 Okay, good morning everybody. A few announcements here at the beginning. So, we're not. There we go. Alright, so homework six, do Thursday by 9.30 in the morning. This was a long assignment, so if you haven't started it yet, good luck with that. Assignments for next class are to read 29-1 and 29-2, and I have a video for you to watch to accompany that. It occurred to me as I was editing this one, that I had completely forgot to give you this one. It has to do with, it clarifies forces on loops of current, magnetic, dipole moments and things like that. So you can go back and look through that. It's about 40 minutes, but it's just going through, step-wise, going through the calculation of how you calculate the torque on a loop of current. So, if you're comfortable with that, you don't have to worry about it. If you're not, you want to see, you know, painfully going through each step to get the total effect and then go for it. Yeah, so, team meetings. Alright, so the icepocalypse screwed up at least one team's two scheduled team meetings. They had to be scheduled and rescheduled and it was cancelled again. So, the teams are agility. So your team is going to get at least an hour if you want it. Okay, so what I need the lead editors to do is as soon as possible, let me know what times your team can meet for 30 minutes, starting this Thursday and going through next Friday. Alright, and if we have to go a little bit past that, that's fine. If we have to go into the next week, but I'd like to get another meeting in before we close out, well, too much of March. I mean, April 1st is next Wednesday. So, any questions? Okay, alright, well, if not then, let's get this quiz over with and then we can move on to more interesting questions. Okay, so magnets. How do those effing things work, right? We're about to learn that, actually. Magnets are really quite fascinating. I mean, you probably, how many people played with magnets when they were younger or continue to play with them now? Okay, great, thank you, Jerry. Okay, thank you. Courageous people for having them. It's okay, I'm outing you as people who play with magnets now. That's okay. So these are just all, you know, cheap refrigerator magnets. You saw me messing around with these in the lecture videos. You know, why is it that they exert this seemingly permanent force on each other or on materials that can be so-called magnetized? Like iron, certain kinds of iron can be susceptible to the influence of a magnetic field. This is a very mysterious force. It's an old force. It's probably the first one we really came to recognize after gravity. I mean, after all, people fall down a lot when they're young. After gravity is one of the first forces you become familiar with. That and the ground, which doesn't feel good when you fall. Magnets are an old thing, but they were only understood in the mid to late 1800s. And in fact, understanding them requires an understanding of electricity. Magnetism essentially gets its origins from the momentum of charge. Any kind of momentum possessed by a charged particle is capable of generating a magnetic field. The one we'll study is motion itself. So real physical motion, which all of us can understand. But there's a deeper kind of momentum that every particle has inside of it that can't be changed as far as we know. And that's something called its spin. So how many of you have ever heard of spin angular momentum or spin? Okay, in what context have you heard of spin before? Jerry? No, quantum mechanics, but also chemistry. Chemistry, chemistry is really just applied, I love saying this. Chemistry is just applied quantum mechanics and electricity. That's really about it. That really demeans an entire field of study right there. Just add another electron, changes chemical properties, how very exciting for you. You can all tell all the folks in chemistry I said that if you like. I've already used the equipment I need from them this month anyway. I got what I needed. So there is an irreducible little bit of, it turns out to be angular momentum inside every particle, except one that we know of. Every particle in nature has a little irreducible bit of angular momentum. And as far as we know, all the stuff that makes up matter, so for instance protons, neutrons, electrons, they all have spin one half. And that one half is a multiple of the tiniest amount of angular momentum anything can have in the cosmos as far as we know. So they get this little half unit of angular momentum. So they're called spin one half particles, and rightly so in chemistry. The fact that electrons have spin a half means you can't put them all in the same energy levels in the same way. One of them for instance has to have their spin pointing one way and the other one has to have their spin pointing the other. And then for instance in the lowest level energy shell, that's considered to be a filled shell. And if you want to pile more electrons into that atom, you've got to put them in higher orbits. And so that's actually just quantum mechanics written out in sort of a cartoon picture of atoms. So you actually learn quantum mechanics inadvertently when you're learning chemistry. You just don't realize it perhaps until much later. It's the little irreducible pieces of angular momentum inside of atoms that creates this permanent force that we see here. You can get rid of it. Does anybody know how to get rid of the magnetism in a material? Some kind of mechanical thing, actually hitting it. What you want to do is the reason that these things are magnetic is that some fraction of their atoms all have their spin angular momentum pointing in the same direction. And when that happens you get a net magnetic field from it. You'll see why that is as we go into the next few sections of the course. So all you have to do is hit the material really hard to randomize the spin orientation. So just dial in with a hammer, basically. So if you ever accidentally magnetize a screwdriver, which can happen, you just have to hit it against a table. Don't break the screwdriver, don't break the table. And don't tell your parents I told you to do that, with their nice screwdriver. These are pretty dinky magnets. I mean as things go there's not a whole lot going on in these. And I'm going to put them safely away from me. This thing is much more dangerous. So this is a rare earth magnet. And it's not going to look like much, but this is a ridiculously strong magnet. If I had another rare earth magnet near me, I would have to be very careful that they don't get too close or I could break a finger. If you get in the way of two rare earth magnets when they come together, pain will ensue. So at the very least a blood blister and at the very worst a broken finger. So these are rough, these are strong. In terms of Teslas, the units of magnetic field strength. The earth's magnetic field has strengths of the order of tens of milli Tesla. These are more like order tenths of a Tesla or Tesla. So it kind of depends on how close you are to the magnet and so forth. But these are big honking magnets. They're made from rare earth elements which is an interesting topic these days because