Calculating E from V(x,y,z): E = - potential gradient

suppose we are given what the electric potential at every single point in spaces it's given by this function let's say and this this equation is basically saying you put in the value of any coordinate you want and then it will tell you what the electric potential at that point is going to be in volts our goal is to figure out what the electric field vector is going to be at every point in space how do we do that okay so to do this we need to find some connection between electric field vector and the potential function and we've explored something like this in our previous videos we've seen that electric field is the negative potential gradient and what what it's basically saying is that imagine you have an electric field and let's say you want to find what the strength of that field is at point a then it's saying that all you have to do is take a step forward to say point b um some some distance forward and you find out how much the potential has dropped the potential will drop if you move in the direction of the field right and that potential drop is delta v that ratio then the negative ratio how much the potential is dropping per meter that itself will tell you what the strength of the electric field is and if you're not familiar with this and we've talked a lot about this in great detail in previous videos feel free to go back and check that out but there are a couple of problems um if you try to directly use this over here one is this will only work for uniform electric fields i mean think about it if you want to step forward and then calculate how much the potential has changed and divided by delta r then that electric field value should stay same everywhere in between but in our case the electric field may be changing in fact it might be it might be a non-uniform that's what i mean changing means it may be a non-uniform electric field let me draw that so let's say our electric field is a non-uniform then what do we do now as the moment i step forward immediately electric field might have changed then if i do this i will get an average value of the electric field between point and point b i don't want that i want the electric field at a how do i do that well we can do something that we've done a lot a lot of times before in physics we can use calculus we can say imagine instead of taking a big step forward take an incredibly tiny step forward imagine point a and b are incredibly close to each other so close that practically electric field value is going to be the same in that case this distance delta r becomes an infinitesimal and we often write it as dr and then you calculate what the change what how much the potential drops that will be an infinitesimal drop and it'll be dv and then you do a negative dv over dr and that will give you the electric field at that point and so now the only change we have to do is make dv over dr and now this will work for any field but there's a second problem see i don't even know what direction the electric field is at this point so how do i know in which direction i should move to calculate this gradient because i should always move along the field or in the opposite direction i don't even know what direction the field is so how do i know should i move to the right left up down how do i know that well what we can do is we can ignore what whatever we can say it doesn't matter what direction the electric field is let's calculate the x and the y and the z component separately so here's what i mean okay so i don't know what the direction of the electric field is it doesn't matter what i will now do is i'll take a small step only in the x direction so let's say our right side is the positive x so this then would be this would be our dx so this is going to be our dx and as a result there will be some change in potential and now let me calculate that gradient so if i calculate that gradient negative gradient dv divided by dx what does that give us i am now calculating how quickly the potential how much the potential has changed per meter along the x direction so that gives me the electric field strength along the x direction meaning it tells me the x component of the electric field so to calculate the x component i have to differentiate voltage function or potential function with respect to x but we have to be very careful see this under a condition the condition is i'm dx is the only variable that's changing i am not moving there is no dy i'm not moving in the y direction i'm not moving in the z direction i'm only moving in the x direction so the condition is d y is zero there is no change in y and d z is zero and why is that because i want to calculate the component only in the x direction so for that i make sure i should make sure that i'm not moving in the y i'm not moving in the z does that make sense and so that will give me the x component and similarly what i can now do is move a small step in the y direction and calculate what what that dv is and then take the gradient and will that give me that'll now give me the y component of the electric field so to calculate y component i should differentiate voltage potential function with respect to y but again it's under the condition that now i'm only moving in the y direction okay because i want to calculate the y component i have to only move in the y direction that means now my dx should be zero is that making sense my dx should be zero and my dz should be zero and i know this is a little abstract it might take a time it might take some time but i can repeat this why am i making sure my dx and dz was zero what would happen if they were not zero if my dx was not zero let's say dx had some value then that means i did not move this way i might have more at some angle and if i move at some angle then i'll get the electric field component along that direction i don't want that i only want along the y direction so i only need to travel this way meaning dx should be zero dz should be zero is that making sense the same thing i can do along the z direction and that'll give me the z component of the electric field