 We've already seen that spinning a coil inside a magnetic field induces a voltage which you can use to light up things. In this video we're going to see exactly what that voltage depends on by building an equation for it and we'll see how that voltage changes in time, changes with time by drawing a graph. So here we are seeing the snapshot of the coils at different positions and we can start by asking ourselves why is there a voltage generated in the first place? Well from Faraday's law we know that whenever there is a change in the magnetic flux linked through the coil, there is an EMF induced. And over here you can clearly see the magnetic flux over here is changing. It's maximum over here and then the flux goes to zero. You can think of this as wind. All the wind is passing through the coil over here but over here none of the wind is going through the coil. So flux is changing as the thing spins and this changing flux induces an EMF which acts like a voltage. Now if you want to know exactly what the voltage depends on and what the equation for the voltage is, we need to first figure out how the flux is changing. So you know what we're going to do? We're going to first plot a graph of the flux and see how it changes with time. Yay! So let me plot a couple of axes. So let's plot flux on the vertical and the time along the horizontal. So here is our time. Now to do that we need an equation connecting the flux and the time. And I encourage you to pause the video and figure out that equation yourself because you probably already know how to calculate magnetic flux yourself. Alright so how do you calculate magnetic flux? We calculate magnetic flux as whenever you have flat areas and uniform fields we calculate as just N times the field strength B times the area of the coil A times cos of the angle between the magnetic field and the area. And be very careful with that. So the angle theta is the angle between the field vector and the area vector. Over here it looks like the angle is 90 but that's not theta. So you have to first draw an area vector and if you draw that the area vectors are always drawn perpendicular to the area. So we could draw an area vector like this and so notice the angle over here is zero because they're both parallel. So this is zero angle. Alright so we can draw the area vectors everywhere and then we can see what the angle between them is. And if I draw that over here it will be down, so let's draw that. See as it is spinning the area vector is turning as well so it's down, left, comes up and comes to the left, right. And so the angle is clearly changing, theta is changing with time and that's how the flux changes. So if I can now figure out how theta is changing with time I get my equation I can plot. So how can I connect theta with respect to t? How can I make theta as a function of time? Well this goes back to our rotational motion. Just like in linear motion we can say distance covered equals speed into time. In angular motion we can say angles covered equals angular speed into time. So what we can do is, let me rewrite this. So we can write phi b, oops, phi b equals nbA times cos of, I can say that angles covered equals angular speed into time, just like distance equals speed times times. So angular speed into time. And now I can go ahead and plot this over here. So let's do that together. Let's first put some specific points. So let's say that this is the moment where t equal to zero, just to keep things simple. So let's assume this is our time t equal to zero. We start our clock at this point. So right now what is our flux? Well right now theta is zero, I can directly look at theta is zero, cos zero is one and so the flux right now is going to be just nbA. And guess what, this is the maximum flux you can ever get because the maximum value of cos is just one. And so if I call that maximum flux, this nbA represents the maximum flux. If I call that as phi, you know, I can just say our phi naught. Then right now at time t equal to zero, I get maximum value. So let's say this is our maximum value. This is our phi naught. Okay. What about at this position? What is the value of flux over here? Well, notice the angle between the area vector and the field vector is 90 degrees. Cos 90 would be zero. Right? So I can just look at this. I know the angle immediately. It's 90 degrees. The angle between the area vector and the field vector is 90 degrees. So cos 90 is zero. So I get zero flux at this point. What about over here? The angle between the area vector and the field vector is 180 degrees. And therefore I get negative one. Cos 180 is negative one. And so I get flux equals negative maximum. And so it will be somewhere over here. Now you may be wondering, what does it mean to have a negative flux? What does that mean? Well, the way to think about it is you can imagine that this window has an orientation. So in this case, you can kind of imagine if the window, if the air was flowing in one direction, when the window has turned by 180 degrees, now it's facing the other way. So from the window's perspective, the air is flowing in the opposite direction. Does that make sense? And so that's why we say there's a negative flux. So this is positive flux. This is negative flux. Over here, the flux becomes zero again. And finally over here, the flux goes back to maximum. So how would that flux? How would this whole thing look like? Would it be just like straight lines? No. It's a cos function. And you might know, cos and sine functions have that