What I'm going to do is start filling in details from the video lecture, going into things like notation for magnetic fields, the units of magnetic field strength, which I didn't cover in the lecture. The lecture was meant to be fun and punchy. How many people watched it? And how many people can identify the obnoxious band featured in the third minute of the video? That's sad. James, really, truly sad that you know who they are. Okay. Who are they? Yes. Which means? Insane Clown Posse? Yes, Insane Clown Posse. They have a very famous video called Miracles, where they talk a bunch of things that they consider miraculous and inexplicable. Most of them have actually been explained 200 years ago. So there's actually a whole video of, it's called, Science for Juggalos. Juggalos are supposed to be the fans of the Insane Clown Posse. And there's a bunch of people that dressed up as Juggalos went to a concert and did science demonstrations and explanations outside the concert so that their fans would not continue to be so ignorant, which I thought was amusing. So, all right. So just to give people a few moments here to show up, let me do a little boilerplate stuff on magnetic fields. So magnetic fields, you know, I'm not supposed to do this. I'm not supposed to verbally prime you like this as a teacher. This is the stupidest thing I can do. Magnetic fields are hard. They are among the hardest subject that you're going to encounter in introductory physics. Sorry, Tina. That doesn't mean I don't think you can understand them. I think you can understand them, no problem. I just want to prepare you for the fact that you're going to have to maybe crank up the brain power to 11 for this part of the course. I promise you a little bit of relaxation when we get into optics. Okay. So I think I've actually convinced myself that I'm going to skip Ampere's Law so that I can spend a little bit more time on optics at the end of the course. I do a special lecture on the human eye and things like that. Okay. So magnetic fields. What have we encountered in the class so far? So far, we have seen that there is this new thing deep down inside of materials, electric charge. The modern theory of the atom that explains the structure of matter as we know it now, it's made millions of predictions. They've turned out to be true. We built technology off the stuff, blah, blah, blah, very successful theory of nature. Okay. Electric charge is at the heart of that. And this causes an electric force to arise. And we describe this with a force field, which we denote as e-vector. So positive charge by convention is the source of electric field. It emanates out from positive charges and it goes into negative charges. What we're encountering now is a seemingly new field of force, although that is a deceptive thing. It seems very different. But as we explore this today and in the next lecture, you'll see that there's a deep connection to the electric force. There's a deep connection to the electric field. That connection is actually so profound that it revolutionized our understanding of light. It led to the development of the first things which are now known as field theories. So pure mathematical frameworks that describe interactions only by using fields. And that led to the revolution in the early 1900s in quantum physics and relativity merging together, which now has given us all of the technology we take for granted today. So the fact that in the 1800s, scientists were struggling to understand the magnetic force, magnetism, that very basic bit of physics. Why does this happen? That's it. That's the question. Why does that happen? And why if I flip the pole around do they not want to be together? All right. So I wasn't exaggerating in that video. Here, you can play with those. Pass these around. Be careful with these. Don't degauss your credit cards. Don't put them by things with magnetic strips on them like credit cards. Go ahead and try to push them together or snap them, slip them around, snap them together. They join real easy. Just trying to explain this basic thing led to a complete revolution in our understanding of nature. So rocks in the earth that attract other rocks in the earth. That's weird. I wonder why. And it took a long time. It took almost 2000 years to figure that out from the first observations, first recorded observations of magnetism in the 400, 500, 600 BC to the 1800s to really come to understand that. So we have a new field of force. That force field has a symbol associated with it, B vector. B vector will now be our new friend. I'm going to learn to love it. So B vector stands for magnetic field. Okay? All right. So, yeah. How do we get B for magnetic field? I have no idea. You think it'd be M vector or something like that, but it's not. We could go look that up. Let's have a history project for anyone who's Googling during lecture and ignoring me. Go figure out why it's called B. And on that note, and then you can share it with the class. Because somebody figures it out, I'll give you five points extra credit on the next homework. Whoever gets it first has that sound. So if you have, does everyone have devices? Does anyone in here not have a computer or a mobile phone so that this is a fair extra credit challenge? No? What's that phase four, Jasmine? Like the fastest you can look it up there? Yeah. Okay. Let's see how good you folks are with the web. And while you're doing that, I will hand out the quick speed test for the information age. If you tell me why it's B, then I'll want to do something for that. Okay. And why? Why B? Okay, because that's nice and all. But why? Because James, Clark, and Maxwell, none of those words have B in it. So that's weird. Okay. So where did A come from then? Electromagnetic momentum at a point. Okay. Yeah, that doesn't tell me why it's B though. I think Arianna's got it. Magnetic conduction B and then everything else we'll find out after that. He was able to alphabetically, so total concurrency, displays no B. Okay. So purely coincidental, the second thing he named happened to folks come inside of the second letter of the alphabet. There you go. Yeah, no one uses C vector for anything anymore. Okay. So laptops and notebooks and all that stuff away. Go ahead and start on the quiz and we'll get going at 943 and you get five points extra credit on the homework. So we'll do other opportunities. There's going to be an engaged exercise in class probably on Tuesday of next week. There'll be lots of opportunity to get some extra. B, because it was the second letter in the alphabet. Don't you love it? That's how science gets done. Nothing fancy. It's not named after anybody. We're going to encounter this guy named Jean-Beth Pico. I had a misconception that maybe it was because of him. But I looked that up at one point and it had nothing to do with him. I knew it was Maxwell, but I didn't know why, so I'm sure how lazy I am. So thank you, Ariane. Okay. Well, so the origin of the units for magnetic field will be eliminated a bit better in a moment when we look at its effects on charged particles in detail, that equation that I wrote in the lecture video and then looked at a little bit. But the units, we have a new unit for this, which is the Tesla, just the letter T. There is, well, there are multiple conventions for the units of the magnetic field. The Earth's magnetic field is often measured in something called Gauss. So Tesla and Gauss are famous physicists slash engineers. So this is G. And there's a relationship between these. So 10 for the minus 4, Tesla equals one Gauss. So Gauss is a good unit if you're dealing with weaker magnetic fields. And believe it or not, here it's magnetic field while it's quite useful, is not really that strong. It's strong enough that it's affected the behavior of life on Earth. It actually protects us. Right now, there was a recent headline about a massive sunspot on the sun that was rotating toward Earth. And this all happened around the time of that partial eclipse last week. So you can actually see the sunspot on the surface of the sun. It was so large if you protected your eyes and then viewed it through a filtered telescope or something like that. The concern was sunspots, which represent essentially