All right so today this will be an anomalous class in the sense that there'll be a bit more of me talking than usual. I'm Professor Steven Sepula and again I'm really sorry that I wasn't able to be here on Tuesday. I spent two days at an international conference on supersymmetry and unified interactions which I'm an organizer for. So I flew back yesterday morning I got up at four in the morning got on the 6 a.m. flight got back here at 2 p.m. and then had a bunch of stuff to do for the afternoon. So I'm very pleased to be able to work with you this semester on electricity and magnetism and all the related stuff that goes with it and today what we're going to do is we're going to play around a little bit with electricity and then we're gonna I hope we'll have enough time to do this I kind of want to step through it methodically. We'll learn to set up and solve problems involving electric charge electric charges and the interactions between them so coulombs a lot which is a mathematical explanation for an observed phenomenon we will observe that phenomenon today. Okay so a few things so for today you had to basically read chapter 21 all right and there was no assigned lecture video for today the stuff that would have been in the video I prefer to do in class so we can all kind of get to know each other okay the next assignment you're going to read chapter 22 1 and 22 2 and there's an accompanying video to watch with that. Now these videos are you know they can be between like 30 and 60 minutes it depends on the subject material but the good news is you don't have to sit there and watch it all in one sitting you could watch 15 minutes of it go get something to eat watch 15 more whatever you can procrastinate at your leisure as long as you're prepared when you come to class because eventually you'll have to answer questions about the reading and maybe things that were presented in the lecture video. They're not going to be nitpicky things like at minute 32 what adverb did I use and it's not going to be something like that okay so you know I want you to understand concepts and I want you to pay attention to the demonstrations or simulations that are done in the lecture videos there may be questions on those and so forth it's to your benefit to pay some attention take notes all right so treat the lecture videos like you would falling asleep in my lecture class take notes okay pretend to listen I go back and review your notes before the quiz okay homework one has been assigned it's been available since about 8 a.m. this morning it's assigned mostly through Wiley plus however there is a written question that I generated myself as an add-on to the Wiley plus assignment it's part of your homework assignment if you go to the website which I'll show you in a second you'll you can get this electronically from the website or you can grab a copy before you leave today many of you already have it if you don't grab a copy before you go all right so there's a Wiley plus most of it's Wiley plus some of it is this handout problem the numerical answers to the Wiley plus problems they go into Wiley plus okay and that'll be about half your homework grade for the assignment the other half of your homework grade will be based on the written set of solutions you generate for all of these problems Wiley plus and the handout problem label them order them write all your work show your steps explain what you did staple it together put your name on it hand it in next Thursday I will give those solutions to the teaching assistant noosh and she will pick randomly one problem to grade from the pile okay so it's you have to make sure you do work for everything because you don't know which of those problems is going to be assessed on methodology you are not assessed on your correct numerical answer that's what Wiley plus does your assessed on all the other stuff that goes into a solution what was your thought process did you explain yourself well enough did you write all the steps down can I read your handwriting okay if not we'll work on that through the semester okay there are ways you can even type up homework if you wanted to if you don't like your own handwriting you're free to type up homework if you know how to write equations in a program okay so any questions on those things okay so let me show you the the course website really fast here all right so hopefully you've all visited this or will visit it sometime today this is the class material site all right so when you go to the main course website which is here all right you click on class material and that will take you to this sort of semi-self updating page which slowly rolls out material lecture slides lecture notes handouts homeworks and things like that so past lectures are grayed out the current lecture sort of within 24 hour window is in green and you'll see today assignment do is homework zero the math homework which goes on the corner there solutions will eventually be posted they're not there right now and the assigned homework is homework one so this is a PDF copy of what's in Wiley plus now the numbers in this will be different than the numbers you get in your assigned version so don't use the numbers from this to answer the question this is just in case you you know you don't have a wildly plus account yet you want to get started I'll talk about that in a moment okay you have a reading assignment for next class as I said and there's the link to the video assignment the reading and the videos are already pre-assigned they're already all here so you can skip ahead if you want it's up to you all right if you want to get ahead of things that's your business all right any questions on that no all right all right so go ahead and put your books notebooks all that stuff away please don't start quite yet I know it's just for your benefit let's let's take a look at the answers to the quiz oh yeah thank you let's take a look at the answers to the quiz questions okay so question one all right which of these is false about electric charge so one there is a smallest absolute unit of electric charge which is called the elementary charge and has the value of this thing that true or false true okay there are two kinds of electric charge positive and negative true or false true okay electric charges exert forces on one another in point charge forces are governed by Coulomb's law true or false true okay it is not conserved that is charge is not conserved that is it is possible to create or destroy charge and thus change the total value of charge in a closed system true or false false okay yeah so if you said false to the last one that's correct as far as we have observed it's not possible to just make arbitrarily or destroy arbitrarily charge you usually have to destroy it in pairs like you can take a plus charge and a minus charge and you know add them up to get zero charge that's one way you can get rid of it but you have to get rid of two you can't just take plus charges and magically annihilate them without some cost of course you could get the systems open if things can go in and out of the system it can look like charge is not conserved all right but we've never observed that this happens so as far as we know charges conserved charges exert forces on one another in point charges are governed by Coulomb's law there appear to be two kinds of electric charge in nature we call them positive and negative those are just labels and there is a smallest absolute value of electric charge which is called the elementary charge and corresponds to the magnitude of a charge of an electron or a proton okay question two which of these is a true statement about the electric force as described by Coulomb's law is it one the electric force increases in strength as one increases the separation distance between two point charges anybody for one raise your hand okay to the electric force decreases in strength as one increases the separation distance between two point charges any takers for two okay lots of takers three the electric force between two like sign charges is attractive so two