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so here's something that used to confuse me if you had two charges and we'll keep these straight by giving them a name we'll call this one q1 and I'll call this one q2 if you've got these two charges sitting next to each other and you let go of them they're going to fly apart because they repel each other like charges repel so the q2 is going to get pushed to the right and the q1 is going to get pushed to the left they're going to start gaining kinetic energy they're going to start speeding up but if these charges are gaining kinetic energy where is that energy coming from I mean if you believe in conservation of energy this energy had to come from somewhere so where is this energy coming from what is the source of this kinetic energy well the source is the electrical potential energy we would say that electrical potential energy is turning into kinetic energy so originally in the system there was electrical potential energy and then there was less electrical potential energy but more kinetic energy so as the electrical potential energy decreases the kinetic energy increases but the total energy in this system this to charge system would remain the same so this is where that kinetic energy is coming from it's coming from the electrical potential energy and the letter that physicists typically choose to represent potential energies is au so yu4 potential energy I don't know like PE would have made sense too because that's the first two letters of the words potential energy but more often you see it like this we'll put a little subscript C so that we know we're talking about electrical potential energy and not gravitational potential energy say so that's all fine and good we've got potential energy turning into kinetic energy well we know the formula for the kinetic energy of these charges we can find the kinetic energy of these charges by taking 1/2 the mass of one of the charges times the speed of one of those charges squared what's the formula to find the electrical potential energy between these charges so if you've got two or more charges sitting next to each other is there a nice formula to figure out how much electrical potential energy that is in that system well the good news is there is there's a really nice formula that will let you figure this out the bad news is to derive it requires calculus so I'm not going to do the calculus derivation in this video there's already a video on this we'll put a link to that so you can find that but in this video I'm just going to quote the result show you how to use it give you a tour so to speak of this formula and the formula looks like this so to find the electrical potential energy between two charges we take K the electric constant multiplied by one of the charges and then multiplied by the other charge and then we divide by the distance between those two charges we'll call that R so this is the center to Center distance it would be from the center of one charge to the center of the other that distance would be R and we don't square it so in a lot of these formulas for instance Coulomb's law the R is always squared for electrical fields the R is squared but for potential energy this R is not squared basically to find this formula in this derivation you do an integral that integral turns the R squared into just an R on the bottom so don't try to square this it's just R this time and that's it that's the formula to find the electrical potential energy between two charges and here's something that used to confuse me I used to wonder is this the electrical potential energy of that charge q1 or is it the electric potential energy of this charge q2 well the best way to think about this is that this is the electric potential energy of the system of charges so you need two of these charges to have potential energy at all if you only had one there would be no potential energy so think of this potential energy as the potential energy that exists in this charge system so since this is an electrical potential energy and all energy has units of joules if you're using SI units this will also have units of joules something else that's important to know is that this electric potential energy is a scalar that is to say it is not a vector there's no direction of this energy it's just a number with a unit that tells you how much potential energy is in that system in other words this is good news when things are vectors you have to break them into pieces and potentially you've got component problems here you've got to figure out how much of that vector points right how much points up but that's not the case with electric potential energy there's no direction of this energy so there will never be any components of this energy it is simply just the electric potential energy so how do you use this formula what do problems look like let's try a sample problem to give you some feel for how you might use this equation in a given problem okay so for a sample problem let's say we know the values of the charges say they start from rest separated by a distance of three centimeters and after you release them from rest you let them fly to a distance 12 centimeters apart and we need to know one more thing we need to know the mass of each charge so let's just say that each charge is 1 kilogram just to make the numbers come out nice so the question we want to know is how fast are these charges going to be moving once they've made it 12 centimeters away from each other so the blue one here q1 is going to be speeding to the left q2 is going to be speeding to the right how fast are they going to be moving and to figure this out we're going to use conservation of energy for our energy system will include both charges and we'll say that if we've included everything in our system then the total initial energy of our system is going to equal the total final energy of our system what kind of energy did our system have initially well the system started from rest initially so there was no kinetic energy to start with there would have only been electric potential energy to start with so I'll just call that u initial and then that's going to have to equal the final energy once there are 12 centimeters apart so the farther apart they're going to have less electric potential energy but they're still going to have some potential energy so we'll call that u final and now they're going to be moving so since these charges are moving they're going to have kinetic energy so plus the kinetic energy of our system so we'll use our formula for electrical potential energy and we'll get that the initial electric potential energy is going to be 9 times 10 to the ninth since that's the electric constant K multiplied by the charge of Q 1 that's going to be 4 microcoulombs a micro is 10 to the negative sixth so you got to turn that into regular coulombs and then multiplied by Q 2 which is 2 micro Coulomb so that'd be 2 times 10 to the negative 6 divided by the distance well this was the initial electric potential energy so this would be the initial distance between them that Center to Center distance was 3 centimeters but I can't plug in 3 this is in centimeters if I want my units to be in joules so that I get speeds in meters per second I've got to convert this to meters and 3 centimeters and meters is 0.03 meters you divide by a hundred because there's a hundred centimeters in one meter and I don't square this the R on the bottom of here is not squares you don't square that bar so that's going to be equal to it's going to be equal to another term that looks just like this so I'm going to copy and paste that the only difference is that now this is the final electrical potential energy well the K value is the same the value of each charge is the same the only thing that's different is that after they've flown apart they're no longer three centimeters apart of they're 12 centimeters apart so we'll plug in 0.12 meters since 12 centimeters is 0.12 meters and then we have to add the kinetic energy so I'm just going to call this k4 now the total kinetic energy of the system after