China has most of the rare earth element deposits that are known about in the world which makes them a leader in production and sales of rare earth elements which are essential to modern electronics, magnets and things like that. So that creates an interesting geopolitical dynamic all based around rare earth elements like the kinds you'll find in these, like neodymium for instance, which you'll find in magnets like this. So I'm going to use this rare earth magnet to a purpose today. Let's go through the quiz questions and then we'll see how that worked out. Okay, so answers to the quiz questions. All right, so which of these is true about an electric current in the presence of an external magnetic field? So you have a current and it's traveling in the presence of an external magnetic field. Is it one, the current will always move in a circle in the presence of an external magnetic field. Okay, two, the current will experience a force if any component of its motion, length vector, is perpendicular at a right angle to the magnetic field lines. So if you have B and any component of the current, I points in such a way that that component is perpendicular to the field that experiences a force. Anyone for that? Okay, the current will experience a force if any component of its motion, length vector, is parallel to along the magnetic field lines. That is, when the current, like I say, the current is perfectly lined up with the field, it experiences the maximal force. No one, okay, it's Tuesday. All right, no force at all will be experienced by the current due to an external magnetic field. No one's committing to anything. Okay, fine, good for you. Way to play poker, all right? So the current will experience a force if any component is perpendicular. It's the same rule as for a single charge with a velocity vector. So a current is just made of a whole bunch of charges all moving in the same direction. And so you can apply the same rule. If any component of the velocity, if any component of the motion direction of the current is perpendicular to the magnetic field, it will experience a force from the magnetic field. Okay, the magnetic dipole moment, mu, the Greek letter mu, of a current loop is one, the product of the current, i, traveling through each of n loops with the area enclosed by the current loop. So in other words, mu equals n, i, n. Okay, some takers. The ratio of the current, i, traveling through each of n loops and the area enclosed by the current loop. So mu equals n, i divided by n. The product of the magnetic charge and the separation between the magnetic charges. So like the distance between the north and south poles times the strength of those north and south poles. The momentum of the charges in the current. Okay, all right, well, so very low commitment level on that one. It's one, the product of current, the number of loops of that current and the area enclosed by the loops. And we'll use that, I hope, a little bit today. Okay, this is the analogy to the electric dipole moment. And what you'll learn once we get into the next phase of magnetism where we really begin to see how is it that a moving electric charge creates magnetic field. You'll begin to see why it is that the simplest magnetic fields we know about are all dipole fields. You'll see why it is that we don't think there are any magnetic monopoles in nature, but there could be. We just haven't seen them, okay? All right, so in the lecture video what do we learn about the Earth? One, the quote-unquote north magnetic pole is really a south pole if we think of Earth as a giant bar magnet because compass needle north poles point toward it. Okay, two, the direction of the Earth's magnetic field periodically flips direction weakening during the period when it flips. Okay, three, we are protected from electrically charged particles, the solar wind, raining down on Earth by its magnetic field which deflects the particles, or four all of the above. Okay, good, thank God. All right, so yes, it's all of the above. So all of those things are true as far as we know about the Earth, because we define north poles as being the things that point toward sort of essentially the north of the north star, okay, if you will, they must be lining up with a south magnetic pole, so the top of the Earth up in the Arctic actually represents the south end of a giant bar magnet. And you know, that magnet from the Earth comes from a very specific process which I think will have a problem on coming up in another homework assignment or so. All right. It does flip direction, leaving the Earth vulnerable to a rain of cosmic rays electrically charged particles which can cause genetic damage, so in principle cancer rates will go up when this happens and it's probably due to happen in the next thousand years or so. Could happen sooner or when it happens it seems to happen fast, maybe a century or two. It doesn't sound like a lot, that's like three human generations or two human generations right, that it takes to do that, but that's actually relatively fast on geological time scales and we are in fact protected from electrically charged particles by that magnetic field, so it funnels them toward the south and the north and south poles of the Earth and keeps us relatively low levels of cosmic ray radiation, although this place is alive with cosmic rays right now. We could set up a cloud chamber in here and actually see them. So I want to let you guys know what I was doing last week so I mentioned I was going to a conference in Seattle this is the poster for the conference, it's called LRT 2015 Low Radio Activity Techniques. It is the premier international conference for a community of experiments that are trying to measure extremely rare processes in nature. So some people are looking for a very rare kind of nuclear decay called neutrino-less double beta decay. If we observe it we'll be able to measure the mass of this elusive ghostly particle called the neutrino. We don't know that it really exists, but it's predicted to exist, so people are looking for it. But to make experiments like that, to make experiments that look for the dark matter that appears to make up 85% of the matter in the universe requires very clean materials very great amounts of shielding and usually have to put these things deep underground. So a lot of these experiments are being conducted in underground laboratories, like in old gold mines and things like that. And that's across the globe. So for instance, my spouse, Professor Cooley, works on a dark matter experiment. It's located currently in the Sudan Underground Laboratory. It's an old iron mine. It's moving to the Sudbury underground laboratory in Canada, which is a deeper mine, much cleaner and much more outfitted for doing big experiments. So Dr. Cooley and I and two of our students went to this conference last week to present some work, which I thought maybe some of you would find interesting. So I'll mention briefly what it is. One of the problems that you have when you're building clean experiments is radon. Radon is ubiquitous. It's all around us and it comes from uranium decays in the earth. There's a lot of