in this video we're going to talk about the concept of electric potential energy so let's begin our discussion with a picture let's say we have two parallel plates the first one the one on top is going to be positively charged the second one is going to have a negative charge we know that the electric field will extend from the positively charged plate and it's going to point towards the negatively charged plate now let's say we have a positive charge between these two plates this positive charge will fill a force that will accelerate it in the same direction as the electric field which in this case it's going to be the negative y direction a negative charge will fill a force that will accelerate it in the opposite direction of the electric field now let's say this is point a and the charge is moving to point b how can we determine the electric potential energy at each of these points so let's say that point b is at a position yb above the negatively charged plate and point a is at a position y a above the negatively charged plate so here's what we can do to come up with a formula to calculate the electric potential energy so we're going to start with this the work done going from a to b is going to be equal to the negative change in potential energy so potential energy you could represent as p e or capital u you need to be able to distinguish gravitational potential energy from electric potential energy some textbooks will use u sub g to represent gravitational potential energy which is mgh others will use ue to represent electric potential energy and some would just use p e so be careful with those differences let's use capital u in this video to represent electric potential energy we know that the work is equal to the force times the displacement and going from a to b we have the change in potential energy so final minus initial the final potential energy will be at b and the initial electric potential energy will be at a now the electric force is equal to the magnitude of the electric field times the charge so we can replace f with e times q now d represents the displacement so as we move from a to b the displacement is going to be the change in position so it's yb minus y a by the way let's take this negative sign and let's move it to the other side so i'm going to make this positive make this negative so the displacement is y final minus y initial or y b minus y a now let's distribute negative eq so we're going to have negative eq times yb and here we have two negative signs so it's going to be plus e times q times y a so ub is equal to negative eq yb negative ua is equal to positive e q y a if you multiply both of those by negative one you'll get that positive u a is equal to negative e q y ya so we can thus say that the potential energy the electric potential energy is going to be negative times the electric field times q y i mean q times y so this is the equation that can help you to calculate the electric potential energy notice how it's similar to the gravitational potential energy which is equal to mgh in both cases the potential energy is dependent on the position from ground level in this case the negatively charged plate can be considered the ground level of this particular structure let's work on this problem a 50 micro coulomb point charge moves from point a to point b as shown in the diagram below the magnitude of the electric field is 2 times 10 to the 6 newtons per coulomb points a and b are 80 centimeters and 20 centimeters above the negatively charged plate respectively part a calculate the electric potential energy at points a and b so we know that this positive charge is filling an electric force that's going to accelerate it towards the negatively charged plate and we know that b is 20 centimeters above the negatively charged plate and a is 80 centimeters above it so to calculate the electric potential energy at these points we could use this formula the electric potential energy is going to be negative times the electric field times the charge times the position above the negatively charged plate so what's e we have the magnitude of e is two times ten to the sixth but notice the direction of e e is going in the negative y direction so for this formula to work we need to plug in negative two times 10 to the 6 newtons per coulomb and then we have the charge so q is positive 50 times 10 to the negative 6 and then we need to plug in the height so let's calculate the potential at a first so if we calculate the potential at a we need to use the position at a it's 80 centimeters so that's going to be 0.80 meters so the two negative signs will cancel which will give us a positive answer so the electric potential energy at point a is going to be 80 joules now let's calculate it at point b so it's going to be negative times negative two times ten to the sixth and then fifty times ten to the negative six but this time times point twenty so ua is going to be 20 joules so that's how we can calculate the electric potential energy this is supposed to be ub my mistake so that's how we can get the electric potential energy at those two points now what can we say about the change in electric potential energy as well as kinetic energy going from a to b is it positive or negative and what about the work done on a positive charge going from a to b is that positive or negative well the electric potential energy is decreasing as we move from a to b we went from a value of 80 to 20. so the electric potential energy is decreasing which means that the change in electric potential energy is negative as the electric potential energy decreases the kinetic energy will increase as the electric force accelerates the positive charge towards a negatively charged plate the speed of the charged particle is going to increase so the kinetic energy is going to increase and since work is equal to the change in the kinetic energy if the change in kinetic energy is positive the work done on the charged particle will be positive another way in which you can determine the sign of work is looking at the force and the displacement vectors if these two vectors are in the same direction the work done is positive if they're opposite to each other the work done is negative if they're at right angles to each other the work done is zero the charged particles moving in the negative y direction so its displacement vector is downward since these two vectors are in the same direction the work done is going to be positive but now let's go ahead and calculate the work done in this example so work is equal to the negative change in potential energy so that's going to be the final potential