 Kuhlm's law helps us calculate electric field due to point charges and similarly in magnetism B. O. Savar law helps us calculate magnetic fields due to point current elements. But we've also explored Gauss's law which helps us calculate electric fields in symmetric situations. And we've seen that one can be obtained from the other, they are equivalent. So now the question is, do we have something similar in magnetism? Something that can help us calculate magnetic fields for symmetric situations? The answer is yes. We have something called Ampere's circuital law. And in this video, we're going to ask Mr. Ampere to help us understand his law. So Mr. Ampere, what's your law? Tell us. Ampere says, let's take an example. Imagine we have three wires that carry some current. I1, I2, I3. Now Ampere says, draw a closed loop. And I ask, what do you mean? He says, anywhere in space, any shape you want, draw a closed loop. I say, okay, cool. Let me do that. So let's say I draw a closed loop that goes somewhat like this. I am excluding I3 because why not? He asked us to draw anywhere I want. Fine. What next, Ampere? Next he says, walk around this loop. And I ask, well, there are two ways to walk, either this way or that way. Which way should I walk? He says, any direction you want. And I love that. I love the freedom that he's giving us. So let me walk this way. I'm walking Ampere. What should I do now? And now comes the important part. He says at every point, I'll write this, at every point, find the dot product of the magnetic field and a tiny length DL vector. Okay. What does that mean? So here's what Ampere is saying. At every point in space, the three currents are together producing a magnetic field. Right? We know that current produces magnetic field. So maybe at this point, I'm just drawing random directions now. Maybe the magnetic field is this way. Maybe there's a point over here where the magnetic field is, I don't know, maybe this way. And maybe there's a point over here where the magnetic field is this way. Now, what Ampere says is that as I'm walking, at every point, take a tiny step length, which is DL, and it will have a direction of this, you know, tangential to this path. So over here, DL would be this way. Over here, DL would be this way. And over here, because I'm walking like this, DL would be this way. You can imagine DL to be a very tiny step, like a nanometer or something. It's an infinitesimally small step. And take a dot product of them, scalar product of them. So BDL cos theta, you might know how to take the dot product by now. And do that everywhere, and he asks us to then take a summation of that. So add all of that up over the entire loop. And since we're dealing with infinitesimals, we're dealing with calculus, addition in calculus, summation in calculus is what we call integral. So this is what ampere wants us to do in that closed loop. Take the integral of BDL everywhere. Now ampere is warning us, not warning, sorry, reminding us that this will only work for closed loops. So for example, if I had chosen, say, a loop which looked like this, then it will not work. Ampere says, don't take this. Even here, I can take BDL, right? But Ampere says, no, no, no, only for closed loops. And so to remind us, he's gonna put a circle over here. Oops, let's use the same color. Circle over here, and says closed loop. All right, all right, what happens if I do that? Ampere asks me, Mayesh, what do you think will happen if you took this integral over the closed loop? And I say, I have no idea. You tell us, Ampere. This is where Ampere smiles and laughs and says, ha, the answer is going to be, and this is the Ampere's circuit of law. The answer is going to be always mu not times i enclosed. And this, my dear friends, is what we call the Ampere's circuit of law. It's called circuit of law because it has a path involved. And paths are usually called circuits. You may have heard of that in racetracks. Racetracks are also called circuits. Now before we continue, I'm sure a lot of questions are brewing up in your mind like, why do we need this law? Well, as we will see in future videos, we can use this to figure out the strength of the magnetic fields in certain symmetric situations. We'll do that in the future videos. But for now, let's concentrate on the right-hand side and ask ourselves, what does this enclosed current mean? Well, let's ask Ampere, Ampere, what is this enclosed current? Well, one way to think about this is basically how much current is enclosed, the total current enclosed by the loop. But Ampere is a little bit more specific. Ampere says, look, to figure this out, first step you have to do is attach a surface to this loop. And I don't understand the last Ampere. What do you mean? So Ampere says, OK, imagine this. Imagine you took this and dipped in a soap solution. What would happen? There would be some soap film attached over here, right? Here you go. Let's imagine that's the soap solution. Now Ampere says, enclosed current is the current that punches through this surface. Whatever is punching through that surface is the enclosed current. So the enclosed current is basically the total current that is passing through an attached surface to the loop. The attached surface is our soap solution. So in our example, what would be the value of B.dl, according to Ampere's law? Well, that's going to be mu naught times. What is I enclosed? Only I2 are passing through the attached surface. They are the only ones enclosed. I3 is not. So I3 will not be in the picture. So the total will be I1 plus I2. But the