 In a previous video, we attached our generator to an inductor, a pure inductive circuit and found the expression for the current. In this video, we'll dig deeper, we'll take this expression, figure out from this what's the relationship between the voltage and the current and from that we'll be able to draw plot a graph. So this is the voltage graph that we have seen earlier, we will be able to plot a current graph and see the relationship between them and eventually we'll be able to figure out exactly how inductors behave when you put an alternating voltage across them. So where do we start? Well we can start by trying to find the relationship between the current and the voltage and for that let's try to put the current in the same fashion, same format as the voltage. So the first thing I immediately see is this part over here now represents the maximum current which we can call I0, just like how we call this as V0. And the interesting thing to see which we'll not explore here, we'll do that in a future video, is that even though there is no resistance in the circuit, our current is limited, there's a maximum value which is very interesting, that means inductors also sort of kind of provide some kind of resistance to the current. But how, why, what, we'll talk about that, all of that in future videos separately. But now let's concentrate on this part, okay? Because we want to compare these two and draw a graph so I need to get this part right. So what I want to do is convert this into a very similar function like this. So I want to convert this into sine so that I can compare them easily. So now would be a great time to pause the video and use your trigonometry and see if you can convert negative cos into positive sine so that I can compare with this. So pause the video and see if you can give this a shot yourself. All right, let's see. So let me try that over here. We've seen in trigonometry videos before, trigonometry lessons before that cos theta can be written as sine of pi by 2 minus theta. So minus cos omega t can be written as minus, I'm keeping the negative as it is, put the bracket, minus sine pi by 2 minus omega t. Does that make sense, okay? But I want to get rid of the negative sign as well. I don't want the negative sign. And another thing we may have seen in trigonometry is that you can always put minus sine theta can be written as sine of minus theta, which when I can put the negative sign inside. These are all trigonometric relationships. I'm not deriving any of that over here. I'm just using the identities. And so this now I can write it as sine omega t minus pi by 2. So I put a negative sign inside the pi by 2 becomes negative, omega t becomes positive. And so you put it all together, we can now write the current expression. We can write the current as I naught times sine omega t minus pi by 2. And there we go. We now have an expression which we can compare with the voltage. So what is this expression telling me? It's telling me that the current will also be an oscillation, sinusoidal oscillation, no surprise over there alternating current. But it's saying that it's not oscillating in sync with the voltage. In fact, you can see there's a negative pi by 2 over here. This means it's oscillating 90 degrees behind the voltage. What does that mean? Well, remember 360 degrees or 2 pi represents one full cycle. So 90 degrees, one fourth of it represents a quarter cycle. This means the current is oscillating a quarter cycle behind the voltage. And in a minute we'll look at the animation and it will be all clear. But before we do that, let's go ahead and draw the graph for the current. And I want you to pause the video and see if you can plot the graph yourself. You have the voltage graph. So you have reference for that. With respect to that, try plotting the current graph. Okay, so let's start at this point. I know at this point we have omega t to be 0, therefore the voltage is 0. And so when omega t is 0, notice you get a sine minus pi by 2, which is minus 1. So i becomes minus i naught. So at this point we are at negative maximum. Let's say current negative maximum is somewhere over here. This would be our minus i naught. All right, let's consider this point now. I'm only considering the easy points. This is the point where the voltage is maximum, meaning omega t should be 90 degrees. That's when the sine is maximum. That's the point. And when omega t is 90 degrees, pi by 2, pi by 2 cancels. Oof, that's when you get sine 0. That's when the current is 0. So at this point, the current is 0. The voltage is maximum, but current is 0. Interesting. Then comes the point where voltage goes to 0. And this is the point where omega t is pi. You get that, right? 0, pi by 2, pi, sine pi is 0. That's why the voltage is 0. And when you put pi, pi minus pi by 2 is pi by 2. Again, this time you get maximum, plus 1. So you get i equals plus i naught. Oof, at this point, voltage is 0, but the current goes to maximum. So maybe this is the point where you have plus i naught. And so you can kind of now see, I know it's a sine graph and this is enough for me. So I know that the graph is gonna go like this and then maybe come down somewhere over here and sort of like that. You know what, I already have a better graph. Let me just draw that. So here is our current graph. This looks much better. But one of the immediate doubts I get when I look at this graph is, wait a second, doesn't this say that the current is, I mean, doesn't it look like the current is ahead of the voltage? I mean, if this is when the current and voltage are in sync with each other, shouldn't this be current being ahead? Not really, because remember, this is the future. So think of it this way. Right now, the voltage is at 0, but the current comes to 0 a little bit later. By the time the voltage is already at maximum, but the current comes to maximum a little bit later. So indeed, the current is lagging behind the voltage. But you know what, a better way to visualize this is I have an animation for you. Here it is. It's the same graph, but here instead of going forward, moving our time axis forward, we'll move the graph backward and see what happens. It's the same thing, right? So I'm gonna move the graph backward and concentrate on the here. So we'll dim everything and you can now see the oscillations. And you can clearly see the current is chasing behind the