uranium. There's a lot of thorium in the earth, so radon is everywhere. And if you dig into the earth, like make a basement like we did in this building, if you go down into the basement and you take a radon detector with you, you'll find there's a higher amount of radon downstairs than in here and less radon upstairs. Radon is heavy. It tends to sink. And anyway it comes from the earth itself, from the uranium and thorium that's just all around us. So it seeps in through walls and things like that. The problem with radon is that it decays down to things with extremely long half-lives that if they implant in your detector materials, we'll just sit there and radiate for 50 years. And if you're trying to build a really clean experiment that doesn't have a lot of noise from ambient radiation, this is a killer. You don't want this. So one of the things that people have recognized for a long time is that when uranium and thorium give you radon and the radon decays, the daughters of the radon are all electrically charged. And in fact, 85% of them are positively charged. So you could imagine building a shield using an electric field. The question was, how big an electric field and is it practical? So one of the students in I, so Matthew Bremer who's a dual engineering and physics major, we embarked on a little project to do something that everyone talked about doing, but no one had actually done. And that was to measure the effect of an electric field on radon daughters. So on the cheap, Professor Cooley had worked with a teacher who had an engineering facility to build a bunch of little exposure chambers. They're just pressure cookers, so $20 Walmart pressure cookers, drill a couple of holes in them, put in some connectors and gas fittings and you can put a radon source inside of it, seal the lid, expose material, we used copper to the radon and then try to mitigate the radon exposure. So you could try blowing air over the copper or you could try putting it in the presence of a whopping great electric field. So half the project was moving nitrogen over it to try to get radon from sticking, just carried away before it sticks. And the other half of the project was big electric field. So to do the electric field experiment, we have, there's the copper, little four inch by four inch squares, they're really quite pretty and they're cut from a really long length of copper, so we can just slice them into little pieces. Matthew printed over on the engineering over on the engineering 3D printer, he printed a plastic copper sample holder. So this was designed in conjunction with an engineer that works with Edmund College. The engineer built a metal version, and Matthew printed a plastic version, takes about a day to print it. It comes in three pieces side to side and then these rods to keep it solid and hold the halves together. So we went through a bunch of designs, it took a little while to get this right, that plastic is really brittle, so we learned all about the limitations of 3D printing, it's tricky and it doesn't always work, copper is heavy, so this causes the structure to bow a little bit, nonetheless it worked great. And then he found on Amazon that you can just buy nickel copper fabric and I swear to God, it feels like shirt cloth. It's metal, it's all metal, but it feels it's so finely woven and thin strands that it feels like a real cloth. You can make a shirt out of this stuff. So we took to calling this mithril because we're nerds and that's the famous metal from Lord of the Rings that Frodo's co, well originally Bilbo I guess, if you're going to be specific about it, but the coat of armor that they have that goes underneath his shirt that protects him from stab wounds is made of this dwarvish, anyone want to jump in and help me here? Dwarvish metal called mithril, yeah? Yeah, thank you for admitting that you're a Tolkien fan, okay. So what we did was Matthew cut 5 squares and then Mayisha, the other student that was working on the nitrogen-belowing project she can sew and she can sew really well, so she just used fishing line and insulator to sew all this metal together took about a night or two so she just watched TV and sewed and it was beautiful and it's perfectly electrically continuous all throughout, we tested it with a little probe to make sure that it's a solid electrical surface so that that little cap goes over the holder the holder goes in the pressure cooker that gets hooked up to a big electric potential difference the walls of the pressure cooker are grounded so now there's a big electric field inside the pressure cooker and then we wanted to see what would happen. That's our power supply goes up to 35,000 volts direct current, although we don't care about current, we just want the field there's our pressure cooker and this was all being done inside of a fume hood in one of the chemistry student labs downstairs so the chem department was really nice, they let us use that over spring break as there are no labs going on Matthew did a whole bunch of calculations to see if we expected the rate on daughters to be stopped so he did what you did, he just used introductory physics to calculate whether or not this electric field could stop the rate on we expected some effect but we didn't know how much. Well here's the effect this actually came as a surprise to us so if you do nothing if you just put the electric field experiment together and run it and let the copper get exposed to rate on but you don't put the field on so just let it cook in the rate on you get a certain level of contamination and if you then take the copper plates out you put them in an instrument that can count their contamination levels that contamination dies off over five days as an exponential decay so it's just a nuclear decay so what you're seeing here on the logarithmic scale is the declining radioactive contamination of the copper after it's been exposed this down here there are actually two data points down here although it's hard to separate them even on that inset if you turn the electric field on and we can only get it to 6000 volts after that we got arcing inside of the vessel it was cool but it's like shields up and then lightning just shoots out of the anode into the walls of the vessel but that means the electric field is not really working in there and that's making a ton of ozone so we didn't want that at first so we only operated it at 6 kilovolts so we didn't get an ozone big ozone smell thus the fume hood and it reduced the rate on contamination of the copper plates by factor 35 which was a big deal and it was as good as blowing nitrogen over copper so if you combine nitrogen and electric field this may be a great way to protect equipment in the future for these next generation of experiments so anyway that's what we did so it's just electric fields charge particles and cross your fingers and hope some science comes out of it and it worked out we got the data four days before the poster was to be presented so we were really cramming this in at the end so here's Matthew on the right and Meisha on the left they had a crowd of people at their poster almost the whole night poster session