like signed or same sign charges will attract each other okay no one and finally for the electric force between two opposite sign charges is repulsive so two opposite or non same sign charges they they repel each other no okay good yeah so the correct answer is two to that one the electric force falls off in strength as the distance between charges increases there are actually there's at least one force in nature where that's not true you guys want to know more about that at the end of the semester under special topics we can talk okay all right well you know the answer to that one I might as well just let that one go which of the following are actually this is incorrect anyway right which of the following are true about the kinds of materials in which charges can be placed the correct answer is it turns out are two and three so that was my mistake I only meant one of those to be correct insulators do completely or almost completely prevent the free movement of electric charge will actually play with insulators today and conductors allow for almost perfectly free movement of electric charges will play with conductors today to okay so both of those two and three and I'll fix that in the slides okay bonus question so which of these true which of these is true about this course according to the syllabus attendance during the class period is not assessed and doesn't count toward the final grade anyone for one that true okay to according to the Homer policy all final answers to Homer problems must be boxed have appropriate units and adhere to the rules of significant figures anyone for two okay few takers many takers according to the Homer policy I have to work completely alone conferring with none of my peers in the course about problem solving three true no all right according to the grand challenge problem policy my grade on this project will only be determined using questions on the final exam so only questions on the final exam determine your grade okay good yeah so the correct answer is to box your answer is sig figs and make sure you have units on numbers where they're appropriate okay so if there are no units if it's a dimensionless quantity that is it has no units then don't write units down okay but you can work together as long as you acknowledge each other's collaboration so put the name of your collaborators at the top of your written homework solutions your grade on the homework challenge problem the grand challenge problem is a combination of a group grade in an individual right well we'll do we'll do the team building for that next week and according to the syllabus I do assess your attendance you just had it assessed it was the quiz okay okay so let's get into solving problems using Coulomb's law which really is just about exploring and then describing the electric phenomenon okay so we're going to play with the electric phenomenon today and I want to start by reminding you something I think you should know or I think you do already know but but I really want to drive it home and that is the atomic theory of matter okay so how many of you have heard of the atomic theory of matter none of you have heard of the atomic theory of matter how many of you taking chemistry raise your hand if you've taken chemistry okay then you won't use the atomic theory of matter what what do you do in chemistry what are you studying in chemistry atoms okay and the premise is that everything around us is made from building blocks called atoms and the only thing that distinguishes atoms from one another are the number of electrons well the number of protons that they have determines the kind of atom and the number of electrons determines their chemical properties so if they have exactly the same number of electrons as protons they're electrically neutral but some of those electrons are weakly bound on the outermost orbits of the atom and that can facilitate bonding with other kinds of atoms if you ionize an atom if you intentionally remove or add electrons to it that can also give it different chemical properties than it would have in its neutral state so the atomic theory of matter is nothing more than the statement that everything around us is made of building blocks and if we could figure out what those building blocks are and how they stick together like Legos we could figure out how the world works all right does anyone know when the atomic theory of matter was definitively demonstrated to be the correct description of nature any guesses or do you know the date roughly take a guess yeah well so what when when do you think it was 1850 1850 all right and any takers for earlier than the 1800s that we knew about the atomic theory of matter we stopped with solid by by the 1800s yeah tentative hand so okay what's your name Shannon okay so 1600s anyone for after the 1800s okay so everyone thinks it was settled by the 1800s yeah it was actually about 1905 that it was definitively really taken seriously and does anybody know who sealed the deal on the atomic theory of matter as a description of nature that was correct the name Albert Einstein ringing the bells no yeah crazy hair old guy with mustache yeah okay yeah believe it or not so when he was a young just post graduate school he couldn't get a job because he was kind of a jerk to everybody that he worked with so he had a hard time getting a faculty position and he took a job as a patent clerk and finished up work he had started while he was a graduate student getting his PhD and one of the papers he published in 1905 which is referred to as his miracle year he published something like four or five papers that year one of them showed that something called Brownian motion which if you if you look under a microscope out of fluid that's got little dust particles in it you'll see the little dust particles jiggling around even though there's nothing around them that seems to be hitting them and the explanation for that was only successfully done with the atomic hypothesis that is the water is made of atoms molecules those molecules are in thermal motion they're bouncing off each other and jiggling around and so anything that's implanted in the water will get nudged by these little water molecules that are around it and kind of jiggle around like you know Captain Jack Sparrow and Pirate Pirates of the Caribbean or something like that and a motion thing that he does all right no I mean is that movie too old for you guys is that I've lived many a decade I see all kinds of nonsense all right so you know we look at solid objects at the macroscopic level the macroscopic level is our level you know roughly meter-sized okay and they look continuous to us but we know that if you keep zooming in on materials eventually what you find out is that way down inside those glass tubes I just showed you are silicon atoms for instance so silicon dioxide okay so here are some silicon atoms that form what's called a crystal lattice that's just a regular arrangement of silicon these little blue blurs here those are the silicon atoms and this is actually an image of silicon atoms in a crystal using a scanning tunneling microscope so this is about as close to seeing atoms as you'll ever get and that's about the best picture of an atom anybody can take it's blurry and it's not blurry because the camera is not very good okay it's not blurry for the same reason your iPhone doesn't take good pictures it's blurry because when you get down to size is this small there is no exact sense of where something is and where it is not it's all waves it's all probabilities there's a chance that an electron is here there's more of a chance it's here and there's less of a chance it's over there so nature has a blur built into it and that's what you're seeing the atoms can't be exactly localized I mean you can kind of see well that atom is mostly right there and it's not over here in this void it's not here in that void but there's a non-zero chance of finding it in that void it's just small compared to the chance of seeing it here in its sort of bonded lattice position so this is about the best