vector. The cross product is important. What's the direction of C? Bust out the right hand rule. That's the easiest thing to do. Alright so all you're going to do is you're going to take your fingers, you're going to point them in the direction of A. You're going to curl them in the direction that B is pointing and your thumb will indicate the direction of C. So C here points into the board. Now I'm going to use this symbol, this sort of X symbol, to denote a vector that points into the board. Think of it as the tail feathers of an arrow flying away from you into the board. So vectors can be represented by arrows. This is the tail feathers of the arrow if you will, and it's pointing into the board. If I'm coming out of the board I'll use a circle of the dot in the middle. So this is the arrow head coming at you. So you're looking and on the arrow it's heading right toward your face. That's what that is. So tail feathers going into the board, arrows heading into the board. Arrow head coming out of the board, you see a dot inside of a circle coming right at your face. So this lets us represent three dimensions in two. It's easy to draw the first two vectors, it's a little harder to draw the third one that's going into or out of the board. And this is the shortcut we have to use for that. So this is actually the formula that has been determined by experiment to be the one that gives you the direction of the force acting on a charged particle with velocity V and charge Q under the influence of a magnetic field. A few things we see right away. Keeping this thing in mind the magnitude of the force goes as the magnitude of Q V V sin theta, where sin theta is the angle between V and V. If that angle is zero then the force is zero. That is if you have a charged particle moving along the magnetic field line it will not feel a force from the magnetic field. Again very different from the electric field You have to be moving at some angle with respect to the magnetic field in order to feel the force and that force again points perpendicular to both of these vectors. And this formula will tell you the direction that it points. So you're going to have to bust out I hat, J hat and sometimes K hat to solve these problems. And again I strongly suggest that you review how does one calculate a cross product in two or three dimensions. Well three dimensions is the minimum you're really ever going to get on these from a set of vectors like V and B for instance. Another handy thing we see here, if the charge is negative then the original thing we used for positive charge flips. So let's just review the vector pointing to the right. We had a charge going into the board but now let's make this Q less than zero. Let's make it a negative charge. So in that case you're going to have a flip in the direction of the force. So here's how I like to do it. Take your fingers, point them into the board that's V, curl in the direction of B, everything's good and now you've got the direction of positive charge would move and to find the negative charge flip by a sign. And that's the direction of the force on the negative charge. So that's a little trick I like to use. Your mileage may vary whatever you like whatever you're comfortable with, however you like to remember it, that's what you should do. But velocity, curl toward B, that's the direction of positive charge would move, flip it for a negative charge. That's it. That's all you have to do. Just use the right hand rule and thumbs up or down to figure out your directions. And this will be good for making pictures. This is good for conceptualizing what's going on because these can be very difficult and tricky concepts. Now there's one more thing that I want to motivate for you to help you to set up and solve problems involving magnetic fields. So as I said earlier, it's sort of the gravitational analogy here is that the magnetic field creates a force that is perpendicular to the direction of the velocity vector. So let's just generically look at what the trajectory of a particle would be under that kind of force. So if we imagine that we start off with a particle moving with some velocity V in that direction and we have a force that's acting perpendicular to that. Let's say like this. What that's going to do is in the next instance of time it's going to tip the velocity vector down a little bit. It's going to cause a little acceleration perpendicular to the original velocity vector. And so when you get up here, the velocity vector is not going to change in length, but it's going to have changed in slope. And again at this point in time you've got this force again pointing at a right angle to the velocity vector. And so again at this next moment in time over here the velocity vector has tilted a little bit. And again we have a force that acts at a right angle. And again over here at this next step in time the velocity vector has tilted a little bit and so forth. And so what we see here is we see an object tracing out circular motion from point to point to point in its trajectory with its velocity vector at all times tangent to the trajectory overall. So the velocity vector becomes the tangent to this curve. And it tells you at an instantaneous moment which way the velocity vector points but in the next instance as the force continues to act and bend the particle you get this curved trajectory. This is what magnetic fields do to charge particles. They bend them. And as we'll see a little bit later, this is what makes magnetic fields in conjunction with electric fields so useful. Electric fields accelerate and decelerate electric charges. So if you need to get electric charges up to some high velocity electric fields are the way you're going to do it. You just put up an electric field and positive charges will move with the field lines, negative charges will move against the field lines. But if you want to bend particles, if you want to make them change direction, then use magnetic fields. Use magnets. This thus becomes the optics of the charge particle world. Light can