magnetic storms on the surface of the sun. If they belch out a whole bunch of charged particles during the storm, those charged particles can be ejected straight at the Earth. This is something known as the solar wind. It's happening all the time. We're always being bombarded by charged particles from the sun. But during periods of increased solar storm activity, if you get a big burst of solar radiation like that, you can get a big pulse of electromagnetic energy in the Earth's atmosphere. The electrically charged particles can disturb satellites that we use for GPS, so locationing on the ground, telecommunications. There's always this big fear that if there's a big enough solar storm it'll knock out global telecom basically because we rely now on satellites in orbit around the Earth to do a lot of our telecommunications and things like that. Those storms are immense and we're protected by the Earth's magnetic field, which thanks to the force that it exerts on electrically charged particles, largely funnels electrically charged particles to the poles of the Earth, so the North Magnetic Pole and the South Magnetic Pole, which I'll show you in a picture in a bit. What's the phenomenon that results from all of those electrically charged particles being dumped into the North and South polar regions? Anyone now? Aurora's. Yeah, so those beautiful light shows in the sky, that's due to solar wind being funneled by our magnetic field into the Earth's atmosphere and causing various forms of light radiation to be emitted as they travel through the atmosphere collide with nitrogen molecules in the air and things like that. Really quite lovely, but indicative of solar storm activity, so very strong storms or changes in the Earth's magnetic field and the auroras can move, they can grow in size. All right, so Tesla, Gauss, the Earth's magnetic field tends to be talked about in Gauss, but for the most part we'll talk about Tesla, so we'll stick with that as our unit of choice for all of this. Now the funny thing about magnetic fields is that when you're talking about magnetic fields acting on magnets, they're quite easy to understand. So, you know, I demonstrated this phenomenon in the video, you can demonstrate it to yourself with these craft magnets. Let me kill the lights here for a second. So this is just a single magnet, a bar magnet, all right, so-called because it's shaped like a bar. By convention the North Pole, which I'll define in a moment, but the North Pole is supposed to be the emitter of the magnetic field and the South Pole labeled S down here, is supposed to be the recipient or sink of magnetic field.  They're suspended in water and then you put a bar magnet on top of the container holding the water and iron filings and the little iron filings are little bar magnets and what they'll do is that their North Poles will align to point at the South Pole here and their South Poles will align along the magnetic field lines that point back to the North Pole. So magnets in magnetic fields, they behave just like charges and electric fields. Because the minimum magnetic field that we've ever seen in nature, even going down to something the size of an electron, has been a dipole field, as far as we know there are no single North and single South Poles in the universe. We don't know that that's true, we've been looking for a long time actually. There are still people now that look for the production of what are called magnetic monopoles, so a single North or a single South. For instance at my experiment, the Atlas experiment. So there's a professor here, Professor Farron, and she did her PhD thesis on searches for magnetic monopoles using the very first Atlas data. So there are people that are making their careers off of looking for these things, not finding them tells you something about the probability of finding them anywhere in the cosmos. Alright so even if you don't make a discovery, it lets you rule out a whole bunch of ideas that said, oh you should have seen magnetic monopoles by now if my idea is correct. Okay so that's science in progress. So as far as we know, this is the minimum configuration magnetic object in the universe, a dipole. And that already makes magnetism a little bit harder to deal with. I mean dipoles were tricky, right? Positive charge, negative charge, looping electric field looks just like this between the positive and negative charges. Same configuration here for North and South Poles. What you just have to remember is that just like dipole, electric dipoles in an electric field, they'll rotate to line up along the electric field line. And if the electric field is non-uniform, an electric dipole will accelerate in the field because the two ends of the dipole will feel a slightly different force. So think of the water molecules that I dripped out of here and then used a charged plastic rod to attract. So the water is a dipole. If that field were uniform, the little water molecules would just line up in the field and that would be it. But it's a highly non-uniform field coming off that plastic pipe. It radiates out, it weakens with distance. And so the ends of the dipole closer to the plastic rod feel a slightly stronger force than the ones away and they accelerate in addition to rotating. So magnetic particles will do the same thing in a magnetic field. These little iron filings will line up their north and south ends along the magnetic field lines and they will begin to accelerate. If I leave this alone for long enough, eventually all the iron filings will clump as tightly as they can around the north and south poles and there'll be some layered in the middle as well. So if I leave that little container I showed you in the video sitting for long enough, eventually they'll come back and it'll look like little iron filing hair is grown on the little hole in the center of the container where the magnet is placed. So it's tricky and this picture will help again remind you of the conventions. The north pole, which I'll define for you in just a moment, the north pole by convention is the emitter of magnetic field lines, the south pole is the recipient, okay here come the lights, so brace yourselves. And just like any other vector field, if you want to know the total magnetic field in a particular region, all you have to do is sum up from i equals 1 to n, all the individual magnetic fields bi that are there. So you just do the vector sum, just like Coulomb's law summing up the electric fields from each individual bar magnet, in this case you sum up the magnetic fields at one point in space, you get the total magnetic field. So yeah. Yeah, yeah actually it's pretty, it's pretty wild. If you take a bar magnet, I did this, I tried to do this as a kid because I thought oh I can isolate the north end and the south end. Yes, this is what I wasted my time on. You slice the bar magnet and it will now break, this end will be the southern end and then you'll have a northern end here. So what you have to do is think of bar magnets, they appear to be made and we'll explore the reason why this is after we get through the B. O. Savart law. They appear to be made of a whole bunch, Avogadro's number for instance, of little bar magnets and all their little magnetic fields add up, okay. So if you think of material as being made of Avogadro's number of little bar magnets, many of which are lined up in a particular direction so that you get a net effect from them that doesn't cancel out, that's really what a magnet is. It's got some amount of what's called polarization, that is the atoms for instance inside the material, maybe 10% of them or 1% of them are all pointing in the same direction and that's enough to give you a net magnetic field that the bar will then emit, okay. Rare earth magnets have extreme polarizations, very strong magnetic fields, those can be as strong as order of like half a Tesla, which is a huge magnetic field. MRI magnets are about one Tesla, one and a half Tesla, so it's a big magnetic field, okay.  