image of atoms you can make because nature has a built-in limit to how well you can resolve things now there are lots of cartoon pictures of atoms atoms again are these little building blocks the way they bond together defines the structure of materials okay so you know why gold is gold colored why copper is copper colored why silver is sort of a lighter white gray color all of that is determined by the properties of the atoms and if you understand those properties at the level of physics you can predict the behavior of new materials okay so these are the popular cartoons for demonstrating what atoms look like there's a hydrogen atom it's got one electron and one proton here's a helium atom it's got two protons two neutrons and two electrons in its neutral state here's lithium here's neon these are nice cartoons they're also completely inaccurate this makes the nucleus of the atom look like it's you know in this case bigger than the electron and very close to it and in fact the size of a hydrogen atom is anybody know what the orbital radius of a hydrogen atom is just off the top of your head I'll be impressed if somebody knows this it's a good number to have in the back of your pocket comes in handy and we'll come in handy in this course we'll use it a few times at least okay it's about 10 to the minus 10 meters or one angstrom okay 10 to the minus 10 meters roughly that's the radius of a hydrogen atom is anybody know what the size of the nucleus is what's the size of a proton protons do have finite size they're not point objects but they're teeny tiny for all intents and purposes for faster points well do you think it's about 10 to the 10 anyone anyone want to say it's about 10 to the 10 so it's about the size of an atom not based on what I just told you right do you think it's it's about like one-fifth the size of an atom one-tenth the size of an atom you just shaking your head all the time one millionth the size of an atom okay are you gonna go for a millionth how about a thousandth the size of an atom no no takers it's about it's about a ten thousandth of the size of an atom or a hundred thousandth sorry hundred thousandth the size of an atom so ten to the minus 15 meters is roughly the size of a proton so protons are teeny teeny tiny compared to atoms so these pictures are totally wrong and in fact if you were to look at an atom as that last picture suggested you wouldn't see this neat orderly arrangement of like planetary electrons orbiting a central nucleus like it's a star in our solar system this is actually more what a hydrogen atom looks like so in its ground state that's what a hydrogen atom looks like perfectly spherically symmetric that's just the probability of finding the electron somewhere okay so all that tells you is that there's a probability of finding the electron roughly uniformly around the center of the atom and then you have other states that you can go into you have the excited states of the s-wave you have p-wave you have d-wave you see they start to they're still symmetric but they're not perfectly round symmetric anymore and all of these are just the chances of finding the electron in different locations so it's not the picture of the atom of this orderly electron just cycling around the proton is wrong okay the structure of the atom is actually very complicated and it's understood only with physics it's understood with something called quantum mechanics which is not in this course but it's it underlies everything we do in this course okay alright so that's that's sort of the atomic theory of nature and one of the nice things about this picture is that you see is at the heart of everything that we have around us are electric charges electrons determine the chemical properties of atoms and protons determine what kind of atom it is to begin with and give you the mass of the atom for the most part protons are far heavier than electrons so if you take the electrons off an atom you change the mass by only about one percent all right so electrons contribute very little to the mass of an atom it's really the protons and the neutrons which nearly have the same mass that do it all right neutrons come along for the ride they're electrically neutral they play a very important role in the out of what we won't talk about it in this course all right but it's the protons that determine the atom type in the electrons that determine its chemical bonding properties it's all charges okay so charge plays a really fundamental role in nature how it was discovered is a long story and we'll see snapshots of that story through the course but today what I want you to do is just start playing around with electric charge a little bit just get a sense of of what it means that there's this thing in nature called electric charge and it has effects on the world around it okay so one of the things I want you to do is you know row by row gather yourselves together so move close together take your soda can take your piece of paper and take a wand okay so here's what you're going to do here's what you're going to do you're going to you're going to take you're going to take this piece of paper this is one kind of material right this is paper and you're going to take the wand I gave you I think they're all some variant of plastic all right so they're all basically plastic and you're going to do something called the tribal electric effect you're just going to rub the one with the paper so you're going to rub two dissimilar materials together and then I want you to lay the can on its side so it can roll and I want you to hold the one next to the can and see what happens so rub the materials really good and then hold the one next to the can and see what happens seems like it right this is going to be easier I think PVC is a much better plastic for this kind of stuff okay maybe you're not good at this and then you know let the the plastic rod actually make contact with the can and then see if the force is still strong after it touches it takes a practice you also try getting one of the well what did you observe about the force after you really okay do you have to get a closer to the candle the acceleration to help them well eventually right now well I'll talk about that a moment what's going on even in the air right now there's free water molecules and water molecules are reading electrically and so they'll soak up any net charge off the materials and carry it away you know the water you drink out of the pipes is not pure it's mixed with minerals and things like that that lowers its electrical affinity but if you drink pure water it leeches nutrients out of your body it's actually a chemical hazard so pure water is actually quite dangerous if you dump it ocean will come after you right so I once worked on an experiment that had an ultra pure tank of water and if it ever dumped we had to report it to the EPA because it could damage the ecology of the area it's so electrically greedy that it can actually kill things yeah everyone thinks water is this great thing but water is horrible we're very lucky we get to utilize it for life but it kind of sucks as things go all right so I'm gonna steal the can yeah yeah so water well water has this ability to really soak up charge so let me uh this here all right so you guys want to go back to your seats or whatever that would be that would be fine alright over here okay so yeah so you know you get the idea here I'll use some rabbit fur I didn't want to make anybody use the dead rabbit for this is it I feel guilty that we have this yeah try not to try not to look back yeah okay screw up Zach this is your moment you're gonna be on YouTube all right come on here there we go all right so what's going on let's think about this from a physics one perspective okay first semester perspective what is the velocity of the can right now zero okay and what now it's the velocity the key is it still zero okay so what is a change in velocity signal what's the name for a change in velocity over some unit of time an acceleration okay and if you observe an acceleration in nature what