be bent by glass and we'll talk about optics later in the class. You use this all the time if you have glasses or contact lenses, if you've ever used a magnifying glass, a microscope, a telescope any instrument that changes the relationship of light angles from one place to another. You're focusing light, you're steering light. Magnets are the optical instruments of the electrically charged particle world. Charged particles will bend in response to magnetic fields at right angles to them or at any angle to them. And so the key thing that you need to do if you want to bend charged particles to get them to go in a circle or just veer away is you need to put a magnetic field there. And again you have to get the angles right. But that's what they will do. That's what a magnetic field will do. So in this case if we're thinking about a positive charge with some velocity V and we have some magnetic field, we know that the force for instance points this way. Let's imagine that this is immersed in some magnetic field. So this is the force due to some magnetic field and this is some positive charge with velocity V. Well what direction does the magnetic field point? Well the resulting force as pointed down for this. So we know our thumb has to go in this direction. We would have originally pointed our hands in the direction of the velocity vector. So I'll do that. I'll point my fingers on my right hand in the direction of the velocity vector. My thumb is in the direction of the force. And I know that I would have to curl my fingers in the direction that the magnetic field points. I can only curl them this way. I can't curl them back. I can only curl them this way. So that means that the magnetic field here must point out of the board. So we have a magnetic field that points out of the board at all of these points. And if it's a nice uniform magnetic field it will just make this particle continue to bend. If this whole region was just magnetic field that this charge particle had to pass through. So if I imagine that I just have a magnetic field everywhere here pointing out of the board nice and uniform. And if I can get a charged particle into this region with a low enough velocity I could get it to cycle endlessly in this region. I could make a charged particle go in a circle forever. And in fact this is the principle behind things like the large hand drawn collider. You create a strong enough magnetic field protons will bend in that magnetic field and if you can make it strong enough you can make them go in a circle forever. And so that's why the large hand drawn collider is a big circle because we can make magnets that are strong enough that you can take a whole bunch of protons and you can get them to bend forever in a circle that might have a huge radius. This could in principle be a very huge radius for this circle. Alright well this looks just like orbital motion. And so we know that centripetal, the centripetal force from semester one physics goes as mv squared over r that is if I have an orbital circular path with radius r and I have a velocity vector v like this and I'm just continuing to go in this circle. The centripetal force the one which appears to hold you in place in the circular motion has a magnitude that goes as mv squared over r mass times the velocity squared divided by the radius of curvature. Well if this is due to a magnetic field we know that its magnitude is qdb sine theta where the theta is the angle between the magnetic field and the velocity vector. More to the point between the velocity vector and the magnetic field. Okay and we just have the charge and the velocity and the magnetic field magnitudes out here. We can relate these two when you've got circular motion. mv squared over r equals qdb sine theta. And this is very helpful if you have a problem involving a force that's holding a particle going in a circular path even for a little ways it doesn't have to be a perfect circle it could be part of the way as a circular trajectory during the time it's executing circular motion this rule will apply though. Centripetal force will be equal to the magnetic force they'll balance each other out and keep the particle moving in that circle. Again as long as you're immersed in the magnetic field the minute you leave the magnetic field you're going to execute linear motion again because you're not under the influence of a force anymore. Alright so this is some helpful things that will help you to get started thinking a little bit about magnetic fields and charge particles moving through them. What we've done is we've explored magnets moving in each other's magnetic fields we've seen that the simplest field is the dipole field. There appeared to be no evidence for what are called magnetic monopoles in nature that is singular magnetic charges that are the source or sink of magnetic fields. What we found instead is there's a complex relationship between electric charge and magnetic fields. We're going to explore that relationship going forward as we look at the B.O. Savart law which is the law that tells us how electric charge and magnetic field actually relate to one another in a very complex way. We've explored a little bit about the motion of electrically charged particles in the presence of magnetic fields. We've seen that this involves motion, magnetic field and force all the angles to one another and to solve these problems you need to understand the cross product review that. And we've seen that in circular motion you can relate this intripetal force to the magnetic force through a not so bad equation V squared over R equals QVB sine theta. So if you know some of these things you can solve for others. If you knew the momentum or the kinetic energy you could figure out things involving V's and M's and so forth. You get the idea you're going to have some problem solving to do here using old ideas and new ideas from magnetic fields. So I hope this has been an entertaining and interesting introduction to this material and we'll pick this up next time with the B.O. Savart law.