So you can snip a bar magnet as many times as you like until you get down to the atoms, actually until you get down to the electrons and you'll still have a north end and a south end. And there's a very fundamental reason for this that we simply don't have time to cover in this course, but I'll preview it and that is that all subatomic particles, electrons are a good example, carry an internal number that identifies them as a matter particle and this is known as spin. So how many of you have heard of spin before, for instance in the context of chemistry and orbital filling, shell filling, okay, how many of you have not heard of spin before? Okay, so spin is just, you can think of it as a little bit of angular momentum, picture it as if the electron is actually spinning like a top and as you'll see that means that there's electric charge moving in a circle and that has consequences, that actually creates magnetic fields.  So as far as we know you can't slice an electron in half, no one's ever seen anything rattling around inside of an electron. So as far as we know, the smallest, most fundamental magnetic field is still a dipole field because the electrons, for instance, which are largely responsible for the magnetic fields and materials, they themselves are bar magnets and they appear to be indivisible. All right, so to understand spin, one has to understand quantum mechanics and that's outside the scope of this course, but I will preview a little bit of that at the end of the course and the last third of the course, okay. All right, so north and south poles, how do we know what's a north end and what's a south end on a magnet? The simple answer is, and for this what I will do is drag this over here, okay. The simple answer is that whatever end of a bar magnet points toward the north of the earth is the north pole. That's it. It's really not that much more complicated. All right, so for instance, here I have a compass, okay, and all I need to do to figure out where north is is get that end on the compass to line up with the painted black end of the bar magnet.  It points north, but in order to point north, what must be the actual magnetic pole at the north end of the earth? South. Yeah, so it's a little weird. The convention is that wherever the tip of a bar magnet points, if it's free to rotate like this, that indicates that with it points to the north pole of the earth, it's a north pole magnet, magnetically speaking, okay. That's where the convention comes from. But from physics, we know that that means that the north end of the earth must be a south pole. It must be a sink for magnetic field and not the other way around. It's not a source of magnetic field. So the convention's a little screwy in that sense. Yes? Is that the case? The one that points geographically north? It's technically the south end. Well, the north, here I'll show you on this slide right here. Give me one second. So it's illustrated over here. So this big, let me close the turn off lights again, okay. This green circle is supposed to be the earth, okay. And if you are standing, let's say, in the northern hemisphere where we are, and you hold the compass in your hand, the end of the magnet that points geographically toward the north pole is called the north end of the magnet. But in reality, that means that the field lines must be going into the north end of the earth. So it's a magnetic south pole, even though it's a geographic north pole. That's what often confuses people. So magnetic north is really a magnetic south pole. It's the... Geographic north is magnetic south. Geographic north is really, on a bar magnet, it would be a south pole. Yeah, yeah. Does the same go for south to geographic south is only magnetic north? Yes. Yes, exactly. Yeah, it's symmetric, right. So now the other thing to note is that the actual pole, which is the axis of rotation of the earth, okay, is offset slightly from where the magnetic pole is. So if you were to imagine the earth as a bar magnet, the long end of that bar magnet does not end at the tip of rotation of the earth, okay. So I've mentioned this to one or two of you before, but there's a very interesting phenomenon that happens with the earth's magnetic field. It appears, the geographic evidence is that it appears to flip periodically for reasons that are still not completely understood. There's a whole area of physics and geophysics that's attempting to understand not only the origin of the geomagnetic field, but also its dynamics. So what causes it, for instance, to flip? And the period of the flip is on the order of tens of thousands or hundreds of thousands of years. So our species has been around on the earth in our present form. If you were to go back in time and say, I still, you know, look at human beings and say, oh, I recognize them. They look like us. That's about 200,000 years on this earth. Prior to that are our ancestors who are a bit different from us in appearance, okay. Our species has experienced at least one pole flip and we survived. Now, why would a pole flip be dangerous? Well, remember I mentioned the solar wind. The solar wind is a source of extreme radiation. Radiation can cause mutation through the creation of free radicals and interference in the DNA replication process. So mutation, all right. So you can get mutations. This is happening all the time in your body. A random cosmic ray particle from the solar wind is smashing into atoms in your body and ionizing them and creating free radicals and that can interfere with DNA. And every now and then this can cause a mutation, which then usually gets fixed by the body. So the body has lots of error checking mechanisms in place to keep those mutations from running wild, okay. It's a very complex system. But if you increase the amount of radiation, you increase the mutation rate. And so the body has to compensate for that. And if you reduce the magnetic shield, you would increase the amount of radiation raining down on the tropics, for instance. The lower sides of these hemispheres, which are largely protected by the earth's magnetic field, would become exposed to the solar wind. So there's a lot of concern about what would happen when the poles flip. And they will flip at some point. Our species, assuming we don't kill ourselves off first, is very likely to experience a flip of the magnetic poles at some point in the next thousand or few thousand years. The good news is that as the study of this is developed, it seems like these flips are very short, maybe less than a century, is now the belief based on evidence that maybe these take less than a century to occur. So we would just have to make sure that we understand the consequences for the electrical power grid, telecommunications, and also cancer rates, for instance. And during the time when we're not as protected or protected at all by the by the geomagnetic shield that otherwise keeps us from getting radiated all the time. Now our species evolved in radiation. So we can handle radiation. Lots of species evolved in radiation. Radiation is a driving force and mutation. We need it. But it's just changes, right? Any change in the system can cause pressure from natural selection on the system. The question is, what will we do about it when it happens? So that's a bit of physics that affects a bit of biology, which I like very much. Okay, so I mentioned this before, terrestrial magnetism, like these little guys here. Refrigerator magnets typically have a strength of .01 Tesla. So this is a refrigerator magnet. Okay, and you can stick notes to refrigerators or other magnetized surfaces. Okay, so chalkboards are good for this. I've got a whole optics demo in the back that I'll take out for the optics part of the class that sticks with magnets to the blackboard. So .01 Tesla or about 100 Gauss. Rare earth magnets, however, which are very common these days, they're used, for instance, in the Apple computer magnetic power connector. Those are rare earth and they can get upwards of a half to even up to one Tesla. They're, they're brutes. You do not want to have your hand between two rare earth magnets if they're separated because if they start to accelerate they can crush bone, pinch skin, cut off blood vessels and things like that. It's very dangerous actually. So you want to be very careful if you're ever handling rare earth like niobium magnets. They're great, they're lots of fun, but they're also extremely dangerous if you're not