does it imply exists a force right exactly so f equals m a that is a universally generically true law if you observe a change in the state of velocity of a system a force is being applied to it it may have happened in the past and now velocity is just constant and continuing at a higher number or a lower number than when it started but there was nonetheless a change in velocity in some unit of time and that is an acceleration and if you know the mass of the system whose state was changed you can figure out the force you can figure out its magnitude and if you know the direction of the acceleration then you know automatically the direction of the force okay so if you know where this points right so if the can see if I can get this to work a little charge so if the can is moving this way where's the acceleration point yeah this way right because it was moving here but the change in velocity was this way so the acceleration must have been this way all right whereas on this side now the change in the acceleration the change in the velocity is this way so the acceleration must point this way and so the force must point this way okay is this gravity how do I know it's not gravity how do I know that this isn't explained just by gravity what what causes gravity what's the source of the gravitational force well earth is an example of something with a lot of mass so mass is the origin of the gravitational force for all of you guys that you know all you need to care about that's the source of gravity we know better now what causes gravity if you see the movie interstellar you can see an example of a modern understanding of gravity and force okay how do I know it's not gravity well let's say I just so what I've done is I've kind of rubbed this plastic this is an insulator it can hold charge pretty well it can hold it in place it doesn't move this is a metal it's a conductor it's very good at removing charge and letting it move freely through its body okay so I can remove the charge and I can try to use the mass to attract the can we're gonna wait a long time for this to do anything okay but if I just give that a little rub okay takes no effort to move this thing all right so what have I already observed about the gravitational force relative to this new phenomenon this what's called this the electrical force which one's stronger electrical force yeah that without the charge on it does squat I throw a little bit of charge on this easy no problem okay and in fact as well we'll learn later it's really important that the electrical force is stronger than the gravitational force we kind of owe our existence to that okay in part okay the electrical whatever this phenomenon is whatever I'm doing when I rub this material and then put it next to this thing this is an insulator we know now this is a conductor it's made of aluminum metal very good conductor let's charge move through it anybody have any ideas about you know let's imagine that there is something called electric charge for a second so I built up a net charge on this thing and then I put it next to this I haven't put a charge on the can right I've never rubbed the can so what's going on maybe what do you guys think is happening here that's causing the force okay can has a charge but it's okay let's let's check that out let's check out that's true okay throw me another can you throw that can Sophie okay thank you all right so does the can have a charge apparently not okay so it didn't gain a net charge in that process of being exposed to that okay so what else might be going on any ideas can I be an acceptor to the charge meaning define what that means maybe negative in charge and that's positive so it attracts each other okay but then shouldn't something have happened when I put these shouldn't be repelling or attracting or something right okay so it does seem like the net charge of this thing really is zero but what's true about a conductor what can charges in a conductor do they can run freely so let's say this is positive okay are there negative charges in that material someplace where in the electrons yeah everywhere okay yes but specifically the electrons right and the and some of those electrons are free to move in a conductor even the the outermost you know bounded electrons of aluminum atoms they're not that strongly bounded so if you give them a nudge they'll take off okay so if I put a positive charge here let's try that again thank you water in the air for ruining that if I put a positive charge here what do you think that electrons are doing in that can yeah they're moving over here toward it and what's happening then to the positive what where is there a positive charge then that builds up elsewhere in the can on the other side yeah so what happens is that you get these electrons that race over here they can't get off the metal but they get as close as they can to this and the can moves well the electrons want to stay at that closest point right so they kind of slide back to that place closest point and the whole thing repeats again and rolling happens okay so it's it's like taking a string and and wrapping it around the can and tugging on the top of the string to make it roll except you don't need a string you can do it at a distance with no physical contact whatsoever and that's one of the really cool and useful things about the electrical force okay and then to demonstrate that charge is a moving thing okay I've got here just a little soda can I use this for another demonstration but we'll use it for the tinsel that's on the front so you'll notice there's a bunch of tinsel on the front right tinsel is just little bits of mylar it's little metal it's you know use it for like holiday decorations right so you can get this for like a buck you can buy a ton of this for a dollar at target okay so I have lots of tinsel if anybody needs it all right so I've built up a net charge on this insulator watch what happens when I touch the the tinsel all right so I transferred some of the charge to the tinsel but what I've done is I've transferred for instance if this is a positive charge okay then I just yanked a bunch of electrons off that tinsel but there's no place for electrons to come in and replenish the ones that were lost there it's sitting on a bunch of styrofoam cups that's an insulator so there are electrons in the table but they're not free to move up through this insulator back into the can alright so I've created a net imbalance of charge by taking some of the electrons and putting them on this stick and removing them from there so net charge between the stick and that system is still the same I just moved it around and now I've left a net positive charge on that tinsel well the tinsel doesn't want to be near itself when it's all carrying the same charge on every arm so it spreads out now how about I remove that charge I could put the stick back on yep I could also just I'm a conductor all right so I can touch the can now there's no charge left and I can put charge back on all right so it's like a little fluid you can just kind of move it around it's like water it's like pouring water from one container to another you keep the total amount of water the same you just move it around and if you make water flow just right what can you get out of it what can you do with moving water that benefits society energy yeah yeah you can power things with a moving fluid well if we could make electricity move we can make electrons move at our will we could do things with it and look low and behold we've done that right as a society alright so we'll touch on all that as the semester goes on the last thing I wanted to demonstrate is the existence of two kinds of charge okay so what I'm going to do is actually I prefer the paper for this the rabbit fur tends to come apart on me can I get a good piece of paper yeah great okay all right so let me get demo cam aimed up there we go okay so what I'm going to do here because I need some conductors so I'm going to remove the net charge from this the earth is a greedy sink of charge it's called ground for obvious