they've reached 12 centimeters well if you calculate these terms if you multiply all this out on the left-hand side you get 2.4 joules of initial electrical potential energy and that's going to equal if you calculate all of this in this term multiply the charges divided by 0.1 2 and multiplied by 9 times 10 to the ninth you get 0.6 joules of electric potential energy after they're 12 centimeters apart plus the amount of kinetic energy in the system so we can replace this kinetic energy of our system with the formula for kinetic energy which is going to be 1/2 MV squared but here's the problem both of these charges are moving so if we want to do this correctly we're going to have to take into account that both of these charges are going to have kinetic energy not just one of them if I only put 1/2 times 1 kilogram times V squared I'd get the wrong answer because I would have neglected the fact that the other charge also had kinetic energy so we can do one of two things since these masses are the same they're going to have the same speed and that means we can write this mass here as 2 kilograms times the common speed squared or you can just write two terms one for each charge this is a little safer I'm just going to do that conceptually is a little easier to think about ok so I solved this 2.4 minus point 6 is going to be 1.8 joules and that's going to equal 1/2 times 1 kilogram times the speed of that second particle squared plus 1/2 times 1 kilogram times the speed of the first particle squared and here's where we have to make that argument since these have the same mass they're going to be moving with the same speed one-half V squared plus one-half V squared which is really just V squared because the half of V squared plus 1/2 of V squared is a whole of V squared now if you you might be like wait a minute this charge even though I had the same mass it had more charge than this charge did isn't this charge going to be moving faster since it had more charge no it's not the force that these charges are going to exert on each other are always the same even if they have different charges that's counterintuitive but it's true Newton's third law tells us that has to be true so if they exert the same force on each other over the same amount of distance then they will do the same amount of work on each other and if they have the same mass that means they're going to end with the same speed as each other so they'll have the same speed a common speed we'll call it V so now to solve for V I just take a square root of each side and I get that the speed of each charge is going to be the square root of 1.8 technically I'd have to divide that joules by kilograms first because even though this was a 1 to make the unit's come out right I'd have to have Joule per kilogram and if I take the square root I get 1.3 meters per second that's how fast these charges are going to be moving after they've moved to the point where they're 12 centimeters away from each other conceptually potential energy was turning into kinetic energy so the final potential energy was less than the initial potential energy and all that energy went into the kinetic energies of these charges so we solved this problem let's switch it up let's say instead of starting these charges from rest three centimeters apart let's say we start them from rest 12 centimeters apart but we make this Q 2 negative so now instead of being positive 2 microcoulombs we're going to make this negative 2 microcoulombs and now that this charge is negative it's attracted to the positive charge and likewise this positive charge is attracted to the negative charge so let's say we release these from rest 12 centimeters apart and we allow them to fly forward to each other until there are 3 centimeters apart and we asked the same question how fast are they going to be going when they get to this point where there are 3 centimeters apart okay so what would change in the math up here since they're still released from rest we still start with no kinetic energy so that doesn't change but this time they didn't start 3 centimeters apart so instead of starting with 3 and ending with 12 they're going to start 12 centimeters apart and end 3 centimeters apart all right so what else changes up here the only other thing that changed was the sign of cute you and you might think I shouldn't plug in the signs of the charges in here because that gets me mixed up but that was for electric field an electric force if these aren't vectors you can plug in positives and negative signs and you should the easiest thing to do is just plug in those positives and negatives and this equation will just tell you whether you end up with a positive potential energy or a negative potential energy we don't like including this in the electric field and electric force formulas because those are vectors and if they're vectors we're going to have to decide what direction they point and this negative can screw us up but it's not going to screw us up in this case this negative is just going to tell us whether we have positive potential energy or negative potential energy there's no worry about breaking up a vector because these are scalars so long story short we plug in the positive signs if it's a positive charge we plug in the negative sign if it's a negative charge this formula is smart enough to figure it out since it's a scalar we don't have to worry about breaking up any components in other words instead of two up here we're going to have negative two microcoulombs and instead of positive two in this formula we're going to have negative two microcoulombs so if we multiply out the left-hand side might not be surprising all we're going to get is negative zero point six joules of initial potential energy and this might worry you you might be like wait a minute we're starting with negative potential energy you might say that makes no sense how are we going to get kinetic energy out of a system that starts with less than zero potential energy so it seems kind of weird how can I start with less than zero or zero potential energy and still get kinetic energy out well it's just because this term your final potential energy term is going to be even more negative if I calculate this term I end up with negative two point four joules and then we add to that the kinetic energy of the system so in other words our system is still gaining kinetic energy because it's still losing potential energy just because you've got negative potential energy doesn't mean you can't have less potential energy than you started with it's kind of like finances trust me if you start with less than zero money if you start in debt that doesn't mean you can't spend money you can still get a credit card and become more in debt you can still get stuff even if you have no money or less than zero money it just means you're going to go more in in debt and that's what this electric potential is doing it's becoming more and more in debt so that it can finance an increase in kinetic energy not the best financial decision but this is physics so they don't care all right so we solve this for the kinetic energy of the system we add 2.4 joules to both sides and we get positive 1.8 joules on the left-hand side equals we'll have two terms because they're both going to be moving will have the 1/2 times 1 kilogram times the speed of one of the charges squared plus 1/2 times 1 kilogram times the speed of the other charge squared which again just gives us V squared and if we solve this for V we're going to get the same value we got last time 1.3 meters per second so recapping the formula for the electric potential energy between two charges is going to be K q1 q2 over R and since the energy is a scalar you can plug in those negative signs to tell you if the potential energy is positive or negative since this is energy you could use it in conservation of energy and it's possible for systems to have negative electric potential energy and those systems can still convert energy into kinetic energy they would just have to make sure that their electric potential energy becomes even more negative