the poster session at this conference is a big deal they had something like 30 or 40 posters they put a whole spread of food out and everybody from the conference goes to it so they really try to make this a big deal and they pretty continuously had a group of people asking them questions all night this guy right here he was great he's like I don't know why you're going to want to do a stupid experiment like this no one's going to want to put a big electric field like that around their equipment and we're like okay well fine I guess you're not our competition then so you know naysayers are fine but you know go out and do it right so so yeah this is the middle of a good crowd with Meisha and Meisha and Matthew kind of leading the crowd through the various aspects of the poster so Meisha is a physics bio math triple major and as I said Matthew is an engineering electrical engineering physics dual major and Meisha's been doing research with Dr. Cooley since her first year in college so we didn't only work while we were there the cherry trees were blossoming at the University of Washington while we were there and it was gorgeous like the whole campus just smelled like cherry blossoms so this was great you know we don't get to see this down here and it's a little hard to see but that's Mount Rainier and in the photo it's really neat because there's a draft over Mount Rainier and so there's a big cloud layer that's kind of sliding over Mount Rainier like water over a rock it's really quite wild so this is a walkway at the University of Washington right by the conference venue near the library they have these cherry trees down to a little pool and then Mount Rainier right at the end of the aisle it's really gorgeous so if you're ever in Seattle I highly recommend going to the campus it's really beautiful okay one other thing you might be interested in shameless promotion within my family here so my spouse professor Jody Cooley will be on Science Friday this Friday from 2.20 to 3pm local time she got the pre-interview last Thursday and they decided to run with her on a panel so she'll be on a panel with Nobel laureate Stephen Weinberg and maybe one or two other people and it's a 40 minute segment they normally do 20 minutes per segment so if you want to see what they have to talk about they just did a special on Science Friday on 100 years since Einstein's theory of general relativity came into being and now they're apparently going to do kind of a perspective forward what are the big questions people are working on now going in the next like 50 or 100 years so anyway enjoy I know I will so apparently she's going to do this for some studio here on campus so she won't be on the on a phone with people knocking on her door asking professor Cooley can you tell me where the bathroom is professor Cooley can you sign it for the major so okay any questions I thought not alright so let's solve some problems involving magnetic fields and charges okay so this is to make up for not having done any problem solving for a couple of weeks now so the basic principles here just again refuting the basics electric charge exists into kinds positive and negative these exert influence on each other by a field of force we've now really kind of dug into that field and it's associated potential it's electric potential these fields accelerate charges that's the whole basis of circuitry so you create a voltage you move some current you do some work you get some good stuff out on the other end okay we've looked at capacitors we've looked at resistors we've looked at batteries so we kind of you know looked at basic circuits magnetic fields which seem at first weirdly independent of electric fields their fields of force exerted by certain materials such as magnetite and iron they also though affect electric charges however they don't do this linearly an electric field accelerates an electric charge along or against its field lines depending on the sign of the charge but what's weird about magnetic fields is that the acceleration doesn't change the magnitude of the velocity but it does change its direction and so when a charged particles trajectory will be bent perpendicular to the both the magnetic field and its original direction of motion and I can demonstrate that right now so it seems weird at first and what's tricky about magnetic fields is a little toolkit of comfort with working with things that happen at right angles to one another alright so cross products become a big deal and I'll give you some simple rules to remember for cross products to help you do those today hopefully quickly easily and quietly I have a video on the class website called Confidently Calculating a Cross Product I made it years ago it uses what's known as the matrix method but there's another method I've adopted in the meantime that I personally prefer because it's purely algebraic and you just have to remember one rule and once you remember that one rule you basically can reproduce all cross products pretty quickly so I'll walk you through that in a moment but I thought it'd be nice to do the in class demonstration of bending charge with a magnetic field so let me get rid of this and bring up camera 2 so camera 2 is just aimed at this device this is a crux tube you saw this being operated in one of the lecture videos this was the device that J.J. Thompson used to discover the existence of the electron and essentially it looked very similar to this you have a metal plate that can emit electrons this piece of metal right here is just a little slot let me turn this to the side so you can see that little slot right there that's just meant to filter out electrons heading off at stray angles and create a little almost rectangular beam coming through the slot so it's a filter it's a filter for electrons to make a nice rectangular beam of them the white plate on the back here it's tipped at a little angle normally you can't see electrons with your eyes your eyes are sensitive to visible light not electrons how you see electrons at all is because when they interact with gas atoms inside this tube they emit a blueish light and so your eye sees the light but it never sees the electron and this is something you need to get comfortable with you may be more comfortable than previous generations of people with this but you can't see everything in the cosmos and so you have to use technology in order to bring it into the visible realm for human beings and that means a very narrow spectrum of light that we are sensitive to with our eyes so you have to use tricks to do this you have to use gases or electronics or something to convert the invisible into the visible electrons are not visible but their effects on matter can be visible to human beings so let me just fire this up so this is a roughly 50 kV power supply we need a lot of juice to get electrons to move it doesn't take a big field to move electrons so there you go, as I promised a blueish light that light is the result of a beam of electrons trying to get from the high potential on one side to the low potential on the other and that light results as they smack into gas atoms inside of this