reasons okay and so if you ever need to get rid of a net charge just touch ground touch a piece of metal that connects somewhere like to plumbing okay and then the earth will soak up the excess charge for you you've done this before you've all accidentally gone to ground before but after building up a net charge anybody have an experience with building up a net charge and then touching a piece of metal and going yeah yeah like in the winter when it's really dry you're walking across carpet and you go and touch a doorknob and yeah okay so that you got a homework problem on that actually all right so that's the extra problem I gave you guys so I'm going to use the tribal electric effect and I'm going to put a net charge on this thing so this is just a little balanced PVC pipe okay and then let me yeah just so there's no funny business remove all that charge should be able to affect this all right so these are same materials and I rub them with the same other dissimilar material so I've clearly built up one kind of charge on the system and I can even slow this thing down and I can reverse the direction of motion so same charges repel each other okay we saw that in the textbook but here's a demonstration of it so I've used two insulators to trap the same kind of electric charge and I can make this thing move in my will it's like using the force this is great right I'm all prep for the new star Wars movie this December okay great so that's that's one thing you can do and then here's the other one what if you have a different material so this is glass okay so this is just a glass tube all right and let me drain that again we'll put some more charge on this all right so now I've got paper on plastic okay and now I'm going to do paper on glass okay so when I did plastic against plastic there was a repulsion and now two different materials paper and glass rubbed together I appear to have gotten a different kind of electrical phenomenon built up on this so now see if I can get this to reverse slow it down the acceleration is pointing that way it appears and without touching this one appears to be attractive all right so whatever charge I built up on this it appears to not be the same that was built up on the plastic by Rubin so this is a simple demonstration that there are at least two kinds of charges in nature we call them positive and negative and actually does anybody know who is the person who was signed positive and negative charges based on materials being rubbed and so forth and everybody know the the scientist that that's responsible for that and as you'll see to blame for all of our problems with labeling things and electricity later on something with it like ohm is that what you're doing like George seem George see no no he did other stuff no earlier Franklin did I hear Franklin yeah Ben Franklin yeah Franklin is the one that that assigned the positive and negative designation okay so that's just some nice demonstrations but of course what you need to do is is you need to understand how to utilize a description of nature that captures everything we've just seen and lets you actually calculate the properties of the world and then test those calculations by doing more experiments okay and so after a tremendous amount of work there is a singular equation that describes the force exerted by the electrical phenomenon between two point charges q1 and q2 alright so q will often be used to denote charge and these are point or point like charges point like charges okay what I mean by point I mean that the dimensions of the charge where it's encapsulated or smaller than other dimensions in the problem so you know for all intents and purposes this remote control is a point in this room because this room is huge compared to this thing so if I put a big charge on this I could approximate this as a point charge as long as whatever else I bring near it isn't so close that the dimensions of this thing matter okay you know that there's a point over here and a flat side here those dimensions could matter for the distribution of charge but if I'm really far away from it it won't I won't be able to tell what its dimensions are anyway alright so for instance the proton has dimensions it's about 10 to the minus 15 meters in radius but we're huge compared to that so for all intents and purposes protons look like points to us okay to another proton that may not be true but to us it is alright so q1 and q2 are charges and we've seen that as we build up you know if you guys played around with that the triboelectric effect a bit more you'd observe that the more you rub the more charge you transfer the stronger the force would be on the camp alright so that's a simple experiment you could do you can try like you know one two and then try to attract the can drain off the charge one two three four so double the charge and then try it again and see if the force that goes bigger or stays the same you know there's likely a saturation point plastic can only hold so much soaked up charge but you can play around with like you know just rubbing a little bit rubbing a lot and see if you can build up more charge and you should find that you get proportionally more force what we also find is that the distance between the two point charges affects the strength of the force so the further apart the charges are the weaker the force and this this relationship goes what is said to be quadratically that is it goes as a square okay so in this case one over the distance squared so our one two is the distance okay and it would be in meters so in meters m between q1 and q2 and I should say here the charge is in coulombs coulombs and what we observed is that the force acts on a line connecting the two charged objects you know that if you have a charge here and a charge here like I did with the pendulum this little torsion pendulum so the spinny thing okay when I when I put the other pipe next to it that was charged it didn't lift the pipe up it didn't push the pipe down the force always acted on the line connecting the two of them so I could either attract along that line or repel along that line okay so that little symbol over on the right is a special thing you're going to get to know and love these it's called a unit vector and it just indicates direction its job is purely to indicate direction the length of a unit vector is always one that's what makes it a unit vector so the unit in unit vector means length equals one so if anybody were to ask you what's the length of a unit vector you say one okay wow look at you little sheeple that's just terrible your college think for yourselves come on all right so yeah so that's a good thing to keep in mind right unit vectors always have a link to one I know that seems maybe trivial but it's a really important thing and it's just good to know it's good to have it in your heart and in your head okay because it will come in handy you'll be able to save your bacon with that that information when we talk about coordinate systems okay like a Cartesian coordinate system with an x axis and a y axis which you need to use to describe a situation you always need to draw a coordinate axis and I'll illustrate this in a moment okay x direction y direction we have unit vectors that tell you how you know whether you're going in the x direction or whether you're going in the y direction and those unit vectors are an I with a little hat over it for the x direction so this the I hat means moving in x direction and then we have a j hat which means moving in the y direction if I tell you the unit vector is negative i hat what direction am I moving in left yeah so negative x direction okay so directions you know they can be flipped and if I say negative j hat that's the negative y direction so unit vectors they fulfill in in the world of vectors okay in the world of vectors let's say we have a vector v okay and it's got a magnitude that is a length all right so if v is a velocity and somebody asks you what's the magnitude of your velocity or your speed you would give them a number a pure number something called