careful with them because if they find another magnet or a material to which they can be attracted magnetically and they begin to accelerate that can happen quite fast. Big B equals a big magnetic acceleration to another magnetic material. So roughly speaking, and this is a number I want you to kind of keep in mind, the scale of terrestrial magnetism that is typical magnets that come from the earth, you dig out of the ground is sort of of the order of like a Tesla. .01 Tesla, a micro Tesla, up to about a Tesla. It ranges, but it's somewhere in those three orders of magnitude. All right, so just kind of keep that in mind. That will be a useful thing in a bit. All right, so let's turn to magnetic force on charge. So what do we know so far about magnetism? Well from the video and from the things I've shown you today, we know that magnets attract magnets and they do so just like charges attracting or repelling charges. So if you have a magnetic material, it can be attracted directly or repelled directly by a line of force along the direction of motion, another magnetic material. And this is just dipoles interacting with dipoles. That's what this is. Electric charges, however, interestingly are also affected on their own by magnetic fields and I demonstrated that in the video. I'll show us still from it in a moment. I demonstrated that in the video by taking a cathode ray tube where you can hook that thing up to a high voltage power supply like the Tesla coil I was using to fuel that thing, create a big electric potential difference, strip electrons off the cathode, the little metal plate on the right side in the picture, I'll show it to you in a moment, accelerate them across the potential. They collide with gas molecules in the sealed tube and that causes light. That light traces out the path of the electrons and that's about the closest you'll ever come to seeing an electron in your life is that light. It's an indirect detection but nonetheless it lets you see the path of the beam. You can then expose that beam, we know it's velocity, it's going from right to left. You can expose the beam to a magnetic field and if you've established which is the north end of the magnet and which is the south end by exposing it to the earth's geomagnetic field first then you can establish what direction the magnetic field points with relation to the velocity of the charge and what we found was an astounding thing that charge doesn't accelerate along the magnetic field lines like it does along electric field lines. It accelerates perpendicular to both its original velocity direction and the lines of magnetic field. Alright so let me see if I have the illustration of this here. Right great so this is just a cartoon to illustrate what I just said. If I'm a positive charge and somebody tosses me with some velocity v and some direction, you know v vector, into a region of magnetic field, I will not accelerate along the magnetic field, I will accelerate perpendicular to both the field and my original direction. And this is what makes magnetism at first seems so tricky. We are now unable to avoid describing this using anything other than the cross product, which can seem really terrifying at first but it's not fell for a chemistry prank. I was told this was fixed by the previous speaker. Alright so cross product. The beauty of the cross product mathematically, well let's do this one, A cross B equals C. The dot product of two vectors A dot B equals C. Okay returns a number. Okay so this the dot product it's great when you need a scalar that is just a number. The cross product is essential when you need a vector. Force is a vector. The force that results from a charged particle passing through a magnetic field just like the velocity of the charged particle has direction and magnitude just like the magnetic field has direction and magnitude. The force has direction and magnitude and we are unable to avoid writing down something like this. This means proportional two.  It's a mathematical symbol for proportional two. V cross B. And I illustrated that last time with the cathode ray two. Alright so let me kill the lights here again. So there's the tube. The beam is that bright blue light you see there. It's already being affected by the magnetic field but originally it pointed along the arrow indicated by this blue arrow here. So the blue arrow indicates the original undeflected electron velocity. I then take a magnet and I aim it at the cathode ray tube and the beam diverts from its original trajectory. It bends like this. That means that the force due to the magnetic field originally points down and deflects the electrons from their original path and by the right hand rule and because electrons carry negative charge you can work back from this and figure out that the south end of the magnet is pointed at the beam. So let me walk through this. Alright so from the experiment what we know is that velocity points to the left magnetic field either points into or out of the picture and the force points down perpendicular to both of those original directions. So whatever the relationship is between V and B that gives you F that force it's a cross product. It's unavoidable. You have no choice but to write down something like this and that's because what the cross product gives you is if we go back to A, B and C and if I have A cross B equals C C is perpendicular to A and C is perpendicular to B. So the cross product yields a vector that is by construction perpendicular to both of the original vectors. So if you were to then do something like this C dot A it would be, what's that? Yes it would be a scalar what would its value be? B, no. Now we're just throwing out random letters here just like Maxwell this is terrible. We'll go this way if I if I take two vectors that are at right angles to one another and I take their dot product the magnitude of that would be like C A cosine theta and theta is the angle between them which is what when they're perpendicular from one another 90. So what's the cosine of 90 degrees? Zero. So the dot product of two vectors that are perpendicular to one another is always zero. So this is nice because if I ask something like what's the dot product of the force and the velocity in this equation without hesitation you can just say it's zero because the relationship between let's say the magnetic force FB FB and the velocity is that at the end of the day that F vector is totally perpendicular to both V and B. So V and B can point like this okay and the force might point actually to point this way so it's a V B okay the force will point down I can close that angle a little bit it'll still point down what happens when they're parallel to each other what happens when V and B are parallel to each other what's the magnitude of the force then zero and why is that because the magnitude of the cross product is equal to AB sine theta and if you have a zero angle between A and B sine of zero is B weighted dramatically oh come on commit I heard what is that come on don't be afraid it's just a number just choose a number between zero and infinity yeah zero okay so this this if um if well so if theta equals zero let's make that a little bit less crappy there we go the magnitude of A cross B is by definition zero they don't in other words if they don't cross there's no cross product that's an easy way to remember it right no crossing no cross product any crossing cross products okay and wherever that vector points it will be perpendicular to the original two okay all right so to review because of that happening in nature we are forced to use mathematics which is the language we use to describe nature we are forced to use a cross product to relate the magnetic force on a charged particle to the velocity of the charged particle and that external magnetic field acting on the charged particle if I could have found a way and I did try this but I electrocuted myself twice uh if I could have found a way to get that magnet to point right along the axis of the beam uh we would have seen nothing happen but what I found out was because the the tesla coils hooked up to that end anytime I attempted to bring the magnet close to the electrode a spark would jump and I I would I would it