sealed tube so this is just filled with some gas that gives off a blueish light when electrons interact with the electrons in the atoms excite them and then de-excite them and when they de-excite they give off light so very simple I have the camera at an angle so the beam looks like it's at a little angle let me intensify that a bit so we just crank up the voltage a little bit get a nice bright blue light out of this now let me demonstrate bending with a magnetic field I have the rare earth magnet in my hand I'm going to put one pole it's such an orientation that the magnetic field lines that come out of this end of the magnet they will point into the plane of that screen so the velocity of the electron beam is to the left along the horizontal I'm going to point the magnet so that its magnetic field lines are perpendicular to the plane of the screen and thus perpendicular to the motion of the electron beam and when I do this you should see the beam bend it's easy to see up on the screen that is the magnetic force on charged particles and in fact if you couple this with an electric field which we have done here you have just made a particle accelerator this is essentially a small version of a medical linear accelerator if you need to accelerate electrons to get them up to speed to smash into a cancer tumor you use a power supply like that although it will be on steroids ok if you want to bend the beam to aim it better or to steer it around corners into the treatment area you need magnets magnets are the lenses of charged particles they allow you to steer beams now if I flip the magnet around and put the other pole I can bend down so with just this simple change flipping the pole around I can change the direction that I'm steering this beam of subatomic particles that is the entire basis of my field I can summarize it in exactly one demonstration electric fields to accelerate particles magnetic fields to bend them and this allows fine-grained control of the subatomic world as long as it carries electric charge the beam of neutral particles like neutrons, nothing would happen they don't respond to magnetic fields because they don't charge now, so any questions on that that is the magnetic force on charged particles right there and we can summarize that at the particle level as the force due to a magnetic field is the charge times the velocity of the particle crossed with the magnetic field direction I'm going to do cross product in a second here if you are talking about a current this just changes to this formula the current, coulombs per second times this vector L and all L does is it's a vector its magnitude is the length of the wire let's say that's carrying the current and its direction is the direction where the current is traveling and remember current is the direction of positive charge flow current is the direction of positive charge flow so if you have an electron beam going to the left then you have a positive charge beam really going to the right so electrons going to the left are really like positive charges going to the right keep that in mind as you work problems with this let's look at this cross product so let me get rid of this okay so let's look at the cross product this is something which seems very scary at first the basic properties of the cross product are that if you have two vectors A and B just generic and you take the cross product you get a third vector from this C now the third vector C will always be perpendicular to A and perpendicular to B that is if you take C dot A you will get 0 so if you take the dot product of C and A which tells you the projection of C along A the shadow that that vector casts along A the answer is non no shadow there is no component of C that lies along A that's the definition of the cross product C dot B will equal 0 so the great thing about the cross product is it's guaranteed to return a vector that makes exactly 90 degree angles with the other two they may not make 90 degree angles with respect to each other but as long as they're not parallel then you will get a third vector that results from it and that third vector will always be perpendicular to the first two so it's a great little piece of technology in mathematics and the magnitude of the cross product so if you want to know the magnitude of C that's the magnitude of A so the length of A the magnitude of B and the sign of the angle between them so if I have here A and B and that's theta in between them if I want to know the magnitude of C so if I have A and B and then C C makes right angles with both A and B so those are 90 degree angle 90 degree angle and the magnitude of C is given by this it's the length of A, the length of B times the sign of the angle between them so it looks very similar to the dot product which is if you do the dot product of A and B you get A times B times cosine theta so again if A and B have no angle between them if they are parallel to each other the sign of 0 is 0 so if they are parallel you get no cross product there is no vector that is perpendicular to both of them at the same time actually there's an infinity of vectors that's perpendicular to them where they both point along the same line because you could have a vector here or here or here so the cross product essentially returns nonsense that says no there's no vector you can make that's perpendicular to both of those so I'm going to try to trick me so the cross product is smarter than me let me put it that way now how do you figure out whether C points this way or up like that that is where it's helpful to have a little thing called a right hand rule so you can figure out using the fingers on your right hand you can figure out which way that vector C is going to point now there are many variations on the right hand rule let me walk you through two of them I'll do the second one but I'll do the first one first so take your index finger point it straight forward take thumb point it straight up take your middle finger and point it at 90 degrees to both of those so X, Y, Z you basically have a little this is how you know you're in the physics club I told you when you start doing this this is how you know you've made it so you start doing this to solve problems if you're left handed I apologize but this is a mirror universe where everything is reversed and screwed up don't do this this is the mirror image of this don't do the left hand rule you'll get wrong answers from it so it's a convention you have to stick with it I apologize to the left handed people so you have this little thing X, Y, Z this is how you can remember Cartesian axes so your finger points along X your middle finger points along Y your thumb points along Z that's a Cartesian coordinate system we'll use that in a second if your thumb points down or up you take your index finger point in the direction of A then you take your middle finger point in the direction of B so I have A like this B points out of the board that way and then your thumb indicates where C would point and here it points down that's why I drew this down if A were here and B were here and I did A cross B then I would point this way my middle finger