 Electricity is a force that powers our modern world. It lights our lights. It drives our computers. It is the source of all of the work we do in the modern era. But where does electricity come from? It's at the heart of lightning bolts. It shocks us when we touch doors on a dry winter's day. It reaches out across empty space, exerting forces on matter. To understand this particular force of nature, we have to understand its origins and concepts like electric charge. Let's begin to explore this now. In this lecture, we're going to begin to explore the concept of electric charge and the force that is associated with this property of nature. To begin with, let's review a simple question, one that bears on essentially the entirety of physics one. And that question is the following. What exactly is a force? Take a moment, pause and reflect on what you learned in physics one before continuing on in the video. Fundamentally, a force is any action which causes a piece of matter to change its state of motion. That is, to accelerate, either positively increasing its velocity or negatively decreasing its velocity. This stems from effectively Newton's second law of motion. Force equals mass times acceleration. If you observe in nature that a piece of matter is changing its state of motion, that is experiencing an acceleration, the cause of this, according to Newton's second law of motion, should be a force. Now we can perform a very simple experiment that will tell us a little bit about electric charge and show us that it has an associated force that goes with it. One of the earliest phenomena that human beings experienced is static charge. This is when, for instance, you shuffle your feet across a surface on a particularly dry day and you notice that when you then go and touch something an external object, you feel a jolt in your finger. It hurts. You'll often hear a popping sound that associates with. This is a spark of electricity jumping, for instance, between your fingertip and the surface of the external object that you are about to touch or reaching toward. A common experience in modern life might be when shuffling your feet across a carpet on a relatively dry winter's day and then reaching for a metal doorknob, you may experience this electric jolt, this little spark. It hurts. This is one of our first experiences with electricity apart from, say, frizzy hair on a dry day where your hair seems to fly away and won't stay put. This too stems from electric charge and the force associated with it. Both of these phenomena have something in common, both frizzy hair and rubbing your feet on, say, a carpet and then reaching out toward a doorknob and feeling a jolt. These phenomena have one thing in common and that is something known as the triboelectric effect. That is, rubbing, friction, leads to a buildup of electric charge and that charge can then be released by placing a surface in contact with another surface that can take up the charge. Triboelectricity is the act of rubbing a surface and building up an electric charge. So we can do this. Let's imagine we suspend an object, like a glass rod from an insulated material, like a string, and we hang this in mid-air. And if the air is sufficiently dry, what we can do is we can take a common object, like a paper bag, and we can rub the glass. In rubbing the paper against the glass, we transfer charge from one object to the other and we build up electric charge on, for instance, the glass. Now, with the glass suspended and charged in the air, we can take another object, an independent piece of material, for instance, a piece of PVC plastic pipe. We can take a different piece of paper bag and we can rub that paper bag against the plastic pipe. We can then hold the PVC tube close by but not touching the glass. Let's observe what happens. If you watch carefully, you'll observe that the PVC pipe appears to attract the charged glass toward it. So whatever we've done in rubbing the glass and then separately rubbing the PVC pipe, the plastic pipe, we have built up charges on those that leads to an attractive force between the charged glass and the charged plastic. The triboelectric effect applied to these two materials seems to have caused an attractive nature to result between the two of them. We are clearly changing the state of motion of the glass tubing. In fact, by putting the PVC pipe on the other side of the glass tubing, again, never making contact, I can reverse the direction of motion of the glass pipe because the glass is attracted to the plastic now that they are both separately charged. So we observe that one aspect of the electric phenomenon is an attractive force. This is perhaps reminiscent of something like gravity where two masses attract each other, even through empty space, through this force we call gravity. But here's where electricity shows a distinct nature from our observations of gravitational forces, and that is that electric forces can have a repulsive aspect to them as well. For this experiment, let's take a piece of plastic pipe, PVC pipe, suspend it again from a string and insulating material, something that keeps it from touching other materials and conducting charge. We rub a paper bag against this PVC pipe and we build up a triboelectric net charge on the plastic pipe. Now again, we take a different plastic bag, we rub it against the other PVC pipe that we've held in our hand for the previous experiment and build up a charge there. If electric force is the same as gravity, we expect that we've built up charges on both of them, they should both still attract each other. But in fact, what we observe is that we have built up a charge on the suspended plastic and we've built up a charge on the plastic I'm holding in my hand and now they repulse each other, they repel each other. One plastic pipe with its net electric charge pushes away the other plastic pipe with its net electric charge. The electric force has both an attractive aspect and a repulsive aspect. And this is how we know, observationally, apart from other things that we can do with experiments, that it is distinct from gravity, at least at first blush, that seems to be the case. It has two aspects, attraction and repulsion, and they're both clearly very strong. Gravity has a very strong attractive feature