was painful but that's a low current device but it's a high voltage devices it's thousands of volts that are plugged into the end of that thing it stings a little when you get hit by a spark from it so um I really couldn't without threatening my life uh get the magnet to line up close enough to the beam so that we could see that it has no effect on the on the direction of the beam all right so for that one unfortunately you'll have to take the word of all the French and German and other European scientists that played around with this stuff in the 1800s and 1700s uh then in fact if you put the magnetic field right along the direction of the of the the moving charge uh it has no deflective power whatsoever okay so we're forced to use a cross product a few properties of the cross product again just to review the resulting vector is always perpendicular to both of the original two vectors such that if you were to take a dot product like between fb and v you'd get zero or you know f this is also zero fb and b okay it doesn't necessarily have to be true that uh v dot v does not have to equal zero okay let me turn this light on here we go okay it doesn't necessarily have to be true that v dot v is zero they can have uh one of them could have a component that lies along the other all that matters is is that the opening angle between v and b is not zero okay if it is zero the cross product is exactly zero that means physically there's no force okay so if the angle between v and b is zero there is no force and the electrons just continue on their merry way unaffected by the magnetic field so a couple of other useful things about the cross product that will be very helpful i have a question yes um is there a specific name for the proportionality meaning oh yes there is well i'll get to that in just a second it's actually it's electric charge it turns out so actually let me just write that down oh you for that symbol you mean no no just like for that the constant of proportionality yeah it turns out it's electric charge experiment revealed that the constant of proportionality is just the number of coulombs of charge being exposed to the to the magnetic field so to write that formula in its exact form for a let's say for a point charge it's qv cross b all right now so it's not to worry people too much here the cross product is is like any other arithmetic multiplicative thing okay so um i could write it this way this is just exactly the same mathematical expression okay i could write it this way those are all equivalent i'm just distributing i'm putting the q in a different place but v and b are multiplied through the cross product q is just a number positive or negative it's just a number and i can associate it with b i can associate it with b i can pull it out in front of the whole thing like this okay these are all the same so these are all equivalent all right and so if that's a little scary just go look at some con Academy videos on this stuff they're they'll walk you through it okay some nice things all right let me draw some coordinate axes here so here's an x axis here's a y axis here's a z axis we're going to start dealing now more directly with the fact that we live in three spatial dimensions we've been avoiding this for most of the course so far but now we really don't have a choice we really have to acknowledge that there are three spatial dimensions that we have to worry about we use unit vectors to indicate direction along each of these i hat j hat k hat okay i have for the x direction j hat for the y direction k hat for the z direction or z if you're from Canada or Europe okay cross products cross products will yield other coordinate axes so as an example let's look at i hat cross j hat and this will come in handy if you have been given for instance on the homework uh some vectors involving i hat j hats and k hats and you have to take cross products this will help you to attack those problems piecewise okay so let's just consider the cross product of i hat and j hat now we're going to practice the right hand rule all right so everybody get their right hands up if you have one all right what you're going to do is we're going to point our fingers in the direction the i hat points so just pick a direction okay here it would be like that we're going to curl our fingers in the direction of y so that's the direction of j hat and our thumb will indicate the direction of the resulting vector all right so if i take a uh see here if i have my x y okay so point in the direction of x curl toward the direction of y thumb indicates the direction uh of the cross product points out of the page coincides exactly with the positive z direction so this is positive z i hat cross j hat yields k hat a vector pointing a unit vector specifically pointing along the direction positive on the z axis this is a really nice formula to remember because if you remember this i cross j equals k x cross y equals z okay you can derive all the cross products involving unit vectors to get the ones that are positive that look just like this and don't have a minus sign floating around in front of them think of this equation like a conveyor belt if we were to have k hat fall off the end of the conveyor belt and go back to the beginning we could get an equation that looks like this leave the operations in the same place that is also a positive relationship between these axes so to figure that to see if this is true point our fingers in the direction of k hat curl in the direction of x our thumb should point in the direction of y and it does okay so you point your finger in the direction of the first one you curl toward the direction of the second one your thumb indicates the direction of the third one another way to do this is you can do this other this i like this right hand rule i don't know why but there's this one as well okay so x y z all right so all you have to do is point your finger in the direction point your index finger in the direction of the first vector okay take your middle finger point it in the direction of the second vector your thumb indicates the direction of the third vector from the cross product that's another right hand rule there's like 50 of these things okay be amazed what you can do with just one right hand okay so let's do this again so we kind of if you think of these mathematical operations of just pieces of a conveyor belt but we can slide the symbols through we can have the j hat follow off the end and come back to the beginning of the conveyor belt and we get the last equation so if you ever have a cross product between two scary looking and arbitrary vectors remember that if you can write down let's you know let me give you an example here let's say a equals uh five i hat plus six j hat and b equals uh two i hat plus uh three k hat okay and i'm told to calculate a cross b well the first thing you can do with this is you can distribute the terms in the cross product so we have five i plus six j cross with two i plus three k okay so we'll just distribute we'll write this as a long equation five i cross two i plus five i cross three k plus six j cross two i plus six j cross three k okay so i just wrote this whole thing out and i don't know why i'm putting there now we group them okay so i just distributed each of the terms in the multiplication we got that long equation well a couple of useful things right away anytime you have the cross product of a vector with itself you are taking the cross product of a vector that points in the same direction as a vector and you get zero and gone done that was easy okay next term i cross k so let me let me isolate this term up here we've got five i hat cross three k hat remember i told you it doesn't matter how you write the numbers okay this is equivalent to 15 i hat cross k hat just five times three and pull them out in front okay that's totally legit yes i knew i was going to get a question when you went from a cross b equals the five you only have the two terms and they were added in parentheses and they were crossed between the two that's right on the next line cross is just another multiplication symbol right but then the cross was on the inside and the pluses are between why do we switch signs what do you