moment I write that, I feel uncomfortable, because I know that one current is going up, another current is going down. So one must be positive and one must be negative, right? Ampere says, yes, one must be positive, one must be negative. But how do I figure out which one is positive and which one is negative? What do I do? So Ampere says, we use the same thing that we've used so far in magnetism. Right hand rule. He says, take your right hand and curl it in such a way that the curved fingers are in the direction of your travel. And then the thumb represents the positive direction. So in our example, since I'm traveling this way, if I take my right hand and if I curl my fingers in that direction, my thumb will point downwards. And so this means, according to my right hand thumb rule, downward direction is positive for this loop. So I too would be positive, I one would be negative. So this is now the correct application of Ampere's law for this case. Why don't you quickly try one? So let's say, let's take another loop, which is over here. I'm gonna take a rectangular loop because shape doesn't matter. So let's say we take a rectangular loop somewhat like this. And this time, let's say we walk this way and calculate B.dl. Can you pause the video and think about what will be the closed loop integral of B.dl over here? It's gonna be mu naught times something. What will that be? Can you pause and think about this? All right, so the first step would be to dip this in a slope solution and attach a flat surface to it. And now the current that penetrates through this surface will be our enclosed surface. And you may be wondering, why should we attach a surface? We'll talk a little bit about that towards the end of the video. But the current that penetrates is I2 and I3. And now we need to know which direction is positive. For that we use our right hand thumb rule. In this case, we are moving in this direction. And so if I use my right hand now, let me keep it over here somewhere. Okay, if I move my right hand now, now notice the thumb points upwards. So upwards is my positive. So this is now positive. So what I end up getting is plus I3. So I3 becomes my positive current. I2 becomes my negative current. So minus I2. And I1 is not in the picture because I1 is not penetrating through that surface. And there you go. This is how we use Ampere's circuit law. And before we wind up, I wanna talk about some important characteristics of this law. First of all, this law can be derived from a Bio-Sawar law. And you can derive Bio-Sawar law from this law. So they're both equivalent. And we use whichever one is more convenient in our given situations. In some times when things are very symmetric, we go for Ampere's circuit law because it makes our calculations simpler. Again, something we'll see in future videos. Secondly, on the right hand side, we only consider currents that are enclosed by the loop. So for in this example, only I1 and I2, but not I3. But what about the magnetic field on the left hand side? Is that only due to the enclosed currents? No, that is the total magnetic field. So the magnetic field which we are considering is due to all the currents, enclosed and non-enclosed. So how does that work? Why is it on one side we have total field, but on the other side only the enclosed one matters? Well, that's because again, this is like mathematically we will not get into the details but what happens is what this means is that the contribution of BDL provided by the non-enclosed currents they add up and become zero. So they end up giving zero contribution. So you can imagine as you walk around this loop when you in some cases, the contribution of this is positive. In some places the contribution is negative. So the total contribution of them is zero. This is very similar to what we saw in Gauss's law. The total electric flux only depends upon the charges that are enclosed by the surface over there, right? The charges which are outside, they will contribute to zero flux. Same thing is very similar happening over here. Finally, when it comes to the surface, we said imagine a soap solution attached to it, right? But here's the thing about soap solutions. You don't have to have them flat. If you blow on a soap solution, you will end up having an open surface attached to our loop. So the loop becoming the opening to that surface. We can also attach such surfaces over here. So imagine somebody is blowing from the top. What will happen to that surface? We might get something like this and now the ion closed becomes the current that is penetrating through this surface. And so you can attach any open surface you want to your loop, flat being the simplest one, but you'll always end up with the same value of ion closed. And finally, you may ask, why should we even attach a surface? I mean, what's this business of attaching surface? Can't I just look at this loop and tell what is the enclosed current? Sure, in most simple situations, yes. But in general, we can have very complex situations in which it may not be so obvious. And I'll not dig too much into it. We will look at one such situation sometime in the future where we deal with calculating the magnetic field when there is a capacitor, things will become much more interesting over there and this attachment of surface will make a lot more sense over there. But for now, it's completely fine. In most of our examples, it's fine if you don't attach it, but Ampere suggests you attach a surface to our loop and find the current that is punching through that surface that becomes our ion closed.