energy minus the initial potential energy the final or ub is 20 the initial is 80. so it's going to be negative 20 minus 80 is negative 60. so as we can see the work done is positive 60 joules so that's the answer for part b so that's how we can calculate how much work is done going from point a to point b now let's move on to part c what is the electric potential at points a and b before we answer that question let's erase a few things and rewrite some of our values above so with this information and everything else that we know how can we determine the electric potential at points a and b the electric potential is equal to the electric potential energy divided by charge so make sure you understand that the electric potential energy i'm going to write epe that's capital u v represents the electric potential the electric potential energy is measured in units of joules electric potential is measured in volts one volt is one joule of electric potential energy per one column of charge so electric potential measured in volt is the ratio of electric potential energy in joules per unit charge in coulombs so if we wish to calculate the electric potential at point a it's going to be the electric potential energy which is 80 joules divided by the charge and the magnitude of the charge is micro coulombs or 50 times 10 to negative 6 coulombs so this is a very large number it's uh 1.6 million so we can say 1.6 times 10 to 6 volts or just keep in mind 10 to 6 is mega so we could say 1.6 mega volts you could also say 1600 kilovolts maybe that will be a better value to deal with here now let's calculate the electric potential at b it's going to be 20 joules divided by the same charge so that's 400 000 volts which is 400 kilovolts so that's how we can calculate the electric potential at each of these points if we know the charge and the electric potential energy now we can also confirm this answer another way in which we can calculate work is by using this formula the work required to move a charged particle through a potential difference is equal to negative q delta v so q is positive it's 50 times 10 to the minus six and we're going from a to b so going from a to b the change in voltage as we go from 1600 kilovolts to 400 kilovolts the change in voltage is negative 1200 kilovolts it's final minus initial 400 minus 1600 is negative 1200. so this is negative twelve hundred kilovolts or negative twelve hundred times ten to the three volts and this will give you the same answer positive 60 joules so in all cases we can see that the work done on this charge to move it from point a to point b is going to be positive now i'd like to take a minute to highlight the similarities between gravitational potential energy and electric potential energy so let me just clear away a few things so gravitational i mean electric potential energy is dependent on the height in this case we'll call it y now when we analyze an object that could fall under the influence of gravity the gravitational potential energy is also dependent on the height the gravitational potential energy is equal to the mass times the gravitational acceleration times the height the electric potential energy is equal to the charge times the electric field times the height so note the similarities between these two equations these are both forms of potential energy so here we have charge here we have mass here we have the electric field and here we have the gravitational field or the gravitational acceleration and then both of them are dependent on the height so if you can remember this equation that will help you to remember this one as well number two a 70 microcone charge is located at a point where the electric potential is 300 volts what is the electric potential energy of this charged particle know that electric potential is equal to the electric potential energy divided by charge but in part a we want to calculate the electric potential energy so rearranging that equation or multiplying both sides by q we get that the electric potential energy is equal to the charge times the electric potential so we have a 70 micro coulomb charge and the electric potential is 300 volts keep in mind one volt is equal to one joule per coulomb so instead of buying 300 volts we can write 300 joules per coulomb so we can see that the unit coulombs will cancel giving us the unit joules so it's 70 times 10 to negative 6 times 300 so the answer for part a the electric potential energy will be .02 joules now let's use the same formula for part b so it's going to be u is equal to qv and this time we have a negative 60 micro coulomb charge so since electric potential energy can be positive or negative we're going to plug in negative 60 times 10 to the minus 6. and this time the electric potential is 500 instead of 300 and so this is going to be negative 0.03 joules so that is the electric potential energy of a negatively charged particle with an electric potential that's positive if we had a negatively charged particle with a negative electric potential the electric potential energy will be positive because the two negative signs will cancel so for instance let's say we have a negative 90 microcomb charge locate at this point where the potential is negative 400 so using this formula we're going to get a positive result it's going to be negative 90 times 10 to the minus 6. times negative 400 and that's going to be positive 0.036 joules so the electric potential energy can be positive or negative number three a 100 volt battery is connected to two oppositely charged parallel plates that are separated by a distance of 10 millimeters calculate the electric field so here we have the direction of the electric field we have the voltage of the battery it's 100 volts and we know the distance between the two plates the formula that we need to calculate the electric field is this formula it's equal to the change in voltage divided by the displacement for those of you who want to keep it simple you could say it's the voltage across the plates divided by the distance of the plates but for those of you who want to be like more technical there is a negative sign involved but let's keep it simple we're just going to use a v over d the voltage is a hundred the distance between the plates is ten millimeters which is to convert that to meters you need to divide by a thousand there's a thousand millimeters in one meter so ten divided by a thousand this gives us point zero one meters so the electric field is 10 000 volts per meter the electric field tells us how the electric potential changes over distance so for every one meter that you move the electric potential