mean switch signs the next line oh okay so so well let me put it this way here this this will look a little bit this let me give you a put them together like foil yes yes yeah okay yeah it's just first first out whatever the word is that foil first out or last i'm just thinking of like crazy physics stuff and i was confused no no this is just crazy math stuff you can play with my colleagues in the math department for this one okay yeah can you also do matrices if you are more comfortable doing matrices do the matrix stuff that's totally personal i'm i'm just demonstrating some of the very basics here so that people can make some progress on the homework you should do the same thing yes okay yeah yeah matrices can make this process simpler but not everybody is comfortable with them if you are comfortable if you're not then you don't have to and when you said you crossed out the five i two i because you said the vector was crossed for itself is that because you have an i cross i exactly yeah so you could rewrite this as 10 times i cross i and if you ever have the vector cross itself zero that's that so those terms are a gift you just can't cross them right out okay all right well we have in this term here five i cross three k five i cross three k which we can write as 15 i cross k we have a term that looks sort of like this except this is k hat cross i hat equals j the rule of thumb is if you want to swap the order of the cross product you just have to introduce a minus sign so k hat oops i hat cross k hat equals negative j hat so if you can remember this and you can play the conveyor belt game to get the other two then if you want to swap any of these two and make another equation just introduce a negative sign that's it okay so this is equal to 15 of equal to negative 15 j hat and you have one of your terms in the cross product done two to go okay so rinse and repeat so do the cross product of j and i that's great because that's just k hat right there and then uh j and k well let's see uh j and k thank you yes right because j goes here k shifts over one i shifts over one and i can't even do to compare okay great uh yeah okay so then we have j cross k and that's just going to give you something along i and if all you're told to do at the end of the day is just write this as a resulting vector just group your i hat j hat and k hat terms in the resulting cross product and you're done okay all right so you'll have to do that with a f equals q v cross b problem on the homework okay uh let's go one step further now uh let's see here so i mentioned in the video lights are going off again i mentioned in the video that if you were to either get a charge particle too slowly enough uh or get a get a magnetic field that is strong enough you could uh get the electrically charged particles to bend in a complete circle in the magnetic field straight water bottle that's fine um so here we have a gun that emits electrons okay and the beam is entirely immersed in a region with a magnetic field and we're told that the magnetic field is a it's just shown as a dot dot means arrowhead coming up out of the picture at you okay the tip of the arrow pointed at your face uh a little cross like this would be the tail feathers of a you know a fired arrow going away from you all right so the dot indicates the magnetic field and this experiment was pointing out of the page the beam is emitted from this gun and it's exposed to this magnetic field continuously and so it continues to feel a force that tilts it away from its original direction of motion and if you tune the magnetic field just right you can get it to complete an essentially circular path and actually doing this experiment is essential in doing things like measuring the charge and mass ratio of the electron that was a very fundamental measurement that was done early on for it um so uh electron beam so let's exercise q v cross b so we know that the charge of the electron is negative so whatever v cross b is we're going to have to flip its direction at the end all right that's what the negative sign in q does is it flips the direction of the vector just like it did in coulomb's law okay so we have uh v pointing that away we curl our fingers toward b well to do that i have to flip my hand over because b points out of the page now i can curl my fingers toward b f on a positively charged particle should point out of the ring but this is a negatively charged particle so i flip it and that's exactly where the force vector points we see the electron is kept in orbit in a circle that looks like this bent to the left constantly by the magnetic field okay so q v cross b it works so that's another electron beam okay so again about the closest you'll ever come to seeing an actual electron that's a big beam of electrons so this is a really cool phenomenon this is what's known as um periproduction so periproduction involves the creation of matter and its its enemy slash mirror twin slash counterpart antimatter so the electron has a cousin in all ways it's identical to the electron all that is different about it is that its quantum numbers are reversed and charge turns out to be one of the quantum numbers in nature so if there's an electron with negative charge there's an anti-electron in nature that has positive charge and it's called the positron it was the very first form of antimatter ever discovered its existence was predicted by the fusion of quantum mechanics and relativity it was one of the very first predictions of fusing these two independent theories of nature into a single theory of nature predicted inevitably that there was antimatter in the universe uh that prediction turned out to be correct and it was one of the first unique things that was predicted by the fusion of quantum mechanics and relativity uh that's how we knew it was a very good description of nature it predicted in fact the existence of the positron so what we see here is apparently nothing comes in in this picture uh and then suddenly there's an event there's something that happens and what we know from the modern physics perspective is that a photon a particle of light is traveling through this medium it strikes an uh electron in an atom it ejects the electron and in the process of transferring energy to the atom and its electron which it ejects and that's what you see here that's the ejected atomic electron okay its trajectory is lit up by the medium here the photon converts there's nothing that prevents the energy in the photon from converting into other things once it's interacted with an atom it's just energy conservation writ large and we get matter and antimatter and in fact this is how we make antimatter for doing experiments we take normal matter we smash it into other things and release energy and in the release of energy you'll often get equal amounts of matter and antimatter created now if antimatter meets matter the opposite reaction occurs you go from energy in to matter and antimatter out to matter and antimatter in and energy out so you can take an electron and a positron and you'll get light that's the basis of the PET scan you put a radioisotope into the bloodstream it accumulates in blood hungry tissue like tumors you then put the human body in an imaging system usually a big set of photo tubes and crystals that can be used to catch the light when the matter uh when the isotope decays it decays into antimatter that antimatter hits an electron it's an anti-electron it strikes an electron in your body and you get two two flashes of light they go in opposite directions from one another and you can scan the human body to see where most of the light is coming from and by doing this you can see where the tumor is located for instance in the brain or in the chest or throat or something like that so yeah how would you detect a gamma ray because it doesn't well so a gamma ray is just a particle of light and i'll talk a little bit more about that when we do the special lecture on light but a gamma ray is just really high energy light so the light and let me turn these lights back on the light that you're seeing here comes from a very narrow frequency and wavelength range our eyes are adapted to see the most prominent frequencies that that