to it. We've never seen direct evidence that gravity has a repulsive component to it, although that is not completely ruled out by experiment. Nonetheless, if gravity has a repulsive aspect to it, it seems to be far less common, perhaps far weaker, than its attractive component. Whereas in the electric force, which we're exploring here, the repulsive and attractive aspects seem to be roughly equivalent in scale. Now the history of the study of electric charge is filled with many people who looked at this phenomenon, played with this phenomenon, wound up doing careful or not careful experiments with this phenomenon, but ultimately came to a better understanding of this intriguing force of nature, one that is distinct from gravity. Benjamin Franklin, one of the founders of the United States, for instance, but known in his time also as a statesman, a scientist, an engineer, an orator, and a diplomat, among other things, experimented a great deal with the electrical phenomenon, which was not at all well understood in his day, and he illuminated at least some aspects of the electrical phenomenon. So this is a painting of Benjamin Franklin about 1759, he was quite young, and between 1706 and 1790, which is when he lived, he did many different things, including experimenting with optics, experimenting with electricity, and around the 1750s was really the peak of his experimentation with electricity. We owe many things to Benjamin Franklin, not the least of which is that he identified that there were two kinds of electric charge and assigned labels to those two kinds, positive and negative. When two similar charges come near each other, for instance, a positive charge and a positive charge, they repel. This would also happen with a negative charge and a negative charge. Two negatively charged distinct objects would tend to repel away from each other. But an object with a positive charge and an object with a negative charge brought near each other will attract. Revisiting the experiments that we did earlier in this video, we can see that we would conclude that the glass and the plastic had opposing electric charges, one of them was positive, one of them was negative. They attracted to each other. But the plastic rubbed in the same way, but separately and kept separate the whole time, each built up the same kind of charge and they repulsed each other. Electric forces have two aspects, attraction and repulsion, and the origin of those aspects are these charges that you can build up on surfaces, sometimes positive, sometimes negative, according to the lingo that Benjamin Franklin left us and that we still use to this day. Among Franklin's many inventions were things like bifocals, which many people still use today to help with nearsighted and farsighted seeing, but also the concept of the lightning rod, putting a metallic, electrically conductive object on the highest point of a structure so that if lightning is going to strike the structure, it strikes this conductive high object and then the electricity can be channeled away from, say, the wooden structure which would otherwise catch on fire if it were directly struck by a lightning bolt. And these are all still inventions that we benefit from to this day, though they're quite old by modern standards. What we know now in the modern world from centuries of experimentation with matter and the electric force is that the origin of all of this are the atoms. Atoms, for instance here, we have depicted a helium atom, has each a cloud of electrons orbiting a central nucleus, a tightly packed electrically charged object orbited by differently electrically charged objects. The electrons have negative charge and the protons in the nucleus of every atom have positive charge. The electrons have negative charge, protons have positive charge. A remarkable fact of nature is that the magnitude of the electric charge possessed by each electron and the magnitude of the electric charge possessed by the proton appears to be exactly equal although their signs are opposite. In the units of nature that go with electric charge, the Coulomb, and I'll come back to who Coulomb was in a moment, the quantity of electric charge carried by the electron is the negative of this number, which is written E. E is 1.602 times 10 to the minus 19 Coulombs and this is known as the fundamental charge of nature. This is considered to be the foundational unit of measure for all electric charge. Things get complicated when you begin to study the nucleus of the atom. You'll see that it's in fact possible to have an object with a less charge than what we call the fundamental unit of charge but nonetheless for largely historical reasons we still consider this number 1.602 times 10 to the minus 19 Coulombs to be the smallest unit of charge typically present in nature. Electrons have the negative of this number negative E or negative 1.602 times 10 to the minus 19 Coulombs. Protons possess plus E or plus 1.602 times 10 to the minus 19 Coulombs. Now the nuclei of atoms contain one other kind of particle, the neutron. The neutron possesses no net electric charge. It is neither positively charged nor negatively charged. It is an uncharged object and so viewed from a great distance the neutron appears to possess no electric properties whatsoever. The electron however has electric properties and the proton has electric properties and those properties are caused by the fact that they possess electric charge of either plus E for the proton or minus E for the electron. Now this is just a cartoon depiction of an atom and in fact the nucleus that you can kind of see here at the center of this cartoon is far bigger than it actually would be in reality. To give you a sense of scale this whole atom with its electron cloud is roughly one angstrom or 10 to the minus 10 meters in size. The nucleus of an atom is typically about one femtometer or one fermi 10 to the minus 15 meters in size. The size of an atom is about 100,000 times the size of a nucleus. This is clearly not 100,000 times smaller than the extent of this cloud. So these cartoons should be taken with some skepticism about the accuracy with which they depict nature but nonetheless they give you a useful idea to hold in your mind in depicting the fundamental building blocks of matter as we know