a scalar okay so for instance like five meters per second but if I asked for a velocity I also want to know direction right because it's important to know sometimes where you're headed not just how fast you're going there so for instance it's also important to specify that you're doing five mile or five meters per second in the positive x direction or I hat so math is just a way of representing complex linguistic ideas in a simple notation math is the language that we use to describe nature and it works ridiculously well okay it's been developed over hundreds and hundreds of years if not thousands of years and it's a very effective shorthand for saying very complicated things in a compact way so it's just another language okay like English or French or Spanish it's got its own symbology it's got its own rules but once you learn them you can describe all kinds of crazy complex things in nature including weird forces like this one okay so this is the magnitude this is the direction and vectors have both right I always think back to uh despicable me the villain vector right his crimes have both direction and magnitude okay so for me that helps I like that okay so if that helps you to remember it so be it but if you're asked for a vector if I ask you for the force I don't want just the strength of it the magnitude of it I want the direction of it as well okay so that's a vector it has a length and it has a direction and some of them can be quite complicated we could imagine another velocity that's five meters per second in the positive x direction but it also has a three meter per second component in the negative y direction that's also a vector okay so that basically means that I go every time every time a second passes I go three meters in the positive x direction so let me see if I can do this here I'm not five meters in the positive x direction I'm not gonna be able to make this work here we'll go this way we'll call this positive x so I go two three four five meters okay and then I'll see that was this was positive x so that's positive y so I'm going to go great I'm going to go three meters and they have positive y direction okay I'm going to try to go through the wall all right that's all that means it's just a way of representing you know you go one way and then you go the other all at once all right so unit vectors play a really important role they let you in a compact way describe direction alongside magnitude so another way to think about vectors is if you were going to a friend's house and you asked well how do I get to your house I've never been there before they might say to you something totally useless like oh no problem you go 10 miles and then you go three miles and then you're there that's useless directions right because they didn't tell you anything about what direction try to go north and then east or do you go south and then 10 and five what so vectors directions are only useful if you specify both magnitude and direction all right so 10 miles north five miles east boom you're there no problem all right punch it into google maps all right so so that notation is something I'll use a lot it combines both magnitude and direction information and the direction is entirely encapsulated in whatever sign is in front of the unit vector and the unit vector itself okay so that's negative y direction and they they when the other one's a positive number so that's positive x direction there's any questions on on that all right we're gonna use this we're gonna use this and get some practice with this okay okay so this is the form of Coulomb's law it's got a constant out in front as well which I'll just write down here this constant is not something that could be predicted it had to be measured all right and it's just a number it's a number that nature uses it's an important number but it's not obvious why at first all right so it's 8.99 times 10 to the ninth newton meter squared per Coulomb's square since forces are measured in newtons all right kilogram meter per second squared and you've got Coulomb's times Coulomb's divided by distance so meters squared if you want to get newtons on the left hand side whatever that constant of nature is its units have to be newtons times meters squared to cancel out the meter squared in the denominator divided by Coulomb's square to cancel out the Coulomb's squared in the numerator all right so that's where the units come from it has to be just figured out by doing experiment okay okay so that's the constant k just because the book makes this point and we will use it later there's often another way of writing k which is that it's equal to another number epsilon epsilon naugt not epsilon sub zero subscript zero is physicists just call this epsilon not okay so epsilon not is this thing and you can figure out what it is by punching this equation into your calculator and solving for epsilon not but epsilon not is 8.85 times 10 to the negative 12 Coulomb's squared per newton meter squared it flips the units upside down okay epsilon not turns out to be the more fundamental constant of nature nobody knew what it was at the time it was just something you measured and and then you moved on with your life but it turns out that number is extremely deep it's embedded in the cosmos and it's related to something even more fundamental in nature but we'll get to that later okay all right so let's use this thing let's learn how to solve problems set up and solve problems so I'll use the remaining time to kind of get this moving and then I I think we won't be able to get to your problem solving exercise that is why I hate doing lectures because it wasted your time okay but I'll give you an example problem and then you just you'll do an extension of it so you can practice this the solutions will be posted on the website so you can try it yourself see how far you get and then and then go from there okay I want to warn you though these problems that I cook up I'm trying to make them relevant to you as non-physicists all right so I'm looking for physics physics is everywhere in the natural world so I just try to pick it out of things like chemistry and biology and then use examples from those to build up physics concepts but I will warn you that you know this is what biology looks like to a biologist right you know or at least an ornithologist is that bird studies I think that's bird studies uh or is that insects I never remember but anyway you know all the parts are properly labeled right that bird's got a crown an nostril bill lesser coverts what the hell's that uh greater coverts but the hell's that what's a covert I don't know what that is tarsis vent vent all right so but this is what biology looks like to a physicist right there's some bird up there it's got some got some bird in the front there's some bird on the back there's lots of bird on that bird okay so so I'm warning you that I'm giving you examples that have physics embedded in them but they're greatly simplified models of the real world okay they're meant to make a point but it's to illustrate that there is physics down there you know deep in the cell uh you know lurking in the eyeball there's physics everywhere you have to tease it out in the sense that in order to like solve problems with it I do have to simplify things a little bit okay so birds just made a bird all right so let's take a look at the cell membrane of a living cell right if you could zoom in on a cell membrane right down to the atomic scale or the molecular scale at least you know you'd have this nice bilipid membrane here separating a whole bunch of ions on one side from a whole bunch of ions on the other and so the job of the cell membrane is to be non-permeable to these ions but allow them through with ion channels or pumps all right so there are structures embedded in the cell membrane that can be activated and deactivated and allow ions of different kinds to be moved in and out of the cell to maintain or equilibrium or create non-equilibrium situations that are necessary for life processes to proceed okay so you'll notice in this