points in the direction of B my thumb then points up so if you reverse the cross product if you do A cross B it points this way if you do B cross A it points that way so you can flip signs around by swapping the order of the cross product so be careful with that here's the other one this is the one I actually prefer because I feel like it goes faster you take all of the fingers on your right hand and you point them in the direction of A and your thumb points in the direction of C so A, B, C get the same answer A, curl toward B, C if B were into the board I'd have to go A, B and then C would point out so this takes a little bit of thinking in three dimensions and it's tricky practice it if you're not good at it not everybody is, I sucked at thinking in three dimensions until I was like late in graduate school luckily I like to draw so drawing things help me a lot and if drawing things helps you do it, don't rely on something that isn't working for you, do something else so if drawing works better or just remembering the rules I'm going to show you in a moment works better just do those how do you calculate a cross product and how do you do it quickly I'm going to show you the drill and it's based on the coordinate axes so let's imagine that you have a vector A which is equal to let's say A1 I hat plus A2 J hat and then you have a vector B which is equal to B1 let's see J hat and you want to find C vector which is A cross B it's very generic this could be velocity and you have to multiply it by charge this could be magnetic field then you take the cross product or QB cross B this is just a place holder for any calculation like that alright well let's just start by writing this down like algebra so this is going to be A1 I hat plus A2 J hat crossed with B1 J hat now the cross product is like any other multiplication you can distribute so if I had something like this if I had A1 plus A2 B1 you would just say oh well that's A1 B1 plus A2 B1 you would just distribute just make sure you keep the cross product signed when you distribute so this can be written as A1 I hat cross B1 J hat plus A2 J hat cross B1 J hat so all I did was distribute that's it so I went one more algebraic step forward ok now this is where the handy little rules come in ok so let's talk about the handy little rules and I like this because this is just pure algebra ok slide that over for a second ok you've got I hat you've got something I hat crossed with something J hat so these are A1 and B1 they're just numbers they just multiply these little unit vectors you can pull them all out in front it doesn't change anything let me just for this term right here let me just write this as A1 B1 and then in parentheses I hat cross J hat ok so all I did was I pulled those numbers out and that's fine to do for instance if you had something like A B times C D and these were all numbers you would have no qualms about saying oh well I could rewrite this as B D right because they're just multiplying each other you can multiply them in any order and pull them out you have to be careful with vectors so these are the vectors leave them crossed into one another but you can take those numbers that multiply them and just pull them out in front like one big number ok no harm in that it's if you start messing with the cross product without using the rules that you'll get into trouble so here are the rules this is how I remember it I hat cross J hat equals K hat the X direction crossed with the Y direction equals the Z direction and you can see that right with the right hand rule right X cross Y equals Z that's it that's the definition of a Cartesian coordinate system in fact a right handed Cartesian coordinate system is defined by exactly that equation the cross product of X and Y equals Z it's perpendicular to both X and Y and of course X and Y are perpendicular to each other the next thing you do with this to get all the other rules is something it's called permutation in mathematics but you can just think of it as the conveyor belt the conveyor belt imagine these symbols are sitting on a little conveyor belt ok and they're attached to the conveyor belt and the conveyor belt rotates clockwise ok so what happens is K gets yanked off the end transported around here and plopped at the beginning and J gets moved over to its spot and I gets moved over to where J was alright little conveyor belt so this equation is also true K cross I equals J this is called permuting the order that is you just shift them all by one and you take the one on the end and put it back at the beginning permutation that's the fancy math term I like to think of it as a conveyor belt for these little symbols this is how I remember this you can do it again you can move J to the beginning slide K over one slide I over one leave those operators in the same place so J cross K equals I yeah Julie so in this case you have to I times K equals J let's look at that like this one here so what happens if you swap any pair and the answer is you pay a penalty of a single minus sign the first three are easy to remember you just do the conveyor belt I J K K I J J K I if you want to get I cross K K cross J all you have to do is put a minus sign in so J cross I equals negative K hat that's it I cross K equals negative J hat and finally K cross J equals negative I hat and those are the six rules and you can get them all from the first one so to get the next two you just do the conveyor belt to get the last three you swap the order of any two of these and put a minus sign on the other side that's it now there's one more set of rules that they turn out to be sort of the silliest and easiest to remember if you have I cross I J cross J or K cross K taking a vector and asking what's the cross product of it with itself and the answer is what's that zero there is no vector that's perpendicular to two parallel vectors I is parallel to itself there is no single vector that is perpendicular to both it and itself okay so these are all zero zero zero I cross I K cross K zero okay so anytime one of those symbols appears with itself in a cross product that's what's great about this okay so let's go back and revisit our little cross product using these rules I have A1 B1 those are just numbers I hat cross J hat okay well I go over here and I look I hat cross J hat is K hat so this is A1 B1 K hat okay and then finally I have this one I have A2 B1 J hat cross J hat and J hat cross J hat is zero great so that term just goes away gone not even interesting so that's what's nice about this you can right away recover you can take the most complicated vectors and start doing cross products with them and stuff will cancel out pretty fast and hit the group your terms that don't cancel out so we find out that C is equal to A1 B1 K hat that's it done okay no right hand rule needed it's all built into this stuff it's all built into this stuff alright so I J K I cross J equals K do the conveyor belt trick to get these two swap any pair on the left put a minus sign on the right to get the other three and you're done and then any of them cross with themselves zero questions so we can start doing problems now okay