it, the atom. Now that we have established that electric charge in its two kinds, positive and negative are the origin of electric forces let us explore the rules that govern those forces and this ultimately stems from a law of nature Coulomb's law which you should add on top of the existing laws of nature that you've already learned in Physics 1 Newton's laws of motion there are three of those plus the conservation laws that often go with physical phenomena such as the conservation of total energy and the conservation of total momentum and the conservation of total angular momentum. Coulomb's law is another law in what you can imagine to be the great t-shirt of physics laws that you can very simply write down to describe the behavior of the natural world. The namesake of this is Charles Augustin de Coulomb now he was born and later died in France born in 1736 died in 1806 one of the things that you'll observe about a lot of the key figures who were involved in the origins of the first understanding of the electric force is that these people were often living during times of upheaval and revolution and the renaissance and the birth of democracies in the western world so Europe, the United States what became the United States in the late 1700s and so forth. Science and these activities were not separate from each other and it's no accident that many of the people who figured a lot of these key things out about electricity for instance and magnetism were doing their work around this time when exploration of the natural world curiosity driven inquiry was greatly encouraged and considered a normal part of the language art music renaissance of the day. Coulomb first presented his work on electricity and magnetism in 1785 and what's remarkable about this is that in coming to understand the rule that governs the strength of the force between say two electric charges Coulomb laid the foundations for a nearly complete understanding of the atom and how atoms bind together both to form themselves and with other atoms to form molecules and larger structures the foundations of chemistry and biology these explorations of basic questions why do two charged objects attract or repel each other by what degree do they attract or repel each other given different distances that they're separated by these basic questions led to a fundamental revolution and our understanding of material substance in the world around us and the origin of its structure this is the first step to understanding that now we're just going to dive right in here to Coulomb's law I'm going to come back to the pieces of this in a little bit and work through how you should think about solving problems where electric forces are present and you need to apply Coulomb's law to understand those forces their directions and magnitudes a reminder force like acceleration and velocity and displacement is a vector quantity and as such it has at least two things that you have to figure out in order to fully describe a force you must figure out the magnitude of that force the strength of that force and you must figure out the direction that the force points now Coulomb's law puts the force on the left hand side that is to say the force between two electrically charged objects one and two and specifically by the way I've written this this is the force that the second object two exerts on the first object one if we're thinking about tiny little indivisible particles each with their own electric charges then this would be the force that particle two exerts on particle one if we wanted to write the force that particle one exerts on particle two we would swap the order of these numerical labels both the left and the right hand sides of the equation now there are other things that are present in Coulomb's law there are the electric charges which we've explored a little bit in the early part of this lecture video the electric charge possessed by object number one particle number one in Coulomb's is almost always written with the letter Q either little Q or a capital Q that denotes charge in Coulomb's and Q with a subscript one is the electric charge possessed by particle one or object one similarly the electric charge possessed by particle two or object two is denoted Q with a subscript two I will simply refer to these with the verbal shorthand Q1 and Q2 going forward Coulomb figured out through careful experimentation that the force is proportional to the product of the charges now if Q2 is a negative charge then Q2 will be a negative number if Q2 is a positive charge then Q2 will be a positive number a positive or a negative sign can be lurking in either or both of Q1 and Q2 I'll come back to that in a bit Coulomb also observed through careful experimentation that the strength of the force decreases with the distance between the two objects if I double the distance between two charged objects the strength of the force either attractive or repulsive is reduced by a factor of four if I increase the distance by a factor of three the strength is reduced by a factor of nine always the square of the increase in the distance so what Coulomb observed was that an inverse square law applies to the electric force just as it seems to apply to the gravitational force now the origins of gravities 1 over r squared law and the electric forces 1 over r squared law are fundamentally distinct from each other but nonetheless it's an interesting coincidence of nature that they both appear to follow this inverse square law so Coulomb of course would then be forced to write in his equation describing the strength of the electric force a one over the distance between the two charges squared piece in here there's this little guy sitting out over on the right this is a unit vector that tells us the direction ultimately that we will associate with this force forces are measured in newtons kilogram meters per second squared but we have the product of Coulombs here so we have Coulomb squared in the numerator we have meters squared in the denominator and the unit vector is dimensionless it only tells us which way something points it doesn't tell us anything about the length of that thing length is one by definition with no units so q1 times q2 over r squared has units of Coulomb squared over meters squared well that's certainly