penetrate the earth's atmosphere reach the surface of the earth from the sun but there's all kinds of light in the universe there's a radio waves which we can't see infrared which we can't see those are long wavelength and then there's x-rays and gamma rays and those are really short wavelength light above ultraviolet that we also can't see light is like any other particle and it turns out that it has a strong interaction with electrons and so it will if it encounters an electron it can interact with it and it and lose energy so you're actually seeing here what is probably an x-ray or a gamma ray coming into this detector medium it just happens to hit an atom in view you eject an atomic electron and you get matter and antimatter created in the wake of that happening so this in fact is how you detect a gamma ray right here you convert its energy into electrons and positrons and sometimes other forms of light lower energy light so for instance we have today a special visitor from the University of Paris Orsay and he's actually going to spend the next two days giving us pedagogical lectures on the means by which you detect light and electrons using large detector systems and the trick is put a lot of material in their way so if you want to stop a photon of light put a big block of material in its path and instrument the material so you can see the resulting shower of energy that comes from it smashing into atoms that's the basic principle so that's for instance how you detect the gamma rays from a PET scan you usually put these dense crystals in the path of the light they have some kind of readout system on the back when the gamma ray strikes the crystal it will start creating what's called an electromagnetic shower it looks like this but millions of particles and then all you have to do is read out the particles that come from that using some device at the back of the crystal it's a very common technology medically and also in fundamental science so the trick with photons is stop them at all costs put something in their path and guarantees they're going to dump all of their energy at some point right it's it's actually the sort of the same principle that one would use for instance to protect yourself from UV radiation you need to put a chemical on your skin that will preferentially absorb the UV and prevent it from interacting with the actual cells in your skin creating free radicals and causing skin cancer for instance right which I had here and had removed for instance so it's good to wear sunscreen all right good questions all right so this just this slide just reviews what I what I just showed you um I'll make sure that these are up on the website they've got the handy little cross product rules which I got right okay excellent so past me was smarter than present me which is usually true all right so that's the force on a single charge and what I'll wrap up with today is the force on a whole bunch of charges so uh awesome you take one electron you fire it into a magnetic field it bends that's super cool but what happens if you take a whole bunch of electrons and fire them into a magnetic field let's say through a conductor all right so the question we want to kind of answer here is if I were to take a material expose it to an electric potential difference and drive a current through it which is just charge moving through a volume and then run that current through a magnetic field what would I expect the current to do more importantly because the current is trapped on the conductor it can't escape the conductor what will the conductor do in response okay and I'll show you a video of what this looks like but we can we can sort of figure it out so let's begin with the microscopic picture so we have a conductor let's think of it as a cylinder of material conductor and it's got charge moving in it I'm going to draw the most positive just to stick with our convention that current is the direction that positive charge moves although the electrons will be moving in the other direction from where I draw these and these positive charges if I have managed to put some kind of delta v some kind of change in potential between the two ends of the conductor like looking at up to a battery for instance I would expect then that the charge would drift in response to this electric field set up by the electric potential difference okay well the conductor all right fine let's think about the conductor it's got a length l so it's going to link l it's got an area which is called that a okay nothing remarkable here I'm just writing letters down and our electric our charge okay it's some charge q and it has a velocity on average that is the drift velocity e drift okay well let's imagine now that I expose this bunch of charge to an external magnetic field so I immerse the conductor in an external magnetic field and what I'm going to do is I'm going to just because I have pluses already written up here I don't want to put crosses in here so what I'm going to do is I'm going to say that the the magnetic field points out of the board all right so it's arrows coming out at you okay so this is b and it's everywhere I'm just going to draw a few magnetic field lines in a few places but it's everywhere it's uniformly envelops this entire material so every charge in here everywhere it goes there's a magnetic field that points out of the board okay well let's just analyze oh yes I have a picture um we talked about positrons but if you drop plus lines from now on do we have to assume the electrons are moving to the left right we don't need to worry about is that a positron or is it a positive charge again we're going to conduct it again so I should I should have said earlier there's not a whole lot of anti-matter in the world around us if there were we'd all be dead okay actually there was a whole lot present just after the big bang and most of it went away and we don't know why we shouldn't actually even be here having this conversation today because equal amounts of matter and anti-matter were forged in the big bang and yet the universe appears to be like 99.99999999999% matter now we don't know why that's a big mystery that we're trying to figure out so we're talking about conductors they're made of matter they're made of protons and neutrons and electrons the electrons are what do the moving but we talk about the direction of positive current is being the direction positive charge is moving so if the electrons are going to the left the positive charge is going to the right that's all okay that's it so yes don't worry too much about anti-matter there's not a whole lot of it around we can make it but not in large quantities anyone read the book or see the movie angels and demons yeah so okay about no no one okay one person yeah all right so there's the hallmark of the book angels and demons is an anti-matter weapon right and the terrible thing is that the first ten pages of that book the main character gets a demonstration of the annihilation of electrons and positrons in the anti-matter storage containment device and he says there's a big flash of light well that flash of light the flash of gamma rays they should have all died right there at the beginning of the book everybody should have been dead so it should have been a short book but unfortunately it wasn't it was not not a great book but the movie was punchy so go see the movie they have nothing else to do with your life go see the movie angels and demons it's a fun ride the very beginning of the movie is the atlas experiment where i work although we are never allowed to sit that close to the atlas experiment again we'd all be dead because there's so much radiation coming off that thing when the beams are in operation so okay more mundane conductor positive charge magnetic field so if we're just talking about the magnetic force on a single charge okay so here's a single charge q we would just say no problem that's just qv cross b uh and this is v drift okay and then the v field comes out of the board so right hand rule uh let's see if we can do this one so we've got velocity magnetic field force is down so i would expect the force on a single particle any one of these due to the magnetic field to point down again velocity