particular snapshot of a moment in the life of the cell wall uh there's a whole bunch of sodium ions outside a little bit of potassium uh you know lots of chlorine and then inside you have some anions so these are just you know in this case there they have four minus charge four minus the elementary charge a whole bunch of potassium and a little bit of chlorine okay and so you can imagine well I you know I've got charges in here I've got I've got potassium next to potassium what are those two things going to do are they're going to attract or repel each other they're going to repel because they're the same charges right and you can see here ah but the potassium is attracted to the anions it kind of bonds with them right so the anions will soak up the excess potassium here and you can kind of see that that's the picture along the wall here okay so you can already see in the in the processes of basic chemical and biological phenomena there's charge down there and that's how anything sticks to anything else at the molecular level it's all electric charge gravity doesn't do anything for you down there it's way too small okay so let's build a physics problem off this that we can use to explore Coulomb's law so here's a very simple problem involving some elements of the picture I just showed you right that's really even that cartoon was fairly complicated let's boil this down to something ridiculously simple so let's imagine I have an anion sitting here a distance twice L where L is given here 2.8 times 10 of the minus 9 meters from a potassium ion with a plus one elementary charge on it and then up there equally distant but in the but in the y direction so we can call that the x direction we can call that the y direction you've got a sodium ion also with a positive one elementary charge so negative four elementary charges plus one elementary charge plus one elementary charge and again the elementary charge is e which is 1.602 times 10 to the minus 19 coulomb's what we want to find out is what is the force direction and magnitude that the potassium ion k exerts on the sodium ion all right all the way up there so there's k there's an a we want to figure out what's the force that that exerts on that through its electric charge coulomb's law is the way we will figure that out we will begin by making an assumption and that assumption is that the potassium ion is very small compared to the dimensions of the problem like l here and that's probably true i mean i told you that you know the radius of an atom is roughly an angstrom 10 to the minus 10 meters and this is you know 10 times bigger than that this distance so that's probably not a bad approximation for this all right so for the problems we're going to do at the beginning you you should feel free to make that approximation we're not going to break that approximation quite yet all right in approximation does everybody know what an approximation is if i say that word does that make sense anyone not know what an approximate when i say the word make an approximation does anybody not sure what that means in terms of the context of problem solving it's basically making an educated guess okay it's okay to do that as long as you can justify all right so we're going to make an approximation we're going to make an educated guess and we're going to assume that these are all basically point charges so we can just use coulomb's law okay that's great but what the hell does it mean to use coulomb's law for this problem well let's write coulomb's law down so i'm going to get rid of this stuff and i'm going to try to work from left to right here all right coulomb's law the force that the potassium exerts on the sodium so f vector with a subscript k comma n a so i'll take that to mean the force that k exerts on n a is equal to some constant of nature the charge of the potassium ion the charge of the sodium ion divided by the distance between the two of them squared and then this whole thing is multiplied by this totally f that funny looking thing up here with your hat on top of it okay the unit vector is often the most difficult part of these problems okay so i'll save it for last when you are attacking a physics problem the first thing to do is look at the problem and ask what principles of nature could i apply to this okay well it involves a question of force so maybe i'll need f equals ma although we've not been told anything is moving so there's no a yet so maybe we don't need that this involves electric charge okay these are small charges compared to the dimensions of the problem so coulomb's law should apply so okay i should probably apply coulomb's law to this to answer the question i've got charges i can figure out distances and then i can get f and i need to get direction and i need to get magnitude okay so you so what you want to do is start being able to look at a problem and think first what physics ideas could be used to address the problem not all of them will be necessary but think about it for a moment the next thing you want to do is because this is a problem that involves direction you want to establish a system of coordinates in the picture now i've done that for you here i've implied by drawing these dotted vertical and horizontal lines that you could place the origin of the coordinate system sort of smack in the middle between the anion and the potassium ion and the anion and the sodium ion in the vertical direction okay but you could easily have put the origin of the coordinate system this point down here so you could make this the y-axis and this the x-axis it's sort of totally up to you which you want to do there okay it does change potentially the numbers or the directions you'll get in the end if you move the coordinate system around positive it can become negative and negative become positive but it won't affect the physics of your solution in the end right so i'm going to pick my coordinate system to be i think there um you see if that's what i did here just a simple formula yeah why not that works sure um okay so let's pick the low hanging fruit off this equation do i know k yeah it's just constant of nature take 0.9910 of the 90 meter square per column square done i know that do i know q k do i know the charge of a potassium ion yeah yeah plus one times the elementary charge got that low hanging fruit charge of the sodium ion yeah same thing in fact know that what about the distance between the potassium ion and the sodium ion do i know this number r no i don't not off the top of my head but how could i figure it out how could i figure out the distance between k and n a any ideas make a triangle yeah yeah i can make a triangle right and this is great because it already looks like one it's got a a base of length 2l and it's got a height of length 2l and so it's a right triangle which is nice so i should just be able to use what to figure out the length Pythagorean theorem excellent yes all right so we're going to use the Pythagorean theorem to figure this out and that just says if i have a triangle like this with a right angle here that c squared is equal to a squared plus b squared that's it you're going to get so much mileage out of this theorem in this course okay so a in this case is 2l b is also 2l so i square 2l i square 2l and i get 4l squared plus 4l squared equals 8l squared that's that's the distance squared okay so r k and a will be equal to the square root of 8l squared i'm going to leave it like that for now you can simplify a bit more you can say well i'll take the l squared out of the square root just put an l in front of the square root you can do that uh the square root of 8 is anybody you know what that is off the top of your head yeah two root two right and you said a little less than a little less than three that's right why because the square of three is okay good just checking just making sure we're good all right yeah it's a little less than three and in fact you can does anyone know what root two is off the top of their heads yeah 1.4 roughly yeah 1.44 i think if you want to go one decimal place further yeah so two