not kilogram meters per second squared so Coulomb had to introduce a constant of proportionality k that multiplied the right hand side to give the right units on the left hand side, newtons so k has units of newtons times meters squared divided by Coulomb squared and I'll come back to this constant in a bit this is the form of Coulomb's law the force exerted by particle two on particle one including its magnitude and direction is given by a constant times the product of the charges of particle one and particle two divided by the distance between them squared times a unit vector now I've emphasized this and I'm going to re-emphasize it electric force is a vector quantity it has magnitude and direction and the good news is that we know how to handle vectors and more to the point we know how to handle a situation where there's a bunch of vectors we have to add up to get a total resulting vector so for instance if we have more than one charge acting on particle one if for instance there is a particle two and a particle three and both of them can exert an electric force on particle one then to get the total force on particle one we have to sum up the individual external forces that two exerts on one and three exerts on one and this looks just like the sum of forces from the first semester of physics, physics one if we have a whole bunch of particles acting on particle one then we sum up all the forces between particle two, three, four, five, six all the way up to capital N maybe N is Avogadro's number N is ten it's just some total number of particles we sum up all the individual forces of two on one plus three on one plus four on one, etc and we get the total vector force on the right hand side of the sum forces are vectors magnitude and direction they add like vectors you have to add them together correctly using the rules that you should have begun to master in physics one now I want to remind you also of another force related concept and that is equilibrium this is a special state of a system let's imagine we have N particles all exerting forces on each other if none of them are moving at all that is for instance they're all at rest or none of them are accelerating so maybe their velocities are not zero but they're unchanging then we can state that the system has a net zero force acting on it and that is a state of equilibrium so for instance again consider particle one being acted on by particles two, three, four, etc all the way up to particle N equilibrium is achieved when the sum of all those forces two on one, plus three on one, plus four on one, etc up to N on one when all those forces add up to exactly zero and there is no net acceleration in the system that as a reminder is what is meant by equilibrium and equilibrium is a very useful condition if a system is in equilibrium you have this equation to help you to constrain what's going on in terms of the motion involved in the constituents in the system now one of the other things that is really neat about electric charge is let's imagine having three charges one, two, and three and they're in some box in space and I ask the question what's the total charge in the box well if you want to know the total charge in the box all you have to do is figure out what are the electric charges of each of the individual particles in the box and add them up keeping their signs in place so if some of them are negatively charged and some of them are positively charged I need to figure out the charges of the negative ones and make sure I put the minus signs in front of them and add those up and then figure out the charges of the positive ones make sure there's plus signs in front of them and add those up and then add the negative number and the positive number to get the total resulting charge at the end charges add up and in a closed system where no charges can get in or get out total electric charge is conserved this is another interesting feature of nature that's been observed time and again like conservation of total energy in a closed system conservation of total momentum in a closed system conservation of total angular momentum in a closed system total charge is conserved in a closed system a system where nothing can get in or get out and if you want to know the total charge in the system add up all the individual charges and you'll get the answer so with that in mind here's a warm up question for thinking about electric charges I've depicted particle one with its charge negative E particle two with its charge negative two times E and particle three with its charge positive E what is the total charge of this system imagine I enclose this in a box that has walls that can never be penetrated nothing can get out, nothing can get in and I ask you what is the total charge inside the box you would say what so take a look at this think about the principle of adding up charges to get the total charge apply that principle and see what answer you come up with pause the video here and resume it when you think you've got an answer if you identified that negative E plus negative two E which is negative three E plus positive E which brings us back to negative two E equals negative two E and identified B as the correct answer then excellent job again this was just a warm up question if this was not the answer that you got it's a good opportunity for you to pause here and make up your own example and see if you can arrive at an answer that makes sense given what we've just done in this warm up question now in electricity we use where positive charge seems to be going in a situation as our reference point now this is despite the fact that in reality the atom is made of electrons which have way lower masses than protons certainly much smaller than nuclei the mass of the electron is almost two thousand times smaller than the mass of the proton and so if you start exerting electric forces on atoms and tearing electrons away from protons or tearing electrons away from atomic nuclei the nuclei are less inclined to move as much as the electrons are inclined to move and so in reality in situations where atoms are being ripped apart electrons are moving around in materials it's the electrons that do most of the moving most of the time now Ben Franklin couldn't have known this but when he established the