curl toward the magnetic field thumb indicates the direction of the force this gets a little tricky on exams okay so just practice i don't want you to throw a shoulder out doing this this is the most physical activity you'll get an enam the whole semester so enjoy it so we could just do that but we want to know what's the consequence of the aggregate of all of this charge and what happens to the material in which this charge is trapped so to do this we need a few more pieces of our microscopic picture for instance current density current density okay is the number density of charge carriers n times the charge of each carrier q q times the velocity of drift so v drift and current density is just current per unit area and i'm going to for now put a little unit vector over here remember unit vectors you can write it no penalty they have a length of one i'm just going to write little v hat drift there to indicate that this j points in the direction of v drift captain yeah i thought that you said that the velocity was the velocity before it was affected by the magnetic field so should not just be regular velocity not v drift because v drift is not affected by the magnetic field no no v drift is when it's affected by the electric field that i said okay so i'm making a current with an electric field and now i'm affecting that current with an magnetic field okay so it's sort of two pieces so actually there's a sum of two forces on these charges there's a force q e due to the electric field in the conductor that's causing them to drift in the first place and there's a force due to the magnetic field and so the net force actually have to sum up those two things okay that's a good question though but there is a net force and with any as with any other force you do the vectors some of the forces if you were asked to find the total force due to e and b on these particles okay yeah but there there's an e we're just not focusing on it right now okay it's it's made the current it's done its job and now we're focusing on what happens in response to b for these uh for these charges okay so uh j can be related to a macroscopic thing like electric current which we can measure with instrumentation the area of the conductor which we can measure with calipers and something like that okay and then the direction the charge is drifting so this is a penalty free addition of a unit vector on to the end just to remind us that there's direction in j okay all right well that's nice what we would like to do is we would like to kind of relate the stuff that's going on in this equation to the current and other features of the conductor that's been immersed in the magnetic field so the the thing that we need the thing that we're missing here if we want to know the total force uh on the charges in this conductor we obviously need the total charge q and v drift which is the same for all the charges cross b which is the same for all the charges but we need big q we've got little q that's the charge that's actually doing the moving in the material but we need big q well that's actually not so hard to get n is the number of charges per unit volume so how might i get the total number of charges from a little n i want the total charge or a total number of charges here so how would i go from number of charges per unit volume to just number of charges total you could do that but actually we can we could go much simpler here okay we use the um the avery and then and then the j what what oh like take this and rearrange it yes uh let's save that for uh the next step i'm asking for something much simpler i've got number of charges per unit volume and i just want the total number of charges in the volume so i would do what well out of all the charges yes but i don't i don't know how many there are i just know the number of per unit volume so what's that you want charges i want the total number of charges present in the conductor so what do i do with this number and multiply by volume yeah exactly so i'm just going to take n and multiply by i'm going to write my volume as a v with little hats on top okay so not to be confused with voltage what's the volume of a cylinder area times length exactly so this is n a now and now i'm like a hair's breadth away from the total charge because i know the charge carried by each of the little charges and to get the total charge all i have to take is q times n times the volume and that's big q so let me write that out big q is little q times the number density of charges per unit volume times the volume total and this is just q n a okay so i have now that the total force f is q v drift cross b and i can write this as q n a l v drift cross b and now now we're going to use j or more to the point we're going to use i over a okay so if i go back to this equation i can write that i is equal to i in the direction of v drift is equal to n a q okay so let's just rearranging the current density current densities i over a i added that little unit vector into reminders that has direction in the direction of the drift velocity that's equal to n q v drift so if i want to solve for i i just have to move area to the other side of this equation all right so i wind up with n a q v drift well this is nice i've got n here i've got a here i got q here oh and look i got v drift here okay so i can actually substitute current into the force equation so i'm going to do that fb is equal to l which is the only thing i haven't used so far times i v hat drift cross b vector okay so now it's conventional at this point to do the following uh the book does this all right it's convention to define a vector l vector the length vector which is just l times the unit vector that points in the direction of the drift velocity totally legit to do that you can always define a vector if you like okay and so the way the convention usually works is we define uh l vector and so we have fb equals i l vector cross b so if i know the current passing through a conductor and i know it's length okay and the direction in which say charge is drifting and i know the magnetic field i cannot only calculate the force on each of those individual little positive charges i can calculate the force on the entire conductor and to figure out the force it's easy you just take your fingers and you point them in the direction of the drift velocity you curl in the direction of b and that's the direction that the force on the conductor points and i mean the conductor i mean the whole damn thing because okay so that's just an illustration of what i just said all right so basically if you know current is flowing this way you know the direction of l vector it's just the direction current flows so v drift so check this out this is the so-called jumping wire so you have a 12 volt battery hooked up with a switch it's open right now you have the long wire that goes between the poles of a very strong magnet a north and a south pole so that wire that conductor is exposed to a strong magnetic field you plug in the battery and you throw the switch boom currents in magnetic fields don't play nice together if you're designing something like a power distribution system involving high current and you've got magnetic fields present here come the lights you need to be very much aware of i o cross b because if you're not and you can kill yourselves or other people so for instance it's uh i think i have a video of this i'll show you guys later when i talk a little bit about superconductivity but if you have loose cables next to the mri magnet in the hospital where you're working and you switch on the mri magnet and those cables are carrying significant current to drive the magnet they can not only flail like they look like they're in a breeze i'll show a video of this they just flail like they look like they're in a wind like a flag in a wind but they rip free carrying that current for that moment they can become deadly projectiles and you don't want your patients exposed to that so always have your techs check your mri cabling to make sure it's bolted down all right i l cross b it's not just an equation it's a way of life okay thanks everybody i'll see you on thursday and that office hours 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