 In this video, we're going to take our alternating voltage generator and attach it to a coil of wire, which is also called an inductor, and find out what will be the expression for the current. Now, just to be clear, this is a circuit that only contains inductance. It has no resistance or capacitance, and immediately the question could be, such circuits don't exist, right? I mean, all circuits will have some resistance at least. So then what's the point of studying this? Well, the reason we study this is because if you do it this way, then we can understand exactly how an inductor behaves when you have a voltage signal applied to it, so it'll be easier for us to do, understand that behavior, and more importantly, later on, when we look at more real circuits, this analysis will actually help us. So pure inductive circuit attached to an AC voltage generator. What's going to happen? So let's see. To figure out what the current is going to be, I need to find an expression between the voltage and the current. And let's say our current is flowing in this direction, as of now. So how do I figure out what that current is going to be? Well, one relation that immediately comes to my mind when I think about voltage and current is Ohm's Law, V equals IR. But remember, Ohm's Law only works for resistive circuits, and there are no resistance over here, so I can't use Ohm's Law. So what do I do? One of the things that we've learned about inductors or coils of wire is that they hate changes in current. And we may have learned before that whenever there is a current changing through an inductor, it induces an EMF, which we can treat as a voltage. And we've seen in previous videos that the magnitude of that voltage, which I'm just going to call as VL, that will always equal L times Di over dt. Di over dt. And this expression is basically saying that inductors hate changes in current. And the quicker the current tries to change, the larger EMF or the larger voltage it induces. And of course, if you need a refresher or more clarity on where this comes from, we've talked a lot about that in our previous videos on inductors. Feel free to go back and check that out. But now, let's see if we can use this to build our equation. So the next immediate question I have is what direction is this voltage compared to our source voltage of the generator? So for that, let's say our source voltage is continuously fluctuating. At one point, this will be higher at higher voltage, and then this will be at a higher voltage, and this will be at a higher voltage. That's why it's called alternating. It's continuously changing its direction. So at some point in time, let's say this is at a higher voltage. So I'll call this side as positive, and let's say this is at negative. Now, at this point, because the current is flowing in this direction, and our voltage is also pushing the current in the same direction, in this situation, the current is trying to increase, or the current is increasing, actually. And therefore, because the current is increasing, our inductor says, uh-uh, I hate changes in current. I'm going to try and decrease the current. Okay? Because it wants to decrease the current, it's going to try and push back the current. And to do that, it puts a high voltage on this side and a low voltage on this side. So this is the polarity of that induced voltage. And you can immediately see that the polarity of the source voltage and the induced voltage is exactly the same. And because there are no circuit elements in between, we know that this voltage in this point and this point is at the same potential. This point and this point is at the same potential. In other words, this voltage should exactly equal this voltage, both in magnitude and direction. And so I can use that to figure out what the current is. So let me go ahead and write that. So the voltage across the inductor should exactly equal the source voltage or the generator voltage, both in magnitude and direction. And so if I substitute, I get VL equals L, D, or I over DT. So D of DT over I. And at this point, one confusion I always had is if we go back and go back to inductors, we said that the EMF induces negative LDI over DT. So should I put a negative sign over here? Well, remember, the negative sign is only telling us the direction of the voltage. It's saying it's in the opposite direction of the change in the current. And over here, we've already included the direction over here, including the direction we said we realized that they both have the same polarity, right? So we've already taken care of the direction. So we don't have to worry about the negative sign. And that equals V0 sine omega t. And now we see an equation, which is not just an algebraic equation. It's a differential equation, which basically means there is a differentiation term over here. And some differential equations can be very tricky. This turns out to be one of the easier ones. Because to solve this, all we have to do is separate all the i terms and all the t terms on one side, separate them on two different sides. And then we can just integrate. Now it would be a great idea to pause the video and see if you can integrate this yourself and see what the expression for current is going to be. All right, so I'm going to rearrange just to have DI on one side. So left side will only put DI. And so on the right hand side, I will get, if I just rearrange, I'll get V0 sine omega t. I'll get a dt over here and I'll get divided by L. And now, because I've separated the i terms and the t terms, I can integrate. If I integrate, integral of DI is just I. And that equals V0 over L is a constant. I can pull that outside the integral. And what's the integral of sine omega t dt? Well, our integral, I'll directly write the integral over there. That's going to be minus cos omega t divided by omega. And whenever we integrate, there's going to be a constant. We need to put that constant. And there we go. So this is our expression for the current. And you can immediately see that this is something that we don't know. If you can figure out what the value of that constant is, we are done. So how do we figure this constant out? First of all, what does it even mean? But remember, integrals is basically like integration is like doing a reverse of differentiation, asking what function should I differentiate to get this as the answer? So if you actually take this and differentiate and you can try that, you will actually get this number. Differentiation of cos is negative sine, so the negative cancels out. And the omega pops out and that cancels with this. However, remember, differentiation of constant is a zero. Which means I can also differentiate this plus 100. Or I can differentiate this plus 1000. And then all of those numbers will still give me this value. And so the integral is saying, I don't know what this constant is. That's your job, Mohesh. It's your job to figure the value of the constant out. And so now the question is, how do we figure this out? Maths is not going to answer that. We need physics. And one of the powerful ways of figuring out values of constants whenever I give an equation is we say, hey, it's a constant, which means it does not depend on the values of voltage or V0 or does not depend on the values of omega or L. So we can put whatever values of V0, omega, whatever values we want. And then see if we can figure out what the value of C is going to be. So very powerful method. Let me show you what I like to do over here. Let's say, in this particular example, we put the V0 value to be zero. If this is zero, that means our source voltage is always going to be zero. So I'm basically saying, I'm basically saying, I'm not going to put any generator in this circuit, okay? And I want to know what the current is going to be based on this equation. So from this equation, current is going to be, well, this part will be zero because V0 is zero, but there's a constant. So according to this equation, the current is going to be a constant. But I know in my heart, in my bones, in my stomach, that if I don't put any generator over here, the current in the circuit must be zero. Because if it was not zero, we have problem with energy conservation. Where is that energy coming from? There is no source. So I know that in this situation, my current should be zero. So I know that in this situation, that constant value should be zero. And because it's a constant, if it's zero for these values, it should be zero for any values of V0. And therefore, we can argue that this constant should be zero. And ta-da, we have found the expression for the current. And so for the purposes of this video, we are done. But we still need to dig deeper and figure out what does it mean? How does this explain the behavior of inductors in the presence of alternating voltages? We'll explore all that fun stuff in the future videos.