convention that we will characterize a system by the direction that positive charge flows he couldn't have known about the electron the electron wouldn't be discovered until the end of the 1800s and he was doing his electrical work in the mid 1700s so there's a lot of time yet before technology would advance to the point where the electron could even be identified and it wouldn't be until almost two decades later that the atom would be firmly established being made from protons, electrons and then eventually it would be understood that there are also neutrons down in the nucleus as well so we have a convention we characterize a system by the direction that positive charge does its moving but in reality negative charge is really doing the physical moving so how do we handle that? if I have a bunch of electrons going to the left then clearly the negative charge in a system is headed to the left but because of the conservation of charge if negative charge is going to the left where do you think the positive charge must be going? well in a system where charge is conserved if the negative charge is going to the left the positive charge looks like it's going to the right that's just a consequence of the conservation of electric charge in a closed system and since we're pretty much going to deal almost exclusively with closed systems in this course if you see a bunch of electrons going left then you can immediately conclude that the positive charges are moving to the right even if the atoms are just sitting still alright so keep this rule in mind positive charge always does the opposite of what negative charge does and we characterize a system by where the positive charge appears to be flowing so sure the electrons might be flowing to the left but that means that it looks like the positive charges, let's say the nuclei are headed to the right and so when I ask you which way is charge flowing in a system I mean positive charge and you need to tell me which way the positive charges are going so that's a typical convention in an introductory physics course like this and pretty much in the rest of physics now here is another warm up question this one is an application of Coulomb's law we're not just adding up charges anymore this system is still the same system I showed you on the previous slide it still has a total electric charge of negative 2e but now I want you to think about the forces that particle 2 and particle 3 exert on particle 1 and I want you to tell me if you think that this is a possible equilibrium situation for particle 1 in other words is particle 1 going to accelerate or not in this situation before you pause the video and think about this let me remind you of a few key features of Coulomb's law first of all in Coulomb's law and thinking back to our experiments with electric charge if I have two charges with the same sign they repel each other so for instance I notice that particle 1 has a negative charge particle 2 has a negative signed charge so I would expect particle 1 to be repelled by particle 2 but then I notice that particle 3 has a positive electric charge so particle 3 will cause 1 to be attracted toward it so I have a situation here where 2 repulses 1 but 3 attracts 1 that's not enough information to figure out if this is necessarily an equilibrium situation remember Coulomb's law this is the other thing I want you to keep in mind as you think about this question Coulomb's law says that the strength of the force, repulsion or attraction gets weaker with increasing distance squared so if a charge is farther away its influence is weaker than a charge that's closer by so if I have a big charge that's close and a small charge that's far away then the small charge is already at a disadvantage compared to the big charge if I swap that if I have a weak charge that's close by and a big strong charge that's further away well maybe it's possible that the small charge will have more of an influence because it's closer so taking a look at this situation where particle 1 has a negative e-charge and particle 2 is closer than particle 3 and particle 2 has a negative 2 e-charge and particle 3 has a positive e-charge think about whether or not this could be a possible equilibrium situation for particle 1 think about this and then unpause it when you're ready for an answer to this question if you drew the conclusion that this is very unlikely to be an equilibrium situation for particle 1 then you're correct now why would this be well first of all the charge that's closest to it is the biggest in magnitude and causes particle 1 to be held away toward the left well particle 3 is attractive because of it's positive charge and would tend to want to pull one to the right it's further away so not only does it have the smaller charge but it's influence is diminished by being further away and so it's very likely that in the end the force exerted by particle 2 will overwhelm the force exerted by particle 3 and they won't cancel each other out so this is very unlikely to be an equilibrium situation in fact it's pretty much impossible given the situation that's depicted here it will never be the case that the force from particle 2 is smaller than the force from particle 3 as long as particle 1 remains to the left of these two other particles now here's another question for you to think about is this situation a possible equilibrium situation for particle 2 particle 2 is the blue one here it has a charge of plus e particle 1 has a charge of negative 2e and is very far away certainly much farther than particle 3 which has a charge of negative e so what do we observe 3 and 2 have opposite charges so they attract each other 1 and 2 have opposite charges so they also attract each other but 1 is a lot further away than 3 so could this be a possible equilibrium situation for particle 2 pause here think about this and see what answer you come up with unpause when you're ready to see the following discussion if you concluded that this has a good chance of being a possible equilibrium situation for particle 2 then you're correct is it definitive? but it stands a chance of being an equilibrium situation and here's why particle 2 is closer to particle 3 than it is to particle 1 and so while particle 1 is more attractive from a charge perspective the