times root two is two times 1.48 2.88 which is just slightly less than three so it all checks okay yeah so using a little intuition like that oh it has to be less than three because three squared is nine right but it's greater than two squared which is four okay so you should be able to guesstimate based on that but i'm just going to leave it like that for now okay we're nearly there we just have to figure out what this curious thing is over on the right this so-called unit vector r hat here's what we know about this unit vector we know one thing about it right away what's the one thing we know about the unit vector right away without doing any math whatsoever what's that people what's specifically about it not lucy right yeah what's specifically about it is equal to one yeah magnitude the length yeah exactly those words are interchangeable here right so what we know right away is that this whatever this thing is it's magnitude if i put vertical bars around it i mean magnitude that it better be equal to one that's a check so we're going to figure out exactly what that is in our coordinate system using our numbers but at the end of the day it had better be true that the magnitude of that thing is one does anyone know how to get the magnitude of a vector generically if i gave you a vector hey how do you get its length mathematically square the components add them together and take the square root so i would take ax squared plus a y squared plus az squared and i would take the square root of that okay if it's only a two-dimensional vector there is no z component and i should put a square on that that'd be helpful okay all right yeah and in and in vector mathematics notation this thing is known as a dot a the dot product of two vectors okay so a dot a is each of its components squared and then add it together to get the length of a which is written like this you just take the square root of a dot a okay so in compact vector notation okay we're exercising a little bit of this math language that's how you write the magnitude of any generic vector we'll call it a vector in this case this is what you have to do but that's what it looks like notationally okay what if we wanted to get a unit vector that represented the direction of a and had magnitude of one any idea how we would get that just generically from a vector a like this how could we get an a hat a unit vector that points in the direction that the vector a points but has a magnitude of one any ideas right yeah divide basically the whole vector by its own length or divide each of the components by its by the total magnitude so i could write this as a x in the i hat direction i don't know where that came from plus a y let's just do a two-dimensional vector for now so i could write that vector a as a x i hat plus a y j hat using that notation i showed you earlier and if i divide that whole thing by the magnitude of a okay you can get something that has a magnitude of one let me show you why if i take a and i dot it into itself take the dot product of a hat with itself what do i do up top i get the sum of the squares of the components a x squared plus a y squared down here i get the length of a squared okay but what's the length of a the length of a is the square root of a x squared plus a y squared that's that's the length of a we just did that over here right just a two-dimensional version of the same vector so if i square that if i square this the square root goes away a x squared let's try that again a x squared plus a y squared so i get a x squared plus a y squared divided by a x squared plus a y squared and that is equal to wow there's no confidence in the room on this one practice this okay you should confidently be able to look at a x squared plus a y squared divided by itself is anything divided by itself is one okay so if i put it that way you probably would have felt more assertive right i'm taking the thing up top and i'm dividing it by itself in the bottom i have to get one it's the only way that ever works out okay all right so the only thing left to do here is to figure out what the unit vector is for our problem okay i'm going to wipe this is that okay okay it'll be in the youtube video and it'll be in my notes which i'll post so don't feel bad if there was something up there that you didn't get written down you can always pull it off the notes later okay all right well all we have to do to get our hat i'm just going to write it without the subscript for now is figure out what is our vector and divide it by its magnitude so let's look at our coordinate system we've got um the force that the potassium ion is exerting on the sodium ion so the convention this is the one thing you're going to have to just memorize from coulomb's law and use over and over and over again this convention is going to come up all the time in coulomb's law problems and related problems the convention is that our hat starts on the source of the force and ends on the recipient regardless of whether the force is attractive or repulsive let the charges tell you that just memorize this as a convention you're going to start when you have to figure out what our hat is you got to figure out the components of the r vector and to do that you're going to plant the origin of this vector on the charge that's acting on the other charge so who's acting and who's the recipient of the force in this problem k is acting right so our vector r is going to start here and it's going to end up there and it's going to have components it's going to have a component so that vector is going to point this way right so it's going to have a component along the negative x direction going to have a component along the positive y direction you don't even have to write down any math just draw the arrow and you'll just look at it and go okay it's going to go left and it's going to go up so negative x positive y whatever the numbers are in front of those directions that's what that vector is going to do so our our vector is going to go 2l to the left 2l negative i hat and it's going to go 2l up 2l j hat okay and then i just need to know the magnitude of r but i got that already did that that works done it's just the square root of 8l squared so r hat is just going to be equal to negative 2l i hat plus 2l j hat all divided by the square root of 8l squared you're almost done at this point okay you can simplify that a little bit as i said that could be simplified but that's just an extra step okay but we've got it we've we've done it we've got f vector and we've got magnitudes and directions we got k we got q we got the other q we got that distance squared we got r hat we wrote it down based on that picture we just let the plug in numbers and simplify and we're good to go at this point we're done okay so that demonstrates this process of setting up and stepping through Coulomb's law problems your exercise which we're out of time to do is to calculate the total force on the i believe it's on the anion let's see i think i have it um yeah what is the force that the potassium ion and the sodium ion together exert on the anion that's the problem i was going to give you guys to do but i spent too much time blathering again that won't be a common thing i do in class mostly we'll just kind of get going with problem solving rather than having a little exposition okay do you guys have questions i don't have any more office hours this week i'm going to be traveling tomorrow but you know you can make times to come and meet with me outside of office hours if the office hours aren't convenient all right um make sure you get homework zeroing to me and that's a wrap for today because that question isn't it's not due it's something you can just practice with and i'll have the solutions up on the webpage for you this afternoon okay so try it out yourself from your notes and then see how far you can get and then take a look at the solution okay that's what i recommend oh and if you have like uh coat cans and paper could you bring that down to the front please so i can uh put those back in the storeroom thank you