oops dz making sure dx is 0 and d y is zero and once i have the three components i can put together and i can use i cap j cap and k cap and write down what my electric field is going to be so can you now at this point pause the video and maybe you know take a breather of what we've all done and see if you can now use this use your calculus use your differentiation to figure out what the electric field is going to be so pause and give it a shot all right let's do this so let me make some space in between and start by finding what the e x is the x component of the electric field now we have to do this right and let me show you the notation that we use a shorthand notation to write this whole thing the way we write that is we write negative instead of writing d we write do v over dou x it's just a notation okay don't get scared don't get scared by that all it's saying mathematically all this is saying is that you're only differentiating this function v with respect to x and if there are any other variables like y and z and if there are any other variables they're not changing that's that's what it means to say d y is zero y is not changing d z is zero z is not changing that's how you should differentiate it okay you may be wondering what's the difference and you will see let's do this you'll see now what's the difference between do and this okay so i'm gonna do normal differentiation nothing different so normal differentiation how would you do this so you have the first term if you differentiate it's a uv so first term x into differentiation of y with respect to x so i'll just write as d y over dx let's do normal differentiation first plus the second term into differentiation of the first term which gives me 1 minus let's differentiate this now i get 2 times 2 comes down so we get 4 y and you have to use the chain rule to d y over dx i haven't done anything funky now regular differentiation okay and use chain rule wherever required all right but now remember d y and d z are not changing they are zero which means this goes to zero d y is zero again d y is zero which means this term goes to zero when you apply this this is what we mean by dou v over dou x okay so any d y and d z terms they go to zero and this kind of differentiation is called partial differentiation or partial derivatives because you're only differentiating with respect to we're only making sure changes are happening for x and not y okay so what do you get so let's see we get xex equal to this term goes to 0 this term server this is the only term survive so i get minus y so this is my x component value so now let's do the y component and i encourage you to try that what's y component going to be again it's going to be negative dou v over dou y but let's buy let's start by doing normal differentiation and i encourage you to pause and try this all right so again we use uv rule so first term into differentiation of the second term differentiation of y with respect to y is one plus the second term into differentiation of the first term differentiation of x with respect to y is dx over dy be careful what you're differentiating with your differentiating with respect to y now minus 4y and now let's use this dx and dz is 0. so dx is 0 that means this goes to 0. so now this is partial derivatives whatever you're getting now okay so ey is going to be let's see this term survives this terms are y so i get negative x plus 4y this now is your y component of the electric field and before we go to z component let me show you a faster way of doing this instead of just doing the whole thing a faster way of doing this is we could say look in this differentiation y and z are not changing so you can treat them as constants so while calculating the x component just differentiate this with respect to x and treat y as a constant now if i do this in my head the in the first term y is a constant so when i differentiate this will go to 1 and you'll get y i'll get y and this will become 0 because it's a constant so i'll only get y and then the negative sign comes does that make sense similarly if i when i'm differentiating with respect to y treat x and z as constant there is no z over here but treat x is constant so when i differentiate the first term x is a constant so i'll get x and i'll get minus four y and then the negative sign basically comes from the outside and that's how you do this so whatever the value of e z now you have a differentiate only with respect to z assuming x and y to be a constant well there is no z term the whole thing is a constant for us and what's the differentiation of constant you get zero and so e z is zero and it kind of makes sense if there is no z term in this that means when i take a small z side step in the z direction the voltage doesn't change at all in the z direction and that means that there is no z component of the electric field so finally what is the electric field vector let me make some space let me move down all right so now that i know all the three components i can now say that my electric field vector at any point in space x y z i'll not write that any point in space is going to be the e x which is negative y i cap okay let me let me do it step by step so it'll make there'll be no confusion it's going to be e x i cap plus e y j cap plus e z z component times k cap it's cartesian okay what is e x e x is negative y so negative y times i cap plus what is the y component of the electric field it's negative x plus four y so minus x plus 4y times j cap and what is easier that's zero so this is this is my expression for electric field and now if you give me any coordinate of x y and z i can just substitute over here and i can find electric field at any point in space so long story short if you want to calculate electric field vector from a potential function you have to separately calculate the three components by doing partial derivatives and what that means is you differentiate only with respect to x keeping y and z constant and so on and so forth and then once you have the three components you put them all together and then you get the total electric field vector