force that it exerts is greatly weakened compared to the force exerted by particle 3 by this larger distance between 1 and 2 so 2 and 3 are closer even if 3 has a weaker charge 1 and 2 are further away but 1 has a bigger charge and so I haven't indicated really what the distance is here I've given you no numbers and so you have to draw some kind of intuition based conclusion based on Coulomb's law and thinking about signs of charges and whether the forces are attractive or repulsive and you might conclude that yeah sure this one could be a possible equilibrium situation for particle 2 depends on the details but in a specific problem you would be given more detail than this and then you could really work it out mathematically you just have to draw a reasonable conclusion and a reasonable conclusion is that this could be an equilibrium situation for particle 2 is it definitive? No, but it stands a chance now let's revisit Coulomb's law and here I've been much more careful about writing all the labels down so Coulomb's law tells us in this form the force that particle 2 exerts on particle 1 this is equal to a constant times the charge of particle 1 times the charge of particle 2 divided by the distance between 1 and 2 squared times a unit vector this again tells you the force that charge 2 exerts on charge 1 let's really begin to dive more deeply into the pieces of Coulomb's law and think about a strategy for tackling problems where you need to use Coulomb's law to answer questions about force magnitude and force direction so here's an example of two charges charge 1 and charge 2 now in every problem you need to begin by coming up with some kind of coordinate system some spatial system of measure that you're going to use to describe distances displacements and so forth in a physical system so right away if you have a problem involving two charges and the forces between them either the one 2 exerts on 1 exerts on 2 so I'm going to start with a plain old vanilla Cartesian coordinate system and argue that this looks to me like maybe two corners of a rectangle that's kind of like a square Cartesian coordinates are great for describing squarish things so I am going to choose to put the origin of my coordinate system on charge 2 my y-axis is the vertical line goes right through charge 2 my x-axis is the horizontal line this is the origin of the coordinate system telling charge 2 that x equals 0 and y equals 0 and all the negative distances will be measured relative to that point these are opposite charges so if I am curious about the force that one of these particles exerts on the other say charge 2 exerts on charge 1 charge 2 exerts on charge 1 which way do I expect the force that charge 2 exerts on charge 1 to point pause here think about that for a moment and then resume when you are ready to compare your answer to mine so I asked you what direction you think the force that charge 2 exerts on charge 1 points given that they have opposite charges well opposite charges attract and so I would expect that if I am thinking about the force that 2 exerts on 1 it must be that that force points from charge 1 toward charge 2 along a line that connects them after all that's the definition of an attractive force charge 1 wants to be closer to charge 2 so it's got to be accelerating along a direction that connects charge 1 and charge 2 it has to point from charge 1 to charge 2 let's keep that kind of intuitional answer in mind as we explore the use of Coulomb's law now it's always good practice to label the locations of charges in the coordinate system before we keep going ultimately we're going to need to define the unit vector r with a hat symbol over it or r hat and we're going to need to define r squared and so we're going to need physical distances in this problem so in this problem as I said charge 2 is sitting at the origin of the coordinate system 0, 0 and charge 1 is at 4, negative 2 it is 4 meters to the right and 2 meters below negative 2 charge 2 alright so those are the coordinates of each of these charges charge 2 and charge 1 this is always a good practice to go through when you're setting up a problem that may involve say the forces between two objects, gravitational electric or otherwise now the next step in setting up these problems involving Coulomb's law is to label directions magnitudes and directions so what we're basically trying to write down now is the displacement in space between these two charges charge 1 and charge 2 now since we're ultimately trying to find the force that 2 exerts on 1 we need to draw a displacement vector that points from the thing that's doing the acting which is charge 2 to the thing that's receiving the action which is charge 1 again for this problem we're trying to find the force that 2 exerts on 1 2 is doing the acting 1 is the recipient of the action so by convention we draw our vector our vector with a 1, 2 subscript starting on 2 the thing doing the acting ending on 1 the thing that is the recipient of the action so that's the full vector it starts on 2 it ends on 1 and it has an arrow head that indicates that this is the direction of this conventional vector now all vectors have two pieces they have length and direction the length can be written as r12 which is just the absolute value of this distance here and r hat 12 which is the unit vector that points from 2 to 1 and has a length of exactly 1 the magnitude scales the unit vector's length up to the total distance between the two charges this is something that you would have exercised in physics 1 and first semester physics and if this feels uncomfortable or unfamiliar to you I would recommend pausing here and go and find a video on vectors and basic vector concepts magnitude and directionality in the unit vector and review this concept that any vector can be written as a length times a unit vector the unit vector has length 1 and just tells you the direction of the vector the length scales the unit vector to give you the full displacement between the two points in space but if this is comfortable then go ahead and proceed now all vectors can be decomposed in the coordinate system into components along the independent coordinate axes so for instance this vector has a horizontal component that points from 2 along the x axis and stops right