let's do some problems we're not there we go okay large Hadron collider had to shamelessly sneak one of these in so the LHC large Hadron collider LHC accelerates beams of protons and each beam of protons is taken up to a total momentum of these funny units and I'll walk you through them in a second 6.5 Terra electron volts divided by C where C is the speed of light now believe it or not you can work through it this is a unit of momentum okay you can convert electron volts to joules and C is just meters per second if you work through that you'll see that you get kilogram meters per second which is momentum okay so it's just a funny way of writing momentum but it's very standard in particle physics my field medical physics as well okay estimate the strength of magnets required to maintain these beams in an orbit well here's the orbit the beams make a 27 kilometer circle over the French Switzerland border in a tunnel that's about 100 to 150 meters around so we have a radius for this 27 kilometer circle of 4.3 kilometers alright so 4300 meters we want to estimate the strength of the magnets required to maintain the beams in that circular orbit alright we can assume the proton motion is at a 90 degree angle to the magnetic field lines so let's see what that means anytime you have that the problem has been greatly simplified for you alright so let's take a look at that let's start with the force on a single proton the force on a single proton will be equal to the charge of the proton which is just the elementary charge E times its velocity vector crossed into the magnetic field okay well we don't know the velocity vector but we are given the momentum so we could figure it out but you'll see why actually it's convenient to be given the momentum in a minute we're told that the velocity and the magnetic field lines make 90 degree angles with respect to each other so we already know something if we write the magnitude of FB we have E magnitude of V magnitude of V sine theta is 90 degrees and the sine of 90 degrees is 1 yes it's 1 better not be 0 or this is the shortest problem why would I do this you're right we're done 0 we don't use magnets at all it was a trick question so sine of 90 is 0 now it's 1 now I see you look at that you screwed me up I'm not putting a dollar in the jar that was your fault this is great sine of 90 degrees is 1 so that's just 1 and the magnitude of this is just E magnitude of V magnitude of B well we're looking for the magnitude of the magnetic field ok so we want that thing we want B but we don't know V and we don't know F we don't know the force but the other piece of information here is that this is a circle that these protons are maintained in a circular orbit this is done for 10-12 hours at a time ok so in circular motion there's another force what is that force called centripetal right it's holding you in the circle alright and centripetal force it's magnitude the magnitude of centripetal force is M V squared over R so the mass of the thing in the circle times its velocity squared divided by the radius of that circle we can equate these two things because if it's being held in a circle by the magnetic field these things must be equal to one another ok so we can say well this is just equal to EV B where I've just now taken the shortcut of writing the magnitude of the velocity is V the magnitude of the magnetic field is B well this is great because now we have velocity squared on one side and velocity on the other so we can cancel one of these and we're left with just B equals M V all over E R so mass times velocity divided by charge times the radius of the circle ok what's M times V equal to thanks yep momentum that's right thank you Billy that's just B now a cautionary tale here in classical physics it's ok to exceed the speed of light and actually if you calculate the speed of these protons you'll find out that they in classical physics they're in excess of the speed of light which actually is not allowed but it's ok actually it does change the answer from the correct answer in reality but we're not going to worry about that because I'm not introducing relativity to save this calculation it's not worth it that's like two weeks in and of itself so we're just going to pretend right now but pretend ok so if you put in that momentum we have to convert it we have to convert it to units so 6.5 TeV divided by C what's that 10 to the power of 16 for T oh no T Terra 10 to the 12 ok so this is 6.5 times 10 to the 12 EV divided by C one EV does anybody remember how many joules that is it has a convenient conversion factor but it's been a while 1.602 times 10 to the minus 19 joules is one EV it happens to be exactly equal to the elementary charge ok so I have to do is say 6.5 times 10 to the 12 times 1.6 times 10 to the minus 19 joules well that's what that is joules that's one EV is that many joules divide this by the speed of light which is 2.998 times 10 to the 8 meters per second so that will give you kilogram meters per second and since everything else here is in meters and coulombs and so forth we've got to get this into meters, kilogram, second units ok these are particle physics units but they can be converted whenever you do speed of light do you want to see a 3.998 you can use the right rounding that happens in this problem it doesn't make a difference I'm just being pedantic but that 3 decimal places is the limit of my pedanticism is that a word? I don't even know if that's a word alright so if you plug all that in you'll find out that this is about 5 tesla now in reality to get to that energy it's actually closer to 8 tesla and that's because of special relativity so that's a whole separate thing in and of itself it's pretty close, I mean you're in the ballpark with this answer we actually have 8 tesla magnetic fields but 5 is close enough oh c this is c so c equals this so 1c, 1 times the speed of light is 2.998 times 10 to the 8 meters per second and in the real world it's as fast as we think anything can actually travel but here if you do the math the protons appear to be in excess of the speed of light but that's because I've neglected relativity here completely so using these kinds of things you can very quickly figure out the engineering parameters for a system like this this is pretty close to the correct answer which I said is about 8 tesla the LHC collides many bunches of protons a single proton beam has a nominal current of 0.5 amps so each beam is about 0.6 amps of current but that's 1.15 times 10 to the 11 protons per bunch 2808 bunches around the ring for each beam so it's a lot of protons and every time we collide one bunch of protons about 10 to the 11 with 10 to the 11 of the protons we get a typical 40 proton-proton collisions which we then have to tease one interesting one out of those 40 it's a nightmare it's a big 100 megapixel digital camera that can take 40 million images a second it's to catch the details all that collision stuff so what's the magnetic force experienced by this current? magnetic force for a current that's a related equation and again the current and the magnetic fields are at right angles to each other so the most