Electric potential & work

just when i thought i understood positives and negative electric potentials i looked at the textbook definition and my goodness it had such big big words like work done by external force unit positive charge infinity to that point without accelerating and i was like oh my god what what is this this is this looks so different than what i understand but in this video we'll figure out exactly how we calculate electric potential using work done and by the end of the video this statement would have made a lot a lot more sense to you so let's quickly start with something that we've already seen before in gravity if i say potential at a point a with respect to its sum reference is 11 joules per kilogram it means if i were to keep a kilogram of rock let's say at that point it would have 11 joules of more potential energy compared to how much it would have at this reference point over here it's the ground and the same thing applies here when i say potential at point a with respect to infinity is negative 300 negative 300 joules per coulomb it basically means if i were to keep a coulomb of charge over there a coulomb of charge let's say at that point it would have 300 joules of potential energy less compared to when this is at infinity and we've talked more about this in our previous videos on introduction to electric potentials and potential differences so you need a refresher feel free to go back and check those out but the goal of this video is to figure out how to calculate these values how did we know this was positive 11 for example and how did we know this was negative 300 how did we know it was negative so how do you calculate these potentials all right so to do that we can go back to our definition one more time so potential at any point potential at any point a with respect to some reference is basically how much potential energy that charge has at point a with respect to reference which in our case is infinity so compared to the reference per charge because we are doing it for one coulomb so i have to divide this for a charge and so all i have to do is figure out the potential energy difference if i can do that i can calculate the potential so that brings us to the next question how do we calculate potential energies hmm well i like to go back to my gravity how do i calculate potential energy when it comes to gravity i used to remember the formula mgh it was very famous but i don't want to do that i want to use concept basics uh fundamentals so how would i calculate or what does it mean that my potential energy of one kilogram is 11 joules what it what does it mean going back to basics it basically means that that 11 joules of energy is stored up and if i were to let go of this stone as it comes from here to the reference point which is the ground it loses all the 11 joules and it gains kinetic energy so it gains 11 joules of kinetic energy again per kilogram because we're defining it for one kilogram so what i can say is ah if i want to calculate potential energy at any point just let go of that and see how much kinetic energy it gains when it reaches that reference point whatever kinetic energy it gained that number itself should be potential energy because that must have been stirred up does that make sense so let's write this down because we'll do it step by step slowly so what i can say now is this number should equal how much kinetic energy was gained or just gonna let gain has it moved from a to reference point has it moved from a to reference point per charge so before we continue let's just hold over here and see if you can use this definition to think about to realize when you have potential when the potential will be positive or negative so when you go from your point your preferred point to the reference point if you gain kinetic energy it'll be positive and that's why we got over here positive number as it went from here to here it gained kinetic energy but what if this number becomes negative then the potential would be negative the example for that would be in over here if the reference point was above now think about it now when the rock is going from here to here first of all it doesn't automatically go you might ask how can gravity how in gravity rock would go from here to here how would it go up well you imagine you don't you didn't drop it but you imagine you threw it up that's the that's the trick you imagine it's being thrown up now when it goes up it slows down it loses kinetic energy and so this is negative and therefore potential over here would be negative with respect to this point does that make sense pause the video and think about it okay does it make sense to you well yeah let's follow the chain of thought as it goes up kinetic energy is reducing means meaning potential energy is increasing therefore this point has more potential compared to this and so this point has less potential compared to this and so that means this will be negative potential so make sense right same thing happens over here now can you pause and convince yourself why over here will get a negative number pause and try all right if i were to let go does it go to the reference point infinity automatically no it gets attracted so i throw it away just like over here and when i do what happens to its kinetic energy it reduces because it's of the attraction it reduces and if kinetic energy is reducing you get a negative number here that means the potential is negative and see if it makes sense follow the chain of thought if the kinetic energy is reducing that means potential energy is increasing so at infinity you would have maximum potential energy and so the potential energy over here would be less compared to that of infinity giving you a negative sign okay but now we can take one step deeper and ask hey but how do you calculate this how do we calculate how much energy how much kinetic energy is gained or how much kinetic energy is lost how do you calculate that and for that we use our work energy theorem something that we may have learned a long time ago work energy theorem states work done equals gain in kinetic energy so what i could write over here the gain in the kinetic energy would be same as the work done by which force the force of gravity here or the force of electricity so i'll just write force of electricity work done by the electric force in moving it from point a to the reference point and again per