above charge 1 and then another component which is vertical and goes from that point we stopped at down and ends on charge 1 so this is the displacement vector between 2 and 1 the hypotenuse of a right triangle whose sides are the journey we take along the coordinate axes to get from 2 to 1 we walk to the east to the right this distance 4 meters we walk down south 2 meters and we end our journey on charge 1 I can write the horizontal component of the vector r vector with an x subscript as its length, which in this case is 4 meters times a unit vector that points only and exclusively along the x axis i with a hat over it or i hat the y component of this displacement vector r with a subscript y and a vector hat over it written as the magnitude of the direction we have to go in the y axis which is 2 meters times a unit vector that points down along the y axis that is negative j hat so this is a very typical way to decompose vectors into their x and y components and write those components out as the sides of a right triangle the hypotenuse of that right triangle is the displacement vector now when you have to write down vectors, lengths of vectors and directions of vectors I find it's very useful to attack each piece of the problem one at a time divide and conquer to solve problems involving Coulomb's law where you've got r squared, r hat what are r squared and r hat how do I figure that out? well if I can write a coordinate system like you saw on the previous page and I know where the charges are in space and I can write the magnitude of the x component the magnitude of the y component the appropriate unit vectors for each component in our case i hat and negative j hat I'm pretty much ready to define the unit vector that is that r hat subscript 1,2 that you saw in Coulomb's law a unit vector can always be obtained by taking the original vector and dividing it by its length we'll come back to that in a moment so get yourself the unit vector by doing this once you've labeled your coordinate system get yourself the product of the charges in Coulomb's law q1 times q2 make sure you keep those signs make sure you remember whether these charges are positive negative and keep the signs that go with them when you do the multiplication of q1 and q2 that sign has meaning at the end getting the distance squared in the denominator is often boils down to basically an exercise in doing the Pythagorean theorem that rx squared plus ry squared equals r12 squared which is the length squared of the hypotenuse now when you put all these pieces together into Coulomb's law make sure that your answer has units are they correct check them most of the time students get tripped up by the fact that they are confident in each piece but they assemble them incorrectly and they get the wrong units and they don't understand why they got the problem wrong and the answer is they got the problem wrong because they failed the first and most basic quality assurance law of any physics or math problem check your units and make sure what's on the right-hand side matches what's on the left-hand side it is the most basic tenant of quality assurance in mathematics what's on the left had better have the same units as what's on the right or you've done something wrong and if you don't get the same units you have not discovered a violation of something in nature you've made a mistake if you have discovered a violation of something in nature that's great you'll get a Nobel Prize but you're not there yet you're too soon for that check your math let's take a look at each of the pieces of Coulomb's law now that we've gone through the exercises on the previous slides we basically have everything that we need to assemble the parts of Coulomb's law the constant K, the charge Q1, the charge Q2 R12 squared the distance between the charges squared and R hat 12, the unit vector that points from charge 2 to charge 1 we are ultimately trying to find the force that charge 2 exerts on charge 1 the constant K is always the same number it is in fact a fundamental constant of nature and its value is 9 times 10 to the 9 Newton meter squared per Coulomb squared not a bad number to have to memorize 9 times 10 to the 9 if you have to go one more decimal place 8.99 times 10 to the 9 Newton meter squared per Coulomb squared but this is usually sufficient now what was the charge of charge 1? it was negative 1 Coulomb so we can just put that number in here when we want to get a final numerical answer what was the charge of charge 2? well it was positive 1 Coulomb so that would go in here if we wanted to get a final numerical answer what is the distance squared between the two charges? well we can go back a couple of slides we can look at our coordinate system but remember we went 4 meters along the horizontal axis to go from charge 2 to charge 1 and 2 meters down along the vertical axis to get down finally to charge 1 so by Pythagorean theorem application we can get r12 squared it's the length of the hypotenuse of the right triangle so we just need the x component squared which is 4 squared the y component squared which is 2 squared doesn't matter what the sign is in front of this it goes away anyway so we wind up with 16 plus 4 that's 20 and this thing has units meters squared do not forget your units on these numbers now I didn't carry the units through the intermediate step but I did put them at the end to remind you that you really need to put your units down so you don't forget them so what is r subscript 12 all squared it's 20 meters squared finally we have the unit vector I always recommend getting this denominator before you go for the unit vector the unit vector is defined as the vector divided by its length well we know the length it's the square root of this quantity it's the square root of 20 times meters what's the numerator of this? well the numerator of this is the vector r12 so that's 4 meters in the x direction times i hat plus 2 meters in the negative y direction or negative 2 meters j hat and then we divide by root 20 times meters well I've skipped a step here and I've cancelled all the meters out you'll have meters in the numerator meters in the denominator and they cancel each other out and so this is just about as simple as I'm willing to make this unit vector it doesn't look pretty