 Suppose we are given what the electric potential at every single point in space is, 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, set 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, 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. 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 A 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, how much the potential drops. That will be an infinitesimal drop and it will 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 whatever that, 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. So the condition is 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 dy is zero, there is no change in y and dz is zero. And why is that? Because I want to calculate the component only in the x direction. So for that, 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 that dv is and then take the gradient and will that give me, that will 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 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 moved 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 zero and dy is zero. And once I have the three components, I can put together and I can use iCAPJCAP and KCAP and write down what my electric field is going to be. So can you now at this point pause the video and maybe 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 ex 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 doh v over doh x. It's just a notation, okay? 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 what it means to say dy is zero, y is not changing. dz 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 will see now what's the difference between doh 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 dy over dx. Let's do normal differentiation first. Plus the second term into differentiation of the first term, which gives me one. Minus, let's differentiate this now. I get two times, two comes down. So you get four y, and you have to use a chain rule to dy over dx. I haven't done anything funky now. Regular differentiation, okay? And use chain rule wherever required. All right. But now remember, dy and dz are not changing. They are zero, which means this goes to zero. Dy is zero. Again, dy is zero, which means this term goes to zero. When you apply this, this is what we mean by do v over do x, okay? So any dy and dz 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 ex equal to, this term goes to zero. This term survive, 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 gonna be negative do v over do y. But 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. Differential of x with respect to y is dx over dy. Be careful what you're differentiating with. You're differentiating with respect to y now, minus four y. And now let's use this, dx and dz is zero. So dx is zero, that means this goes to zero. So now this is partial derivatives, whatever you're getting now, okay? So y is going to be, let's see, this term survives, this term survives. So I get negative x plus four y. 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, in the first term, y is a constant. So when I differentiate, this will go to one and you'll get y, I'll get y. And this will become zero because it's a constant. So I 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's no z over here, but treat x as 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 what will be 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 do it step by step. So it'll make, there'll be no confusion. It's gonna 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 four y times j cap. And what is E z? That's zero. So 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.