 In this video, we'll attach a resistor to our AC generator and see what's the relationship between the current in the circuit and the voltage of the generator. Now just to quickly recap, we've seen the voltage equation before. The generator equation can be written as voltage at any point in time, we call this the instantaneous voltage and S stands for source, I'm writing this as source voltage, can be written as equal to some value V0 times sine omega t. V0 represents the maximum value of the voltage because the voltage is continuously fluctuating and this function tells you how the voltage is fluctuating. And again, just to quickly briefly recap whatever we've seen, we've drawn a graph before. It's a sine graph because you have a sine function over here and what is the graph saying? The graph is saying that the voltage fluctuates between some positive maximum and some negative maximum and these values are plus V0 and minus V0. And I always had a hard time understanding what this graph even meant. I thought it was a wave like water but it's not a wave, it's an oscillation and to you know to really visualize that, you have to think about, think about it this way. This is the time axis. Right? So this represents current present and this represents the future. And so if you have to imagine what's happening over here, you can imagine moving the time axis forward. So you move the time axis forward and see what happens to the value. See voltage is increasing, increasing, increasing, reaching the maximum value and then decreases, decreases, decreases. Let me show you in a better way. I will dim everything except for this and now let's see what happens. Alright? As I move forward, you can clearly see how the voltage is increasing. You see that? Voltage increases, increases, reaches positive maximum, decreases, decreases, negative maximum and so on and so forth. Now I want to make sure that you visualize it visualize it perfectly and so what I'll do is I'll show you an animation. Instead of moving the time axis forward, I can just move the graph backward. Right? And see what happens over here. It's the same thing, right? So let me show you an animation of doing that. So here we go. Let's look at that. I'm going to dim everything and you can see voltage going up and down. Can you see that? And we can draw an arrow mark now that represents this oscillation and that the length of this arrow mark represents Vs, the instantaneous value. And so right now it's some negative value goes to negative, positive maximum, zero, negative maximum, positive V0, zero, negative V0. This is how you imagine how it's oscillating and how quickly it oscillates. So let me get rid of this now. And so the number of oscillations per second depend upon this number, omega. Omega gives you the radiance per second. And at first I used to always wonder, what do you mean by radiance? Where is the angle over here? It's oscillations, right? Well, the way to think about it is you can imagine one full oscillation corresponds to two pi radiance. We imagine it to be like a full circle. We'll talk more about that in the future. And so it's called a phase angle because not real angle, okay? But anyways, let's not worry too much about that. So one full oscillation equals two pi radiance. So if we had, say, 10 oscillations per second as an example, then that would equal two pi times 10 radiance per second. Okay, these are all the recap that I'm doing. We've just seen before. So in general, if we have f oscillations per second, we can say omega would be two pi f radiance per second. And so notice, if you know omega value, you can calculate f. You can divide omega by two pi and calculate f. And so what we wanna do now is now that we know about everything about voltage, we wanna figure out the same thing for current. What's gonna current look like? What's gonna be the graph of that current? How the oscillations would look like? And we want to compare those oscillations with the voltage oscillations. So how do we figure out the current? Well, let's assume the current is flowing, I don't know, maybe this way. And let's say the current is I. Now, all we need to do is figure out an expression for current, an equation for current. So where do we begin? Well, since we only have a resistor in our circuit, we can go for Ohm's law. We can say whatever is the voltage across the source, same should be the voltage across the resistor because you only want an element. And then we can use Ohm's law. So it'll be a great idea to pause the video and see if you can figure out the equation for current yourself. All right? So we can directly say from Ohm's law, the current equals voltage divided by R. So voltage across the resistor divided by the resistance. And we need to be very careful. At least right now there's only one component, but in future we might have more components. So it's voltage across the resistor divided by R. It's not just any voltage. But in our example, voltage across the resistor happens to be the source voltage. So that divided by R. And this happens because there's only one component. And we know our source voltage is just we not sine omega t. Sine omega t. So our current is just going to be this divided by R. And ta-da, we have found our current directly from Ohm's law. And what is the equation saying? The equation says that the current is also oscillating. We also had the same sine function over here, which is not a surprise, not really all that much of a surprise. It also tells us, let's look at the relationship between the two, it also tells us that it's oscillating in sync with voltage. What does that mean? This means when the voltage, let's say when the omega t is zero, then our Vs will also go to zero because sine zero is zero. And at the same time, current will also go to zero. Similarly, when omega t is 90 degrees, our Vs goes to maximum, this will also be maximum, current will also go to maximum. So you can see, because they have the same function, they'll be oscillating in sync with each other. Does that make sense? And again, that makes sense to me because Ohm's law, it says when the voltage is maximum, current should also be maximum. When the voltage goes to zero, current should also be zero. So that kind of makes sense, it's nice. And what does this number represent? We not divide by R, that represents our maximum current. And we can call that I not now. And again, it's not a very much, it's not a big surprise that the maximum current is just the maximum voltage divided by R. So now we'll be a great idea for you to pause the video and see what the graph for current is gonna be. So can you visualize or try drawing a graph on top of this itself, current graph? Pause the video and see if you can draw it yourself. Okay, let's do this. This is what it would look like. And so now this peak value represents the positive I not and this would represent negative I not. So this would be our positive I not. This would be our negative I not. And notice how the graph is. The graph is not shifted like this. The graph is exactly this way because they're in sync. They go to zero together, maximum together, zero together, minimum together and so on. And again, if you want to visualize this, we can dim everything and we can more time access forward. So just concentrate over here. Okay, if you move the time access forward, see how they oscillate. They both go up together. They both go to zero together. Both go to the negative maximum together and so on and so forth. That's how you visualize. They're oscillating in sync with each other. And again, I have an animation for you. We can move the graphs to the left and we can decrease it and you can use the same thing. And again, we can draw an arrow mark because it makes it easier to visualize this with an arrow mark and you see they're both oscillating together. This represent, the length of this represents VS and the length of this arrow mark now represents the current I. Okay, and so it goes to maximum, zero, minimum and so on and so forth. I have one question for you and I want you to think about it. I have drawn the current length to be bigger than the voltage length over here. Same here, current maximum to be bigger than voltage maximum. Why is that? Can you pause the video and think a little bit about that? Well, the answer is no reason. It's wrong to say that I naught is bigger than V naught or V naught is bigger than I naught because they are two different units. It's like comparing three seconds and five meters. You can't compare them. You can draw however big three seconds you want as an arrow mark and you can draw however big of five meters you want as an arrow mark. You can't compare them. So these are two different graphs with their own scales and we've just drawn them on the same graph so that we can compare. And even numerically, sometimes I should think, but wait a second, I naught should be less than V naught, right? Because it's V naught divided by R. Even numerically, but think about it, R can be a fraction. So there's no necessity that I naught has to be smaller but of course you can't even, that statement doesn't even make sense. I naught has smaller than V naught. Anyways, this sets the stage for all the future circuits, more interesting circuits where we'll have inductors and capacitors and we'll look at all of that in the future videos.