but it's absolutely correct the unit vector is 4i plus negative 2j that whole thing divided by the square root of 20 and you can verify that if I square this quantity and figure out what that number is you should find out that it's 1 the length of the unit vector is 1 the length of the unit vector squared is also 1 so if I square this thing and I get 1 I've done a good job it's another check that you should often perform especially early on when you're practicing computing unit vectors so finally I can group all my numbers together as much as I possibly can and I can write down the force that particle 2 exerts on particle 1 and I find that this is 2 times 10 to the 8th times the vector quantity negative 2i hat plus j hat Newtons 2 times 10 to the 8th times this vector quantity Newtons and where does it point? well the force vector points along the negative x axis and the positive y axis so it goes from charge 1 to the left and then upwards towards charge 2 and that's represented by this arrow that I've drawn here in magenta pointing from charge 1 to charge 2 remember our intuitional answer what did we expect the force that 2 exerted on 1 to be? we expected 1 to be attracted to 2 we expected the force exerted by 2 on 1 to point from 1 toward 2 along the line that connects them Coulomb's law gives us that Coulomb's law is an accurate description of nature and it matches our basic intuition when we look at this and see that their opposite charges and opposite charges must attract let me make finally a comment on K the constant that appears in Coulomb's law K is 9 times 10 to the 9 Newton meters squared per Coulomb's squared but K can also be written in terms of another constant of nature, epsilon a Greek letter with a subscript 0 which is often simply pronounced epsilon naught epsilon naught can be computed from K by this equation K equals 1 over the product of 4 times pi times epsilon naught so what is epsilon naught? epsilon naught is 8.85 times 10 to the minus 12 Coulomb's squared divided by Newton meters squared this thing is known as the permittivity of free space and by free space I mean what's often called the vacuum totally empty space no matter what so ever in it this tells you about the ability for electric forces to propagate through empty space that's all I'm going to say about this right now you will see much later why it is called this but I just wanted to put this here because interchangeably you might be given epsilon naught or K to solve a problem and this is how you relate them to one another they're both constants of nature K is 9 times 10 to the 9 Newton meters squared per Coulomb's squared it's related to epsilon naught by 1 over 4 times pi times epsilon naught and then epsilon naught has this value which you can derive from K or vice versa so what are some strategies that we've laid out here well first of all in a problem where electric charges may be present for electric forces may be present you need to determine first whether or not this problem is even correctly solved using Koulomb's law for instance are there electric charges that can act on one another if not well then Koulomb's law doesn't apply are you looking for the effect that the electric force from one charge exerts on another charge or charges effect implies some change in motion in the state of motion and a change in a state of motion means a force if both of these things are true well then you can definitely apply Koulomb's law electric charges are present and it's possible for states of motion to be changed by forces between those charges then Koulomb's law is exactly how you want to tackle this and again I want to remind you of this divide and conquer approach to solving problems I've given you a very simple example here the simplest example that you will ever have to exercise involving electric charges and electric forces two charges start by drawing a coordinate system and define where the zero of your coordinate system is located it often doesn't matter where you pick that to be sometimes it's convenient to put it on one of the charges and sometimes it's more convenient to put it on one of the charges than any of the others but only with practice are you going to be able to figure out when the best application of the right origin will occur Koulomb's law one piece at a time the good news is the constant piece K never changes it's always 9 times 10 to the 9 Newton meter squared per Koulomb squared so that one's the easy part can you figure out what the charges are q1, q2 if so write them down don't forget their signs, they're important put that down, we'll keep it very accurately can you determine what vectors are involved can you for instance draw the vector from the source of the force to the recipient of the force, from the actor to the thing that's acted upon do that use your coordinate system to then represent that vector direction in its components, in its x component in its y component, for instance if you're using a two-dimensional Cartesian coordinate system calculate magnitudes of distances compute your unit vector from the rule that the unit vector is the raw displacement vector of the length do all of those things separately and then assemble them together into Koulomb's law mathematically simplify that as much as you can and you'll feel a certain sense of satisfaction and don't forget to check your units always check your units to make sure that they are in fact what they should be which in a force problem is Newton's now that we have begun to explore the very basics of electric charge the usage of electric charge the force between electric charges is the mathematical description of those forces we can begin to build up to more complex subjects for instance, what happens when we have many more charges and what happens if we have so many charges that these simple sums that we would otherwise be able to get away with just utterly fail because they're too computationally intensive well then you may need better tricks like calculus in order to do the difficult work of adding up all the forces so now that we have this very basic foundation using Koulomb's law and a simple example of one charge exerting a force on another we can begin to build up a whole toolkit of deeper application of this basic idea to more complicated situations