changes by 10 000 volts now if we want to get it in volts per millimeter we could have done this we could have done 100 volts divided by 10 millimeters and we could say that the electric field is 100 divided by 10 is 10 so it's 10 volts per millimeter so if we move by a distance of one millimeter in this uh between these parallel plates the electric potential will change by 10 volts so let's say at this plate is 100 volts at this plate is zero volts let me put these numbers over here so let's put 0 over here and 100 over here and let's say this is the distance between the plates so here it's at zero and here it's at 10 millimeters if we were to move one millimeter above the negatively charged plate the electric field will change by 10 volts if we move another millimeter it's going to increase by 10 so now it's going to be 20. in part b we want to calculate the electric potential at a point seven millimeters above the negatively charged plate so seven will be maybe somewhere over here so this is going to be an electric potential of 70 volts to get that answer you could simply rearrange this formula the electric potential v is going to be the electric field times the distance so the electric field using this one because we have the unit millimeters it's going to be 10 volts per millimeter and then if we move a distance of 7 millimeters these units will cancel it will be 10 times 7 and that will give us an electric potential of 70 volts so that's how you can calculate the electric potential at any point between the two parallel plates it's simply the electric field times the distance but you do need to pay attention to the units you want to make sure that the units match if you're going to use that formula now let's talk about the formula above for those of you who want to understand the negative side and how it works let's get rid of a few things and we'll talk about how we can derive that formula so let's say if we have a positive charge and it's moving from position a where the potential is 70 to put on position b where the potential is 20. so as we move from a to b how is the potential changing the final potential is 20. the initial potential is 70 so the change is negative 50. now the displacement relative to the negatively charged plate is what's d the displacement is going to be negative 5 millimeters if we move from point a to point b because we're going down the negative y axis if you take displacement as the final position minus the initial position this is your final position two minus your initial position seven so you get negative five so here we have negative five millimeters so it's negative and then 20 minus 50 is negative so we have a change in potential of negative 50 and divided by negative 5. so these two will cancel so we have positive 50 volts divided by negative five and so we're going to get negative 10 volts per millimeter so what we have here this is just the magnitude of the electric field but in fact it should be negative the reason why it's negative is because the electric field is moving in the negative y direction so that's the purpose of the sign it tells you the direction of the electric field but since we already know the direction we really don't need to worry about the negative sign we can say the electric field is simply the voltage across the plates divided by the distance between the plates you could just use this to determine the direction but if you know the signs of the plates you can easily determine the direction of the electric field it's always going to move from the positive plate toward the negative plate but now you understand how to use this negative sign in that formula now let's talk about how to derive that formula you can start with this equation work is equal to this is the work done by an electric force is equal to negative q times the potential difference to which that charge is accelerated now work is equal to force times displacement and the electric force acting on a charged particle in an electric field is equal to the electric field times the charge f is equal to eq so to calculate the electric field we need to divide both sides by qd qd will cancel on the left but only q will cancel on the right and so we're left with this equation the electric field is equal to negative times the voltage divided by the displacement so that's how you can derive that formula now let's move on to part c what is the electric potential energy of a 400 micro coulomb charge place seven millimeters above the negatively charged plate so we want to calculate the electric potential i mean the electric potential energy at point a so we have a charge at point a but we also have the magnitude of that charge to calculate the electric potential energy is simply the charge times the electric potential so we have a 400 microcom charge and at seven millimeters above the negatively charged plate the electric potential is 70 volts let's convert micro coulombs to coulombs so this should be 400 times 10 to negative 6 coulombs and then we'll multiply that by 70 volts so the electric potential energy is going to be positive 0.028 joules so that's how we can use that formula to calculate the electric potential energy now let's confirm our answer with the other formula that we've used for this situation so if you recall the electric potential energy is also equal to negative times the charge times the electric field times the height above the negative plate for a positive charge that falls towards the negative plate and let's put over here so q is positive 400 times 10 to negative six the electric field and i just got rid of it but it was a hundred volts divided by 0.01 meters so it was 10 000 volts per meter now the electric field is going in the negative y direction so we need to apply that negative sign to get this right and the height above the negatively charged plate point a is seven millimeters above it now this is in meters so we want this to be in meters as well seven millimeters if you divide that by a thousand that's going to be point zero zero seven meters if we use the electric field in volts per millimeter then we can use millimeters here but you wanna make sure that these units they match so it's negative four hundred times ten to negative six times negative ten thousand times point zero zero seven and this gives you the same answer positive .028 joules so both formulas work so now you have two ways in which you can calculate the electric potential energy above some reference point it's equal to the charge times the electric potential and it's also equal to the charge times the electric field times the height with a negative sign you