charge and again let's come back over here and see if it makes sense so if i want to calculate how much potential energy i have over here i need to figure out how much kinetic energy is gained or lost as it goes from here to here but to do that i have to calculate how much work gravity does in moving it from here to here and how do you calculate the work done by gravity or how do you calculate work done by any force you calculate it as the force multiplied by the displacement so all you do is figure out how much the force of gravity would be so maybe the force of gravity is downwards multiply that by the displacement how much the displacement would be and that number would tell you how much kinetic energy is gained and that number would tell you how much potential you would have the same thing would apply here as well if you want to calculate how much potential you have at this point figure out how much work is done by the electric force in this case the electric force is downwards but you move it all the way to infinity so you have to move it in the opposite direction so this is the electric force and in doing that figure out how much work is done now you might say hey but infinite distance means is it infinite work no because remember over here things get a little little complicated uh because as you go away the force itself starts changing and so you can't just multiply force and displacement you have to integrate it and something don't worry we'll not do this in this video we'll do it in a future video but you get the point and notice over here work done by electric force is negative as you go from here to here and we already knew that because it's negative we're gonna get a negative potential now actually we're pretty much done because we now found a way to figure out potentials all we have to do is calculate work done by the electric force and moving it from that point to infinity per charge and we're done but sometimes you will also have external forces acting on you and so there's another way to define electric potential in terms of work done by the external force let me just introduce that because your textbooks tend to do that it's not necessary you don't have to do it i don't usually use that at all but let me do that anyways let me just define it anyway so that things are clear let me just get rid of this all right imagine that when we're moving this rock from here to here imagine we're not just dropping it let's say we are holding it in our hand and we are slowly moving it down slowly and steadily without any acceleration we are moving it down then my external force the force that i am putting on the rock would be upwards would be upwards let me write the external force with this color okay this would be the direction of my external force this would be my external force so it makes sense because if it was not upwards it would just it would just fall down very quickly i want to move it slowly without any acceleration then see what would happen then my force would exactly equal the gravitational force but it'll be exactly negative and so the work that i did would be exactly negative of the work done by gravity so i can also say and the same thing would apply here okay even here what i could do is instead of um instead of throwing it and just allowing it to go slowly what i could do is i could put an external force in the opposite direction exactly equal to the electric force this is the external force and slowly without any acceleration move it from here to infinity and in doing so the work that i did would be exactly opposite of the work that electric field did and so now i could also say that this is also equal to the negative of the work done by the external force in moving it from point a to the reference point does that make sense per charge per charge what is that okay per charge all right but there's a caveat there's a small detail i have to move this without any acceleration so conditions apply no acceleration because if i accelerate let me show you if i accelerate then my force will not be equal to the electric force then my work done will not be equal to the work done by the electric force then this this will not hold true so only if i do it without acceleration the two forces will be equal and opposite and my work done would be exactly equal to the negative but any number exactly equal to the electric field so i can also use this definition so i can also say potential at a point i can also say it is the work done is a negative of the work done by the external force in moving it from that point to infinity without accelerating finally what your textbooks do is they say look i don't want to define in terms of negative so how do i get rid of this negative sign and the way to get rid of negative sign is actually quite simple see if you do when you are moving things down if you're doing a negative work then when you move things in the opposite direction you will do positive work right so instead of moving it from a to r you move it from r to a and then the negative sign will go so let me just write that down we can also say this is equal to work done by the external force in moving it from r to a per charge and i know it's a little crowded so excuse me for that these two are the same things and let's see what this means this means if i start with gravity because it makes sense it says if you want to calculate potential at any point a find out how much x work is done by an external force in moving it from ground to that point per kilogram does that make sense yeah if i want to find out how much potential over here is take a kilogram move it from the reference point to that point a and whatever work you did that would be stored as potential that makes sense right the same thing applies in the charges as well and that's why your textbooks define it as potential at any point with respect to infinity they don't mention that but that's always with respect to infinity equals the work done by the external force in moving a coulomb of charge from infinity to that point without any acceleration now i know there is a lot of things introduced so it's going to take some practice to get used to it but we now have so many different ways of calculating potentials my favorite is just doing this this is my literally my favorite this is the easiest i don't have to worry about external forces um but yeah you can use these definitions as well