voltage. Can you see that? The pink is the current, okay? And so we can now draw an arrow mark to represent both the voltage and the current. And that's how we like to visualize it. So let me draw an arrow mark for the current as well. And if you keep it over here, you can clearly see they're not in sync with each other, current lags behind the voltage. And current is quarter of a cycle, or phase difference of pi by two, behind the voltage. Okay, now we could stop over here, but if you're curious like me, let's go one step deeper and ask the question, why is this happening? I mean, this is the first time we're experiencing where current and voltage are not oscillating in sync with each other. It's really weird. Why is the inductor doing that? I mean, think about certain points over here. There are some points over here, as you can see, where voltage is zero in the circuit, zero voltage. But we have current to be maximum. How does that make any sense? And then you have these points over here, where the voltage is maximum in the circuit, and the current is at zero. I mean, what's going on? And if you think also, there are points where over here, voltage is negative, but current is positive. I've never experienced anything like this in our previous circuits. So I'm really, really trying to figure out what's going on over here. How to digest this? Luckily, I found a mechanical analogy. So here's what I mean. Here we have our current and voltage oscillating one more time. But what we'll do now is we'll imagine the voltage to be similar to height of a plank above the ground, which is pivoted to something that's connected to a wall. So the plank is free to go up and down, let's say. And the voltage represents the height of that plank. So an oscillating voltage that's concerned on that can be similar to, can be imagined to be an oscillating plank going up and down, okay? All right. What we'll do is we'll imagine there's a box kept over here. And the box is gonna slide and we'll assume there is no friction because it's a pure resistance less circuit, right? It's a pure, only inductor is there. So inductance is like the inertia. The box's inertia represents the inductance. And the box's speed represents the current. What I want you to do before I show you the animation is I want you now to consider the speed and the height and look at their relationship. You can see if that follows the same current and the voltage relationship. Think about when the speed of the box would be zero. When would it be maximum? When it would be, yeah. When would be zero and maximum, what heights? So imagine right now I've just kept the box and you imagine the thing goes back and forth. So just visualize this yourself first. All right. So right now, the height is maximum and I just kept the box. My question is, what's the speed of the box? Well, because of its inertia, it's gonna be zero. It'll take some time for it to pick up the speed, right? So we have a situation where speed is zero but the height is maximum. But now let's wait and see what happens. As the plank goes down, notice the box will keep on picking up speed. It pick up, pick up, pick up speed keeps on increasing until the plank becomes parallel to the ground. That's when the height is zero but the box now has a lot of kinetic energy, a lot of momentum. It'll have maximum speed. After this point, the plank will go down but the box will not immediately come back down because of its inertia, it'll continue to keep going up. So even though the voltage is negative, the speed or the velocity stays positive. Ooh, does that make sense now? Let's look at the animation. And as it goes up, it starts losing its momentum and eventually when it comes to the negative height, negative maximum, that's when the speed of the box goes to zero. Very similar to what's happening to our circuit, isn't it? Beautiful, right? I love this analogy. It helps me digest what's going on. And let's now look at the same thing with the arrow mark. The arrow mark of this represents the speed or the velocity and that is equivalent to the current. Zero, a maximum voltage, zero current. And now notice zero voltage, maximum current. The current is positive. Even though the voltage has gone negative because of its inertia. And now somewhere, if you just go just a second back, now the current goes to zero but the voltage is negative maximum. The thing keeps repeating, beautiful, isn't it? Beautiful. Okay, okay. So can you connect all the dots and see what's going on over here? If you consider this point where the voltage is maximum and the current is zero, it's like at that point over here. The way I like to visualize this is imagine this is the point where I just turned on my voltage and the voltage is at maximum. The current immediately wants to go to maximum but the inductor says, hey, wait a second. I have inertia. The current was zero. I will not allow you to instantly go up. And it's for that reason, the current is still zero at this point, okay? But as time passes, even though the voltage goes down, it's still positive. Current is now starting to increase slowly and steadily just like how the speed of the box is increasing. And eventually comes this point. This is the point where the voltage has eventually gone to zero. But because the current was continuously increasing, this is the point where the current has reached the maximum value. After this, what's gonna happen is the current, the voltage says, hey, I'm going to zero. Current says, fine, I'll also come down to zero. And the inductor says, hold on, I have inertia. I can't let you go to zero. I will continue to push the current. And just like how the blocks momentum keeps it going forward, it's the, you can kind of say the inertia of the inductor keeps the current going forward. And that's why the current stays positive. But of course, now, but as time passes by, eventually the inductor loses all its juice, just like how the box loses all its kinetic energy. And finally comes a point where the inductor is ran out of all its juice. Current has gone to zero. But that happens when the voltage has gone to negative maximum. And then the whole story repeats in the opposite direction and the story goes on. So the model of the story is like masses, inductors have inertia, they hate changes in current. And because of that, the current lags behind the voltage by an angle of pi by two.