generic equation is I L cross B ok let's think about each of the pieces here so we want to know the magnitude of the force so the magnitude of the force is equal to I magnitude of L magnitude of B sign of 90 degrees again which is just 1 so just I L B we know B now it's 5 Tesla what's L? well L is just the length of the current you did I just left that out I didn't explicitly put this and I just converted 6.5 TEV over C into kilogram meters in seconds that's all down here you'd have another E which is 1.6 10 to the minus 19 and then this would be 4.3 times 10 to the 3 meters so if you put all that together in a calculator you should get 5 Tesla out of it would it be negative? protons so it's positive anyway it's a magnitude and the magnitude is always a positive number if you were asked however for the direction then you have to worry about sign ok so the length of the current here is just the circumference of the circle so L is 2 pi r ok so it's just the length of that red circle ok and if you plug all that in we were given 5 it's 0.58 amps we're given the radius 4.3 kilometers so 4300 meters and we know the magnetic field from the first part so this comes out to actually be a pretty big force I mean for protons this is immense the magnitude of this is 7.8 times 10 to the 4 Newtons those protons really want to go in a straight line from going in a straight line that energy takes an immense magnetic field inserting an immense amount of force on this alright I'd like to have you guys do some problems now so there's some bonus stuff in here that I'll put in the notes given the current flowing around the large hand drawn collider and its radius what's its magnetic dipole moment I mean after all that current one of those beams is a big loop and the area of that loop is pretty large so you can actually calculate the magnetic dipole moment by taking I times A and it's a big number and then you could do something like well, loops of current are subject to torque from an external magnetic field and there is a big external magnetic field here the earth it doesn't seem very strong but you can calculate the torque due to the 57 millitesimate field present at that location on the earth I looked up the map this morning and it has that magnetic field at 62 degrees so the magnetic field is actually penetrating that area at an angle that makes a non-zero cross product with the dipole moment and you can calculate the torque and it's a big torque but it's also a really huge machine with a lot of rock around it so the question is does it matter but you get a big number out of this it's actually a surprisingly large number that that dinky little magnetic field exerts on that giant dipole moment okay so I just wanted to show you guys this is what the dipoles actually look like the proton beams travel through those holes they're counter-circulating so one goes into the board one comes out of the page and this is what the magnetic fields look like due to this sort of funky superconducting design here so you have all these superconducting wedges and they generate magnetic fields we'll learn about that in the next section of the course and the sum of all those magnetic fields is a nice uniform straight magnetic field that we can use for bending okay so that keeps the protons from wandering out of the orbit okay so here's your problem little biology, little space alright so long duration space flights that is like a mission to Mars which both private and public entities are talking about doing in the next two to three decades it's actually quite a medically challenging thing and not only for the reasons you think of like food, loneliness and so forth but the threat of biological damage to astronauts from the solar wind is actually a huge issue the solar wind consists of ejected charged particles from the sun and we're largely protected from it on Earth because of our magnetic field but if you just hurl a rocket into space far from Earth into the space between Earth and Mars let's say that journey can take upwards of a year or two to happen and meanwhile that ship or the metal it has built into it is subject to a huge wind of charged particles from the sun and estimates have been done that not only would this induce a high cancer rate in astronauts it can actually degrade brain matter because they're just being bombarded by charged particles so your cognitive functioning may decrease rapidly over the years of exposure to the solar wind not the stuff they talk about on Star Trek although they do have this thing called the Sard Ram Scoop that they used to never mind it's a technical detail anyway it's such a nerd so imagine you're an astronaut you're in this habitat that's traveling to Mars people have asked the question could we use really big magnetic fields and then how big to protect astronauts from charged particle radiation so literally shields so imagine trying to use magnetic fields to deflect high energy electrons from the solar wind alright so I took a look at the energy spectrum of these electrons and getting toward the bad end of things these electrons have momenta of about 100,000 electron volts divided by C so 100 keV divided by C so the question is here first of all what direction does the magnetic field in the hull point along X here's X that's X here's Y and here's Z what direction does the magnetic field have to point along X is it positively along X or negatively along X if an electron entering along the positive that's supposed to be the Y direction that's off the screen but the positive Y direction so here comes, or positive Z sorry electron comes in on the positive Z direction and then is deflected back out by the magnetic field what direction does the magnetic field have to point so try to use the right hand rule to figure that out and then finally what strength magnetic field is needed in order to deflect that 100 keV divided by C solar wind electron from the astronaut quarters and you can let V and B be perpendicular alright so they can be initially perpendicular to one another no problem and here's a hint if you're going to deflect this electron you don't want it to get inside and like graze the foot of the astronaut who might be standing there and then give them foot cancer ok so you want those electrons to stop before they hit the inside of the floor so you definitely want the bending radius of the electron when it comes into the magnetic field and then is bent away you want that to be no greater than the thickness of the hull ok or wherever you have the magnetic field so here the hull plating is one meter thick ok so your bending radius is one meter ok so how about it if you go a little bit you've got ten inch minutes to play around with this see how far you can get so try to right hand rule first to figure out the answer to that part A and then for part B do a little calculation and see what kind of magnetic field strength is required to deflect these and keep in mind the biggest magnetic field that humans know how to make right now are LHC magnetic fields eight and a half Tesla that's the biggest magnetic field any human being knows how to make reliably right see what you get