Calculate pot. difference in uniform fields

we're given an electric field towards the left a uniform field which has a magnitude of 100 volts per meter and we're given three points a b and c and we're asked to figure out what is the potential difference v a minus v b and vb minus vc so the whole idea behind this problem is we need to figure out we need to find a way to calculate potential differences from electric fields now in a previous video we've seen what that connection is we saw something we realized that electric field is a negative potential gradient now if you don't know what that means and and you're not familiar with this connection feel free to pause this video go back and watch that video revise the concepts and come back over here all right if you're cool with that let's go ahead with this let's start solving the first one what is the potential difference va minus vb how do we how do we do that well one way is we can use the formula and we'll do that a little bit later but i want to try and solve this logically and for that i ask myself what does this number even mean this number says that if you move along the field then for every meter you go forward along the field you will lose 100 volts of potential that's what this number means and if you're wondering why do we lose potential as you go along the field well if you keep a positive charge somewhere over here and let go of it what happens well it'll accelerate along the field meaning it will gain kinetic energy along the field which means it will lose potential energy along the field just like in terms of in just like in the case of gravity as things fall down they lose potential energy along the field so i know that if you go if for every meter along the field you lose 100 volts or every meter in the opposite direction of the field you gain 100 volts which means if there are two points which are hun one meter apart i know that their potential difference would be 100 volts so that means i just need to figure out how far apart these two are and then i can calculate so can you pause the video right now and see if you can if you can you know just using this information figure out what is the potential difference between these two points give it a try all right so the first thing that comes to my mind is i just have to figure out what the distance between these two points are if i know this distance and let's say i find out that distance to be and you know you can use coordinate geometry to figure that out but let's say you get some number if for example say 10 meter okay just an example then that means for every meter i gain 100 volts so for 10 meters i would gain 10 times 100 so 1000 volts so that would that be the answer no that will not be the answer you can't do it this way can you pause the video and think about okay another thing why you can't do it this way all right so this number that you gain or lose 100 volt per per meter only works when you move in the direction of the electric field or in the opposite direction basically you have to move parallel to the electric field only then you can say per meter you gain or lose 100 volts so you can't move this way so how do i go from a to b the way i like to do it is i mean one of the ways to do it is you can go from a to b along the field first and then you go up this way i'll tell you why see when i go from here to here i can use this number i can figure out how far this distance is and then i can use this number to figure out what the potential how much potential i'm gaining because i'm going in the opposite direction and now i'm going from here to here notice when i go this way i'm moving perpendicular to the electric field and we've already seen before that if you move perpendicular to the electric field there will be no change in potential in fact these two points lie on an equipotential surface remember equipotential surfaces are always perpendicular electric field so long story short from here to here there is no change in potential okay so that means all i have to figure out is what is the potential at this point i mean i just need to figure out what is the potential difference between these two points that's the same as the potential difference between these two points because this and this and all the points lying on this plane would be at the same potential does that make sense so i just need to figure out what this distance is and i can do that by looking at the x coordinate so this is negative one this is plus four so what is the difference between these two points that's four minus minus one that is four plus one so that's five meters and you can pause and you can just check that's right so this is 5 meters and so now i can use this number i know for every meter i go forward every meter i go forward i gain 100 volts so how much would i gain if i go 500 5 meters forward i gain 500 volts so immediately i know that this point which is the same as this point potential wise is 500 volts more than this point so let me write that mathematically vb is 500 volts more than va and therefore what i'm what i need is va minus vb so i can just rearrange this and i'll get va minus vb as negative 500 volt and we're done there we go that's the answer okay now before we go forward to the next one let's just convince ourselves we get the same answer by using the formula this will be an excuse for us to you know get used to the formula in the previous video we also derived the formula we saw that electric field what is the connection between this and the potential difference electric field is the negative potential gradient which means it's a negative difference in potential between two points divided by the distance between the two points and again this distance is only along the field okay so can you again pause the video and see by substituting you end up with the same answer so give it a shot all right so if you rearrange you get delta v equals minus e times delta r now when you're talking about delta v and delta r especially delta v we need to be very careful about which is our initial point and which is our final point right because we're talking about change initial and finals are initial uh important now over here we went from here to here but you don't have to you could have gone backwards as well but just to keep things simple i'll call this as my initial point and this is my final point this point and this point are the same by the way like we just saw potential wise the same all right so what is delta v now delta v is final potential minus the initial potential so this would be vb minus va and that's why it's important if you had called this as your initial point and this as your final point you would call this va minus v will be opposite this equals negative what is the strength of what is the value of electric field now we know the magnitude is 100 but what about the direction it's in the opposite direction of the positive x and that's again that's important when you're substituting over here so that means this is negative 100 volt per meter so negative 100 volt per meter times what is delta r delta r is you're going from here to here so you're going in the positive direction so that's plus phi so 5 meter so meter meter cancels and see what we end up with we get 500 plus 500 so you get 500 volt so vb minus va is 500 volt means vm minus vb is negative 500 so you get the same answer so you get the same answer all right can you tell think about what would be vb minus vc again can you pause and try all right so because we the point b and c lie on the exact you know perpendicular that is the equipotential surface they are at the same potential which means the potential difference between them must be zero and so whether you take vb minus vc or vc minus vb it should be zero and again i could ask does the can we use the formula to get arrive at this yes this time delta r for them is zero because remember delta r is the distance along the field the distance along the field is zero are you getting that along the x direction they have the same coordinate and therefore delta r is zero then therefore you would get the same answer here all right let's try one more okay in this case we are given an electric field pointing downwards with a magnitude of 30 volts per meter we're given a point a whose coordinate is given to us 3 comma 10 and there is a point b such that their potential difference is given to us this time and so our goal is to find what the coordinate is again can you try doing this yourself first all right so again the first thing i want to try and do is see if i can do this logically now i know that the first thing i'm looking at is i'm looking i'm trying to figure out whether b should be above a or b should be below a it's given to me that v b is more than v right it's a positive number that means uh vb has a higher potential than va and so i'm just asking myself if i go down along the field potential will drop oh so it has to be above so b has to be somewhere over here somewhere over here okay so that's the first thing i figure out now i know that for every meter i go up i will gain 30 volt but for us i just have to gain 10 volt so i ask myself how many meters should i go up well i'd like to do it this way i like to write it down so i know for one meter if i go up i know i gain 30 volt 30 volt but it's given to me that i just have to gain 10 volts so how many meters should i go well it's one third and therefore it should be one third over here as well so 1 over 3 so i divide this by 3 i get 10 right so i divide this by 3 and so 1 3 meter so that is 0.33 meter so this distance should be say that down point 3 3 so now comes the question what is the coordinate of this point so this point would be well the x coordinate would remain the same because x coordinate is not changing so that's 3 the y coordinate would be now 10 plus 0.33 so it would be 10.33 and it'll be bar i can put a bar over there now one thing like just like we saw last time this is not the only point that has 10 volt higher than point a i can in fact have any point along this line perpendicular because all of these points on this line are equipotential so point b can be here also point b can be this one so point b can have any value of x coordinate because this is a uniform field it can have any value of x coordinate but its y coordinate has to be 10.33 which means the actual answer that we need to give if somebody asks us this question is we could say hey it can have any value it's not that we don't know the value of x coordinate we're saying it can have any x coordinate once so i'm just going to put x over there pick any point you want and y coordinate has to be 10.33 so this is our answer and i encourage you to try and check this using the formula yourself