typical graph. So let's go ahead and try and plot that graph. So the typical graph looks like a wave. So it's going to look like somewhat like this. Not quite there, but yeah. Not bad. Looks pretty good, right? Next we will do the same thing for the EMF. We'll figure out the equation for the induced EMF and then plot the graph. But before we do that, I want you to make a prediction of how that graph is going to look like. Remember, the EMF induced is high when the flux change is very high. And so at each point, can you kind of predict whether you get a high EMF or low induced EMF or zero induced EMF, and kind of predict what the graph is going to look like? OK. Hopefully you've made your prediction. Now let's check whether it matches. So let's build the equation. How do we build the equation? We can use Faraday's law. Faraday's law states the induced EMF is always equals negative d phi over dt. And so all we have to do is differentiate this equation with respect to time. And again, I encourage you to try and do this yourself. So let's do this. So the induced EMF will be negative. What's the differentiation of this term? Well, n, b, a are all constants. We can pull them out. So you get n, b, a over here. And then you get differentiation of cos omega t. Differentiation of cos is negative sine. So minus sine omega t. And then you have to use a chain rule. Then multiply by differentiation of omega t, which gives you an omega. So an omega pops out. And so if you put it all together, we now get EMF equals negative negative cancels. So I get n, b, a, n, b times a times omega times sine of omega t, which is basically the angle. And there we go. This is the expression for the EMF. And just like before, this now represents the maximum induced EMF, which you can call E naught. So now we know what the induced EMF depends on. It depends upon the number of tons, the strength of the field, the area of the coil, and how quickly you're spinning it. The quicker you spin, the more quickly the flux changes and higher is the induced EMF. OK, now let's go ahead and plot this over here. So and when omega t equal to 0, when t equal to 0, when theta is 0, sine 0 is 0. So you get induced EMF at this position is 0. Interesting. Did you predict that? At this point, you get 0 EMF in this position. OK. Then after it has turned through 90 degrees, sine 90 is 1. So now you get maximum EMF at this position. So let's say this is our maximum EMF. And so the maximum EMF will be somewhere over here. Then at this position, it has turned by an angle of 180 degrees. And so now you get sine 180 is again 0. So you get 0 here. And then at this point, you would have turned by 270 degrees, sine 270 is minus 1. And so you get now negative maximum. So you get somewhere over here. And finally, by the time you come here, you would have turned by a whole 360 degrees, 360 or 2 pi is a 0. And so EMF becomes 0 again. So you have to plot that. Let's do this. Another sine wave is going to look somewhat like this. And so our EMF also is a sine graph. It can fluctuates between positive E0 and negative E0. And that's why we say it's an alternating voltage. So how long does it take to go from here to here? Well, it depends upon how long it takes to finish your rotation. So if you spin this 100 times per second, then our EMF will alternate between plus E0 and minus E0 100 times per second. And this is what we call the frequency. In India, the frequency is close to 50 hertz, which means the coils will be spun at 50 times per second. And so our EMF will also alternate at 50 times per second. Finally, one peculiar thing that you're seeing is that at this point where the flux is maximum, I'm getting 0 EMF. And when the flux is 0, I'm getting maximum EMF. What's going on? I would have predicted exactly the other way around. What do you think is going on? Well, Faraday reminds us, I don't care about flux. I care about the flux change. So at this point, if you consider this as a mountain, then the height represents the flux. Faraday is saying, I don't care about the height. Tell me how quickly the height is changing. So imagine you're standing somewhere over here. How steep is this? It's not very steep. It's very flat, isn't it? So the slope over here is 0. You put one step forward, you don't immediately fall down. It looks very flat to you. And therefore, the flux is not changing very quickly at this point. The slope is 0, and that's why the EMF at this point is 0. Okay, what about this point? At this point, although the flux is 0, if you stand over here, notice it's very steep. Very, very steep. You put one step forward and you go immediately down. So clearly there's a very high negative slope over here. And that's why we're getting a very high positive EMF induced. Why? Because of the minus sign, Lenses Law. The flux is changing. If the doctor is decreasing, EMF is trying to increase the flux and you're getting a positive EMF. Oh, it's all trying to make sense. Same thing happens over here. At this point, you have a very high negative flux. But look at the change. It's nothing. You can stand here easily. There's no slope. You put one step forward, hardly there is any change in the height. And therefore, EMF induces 0. And the same thing over here. If you stand over here, oh, look at the slope. Very high. You know, it's a very steep thing. If you put one step forward, you go high tremendously. And so very positive slope over here. And as a result, very high negative EMF induced. Okay? So hopefully this makes a lot of sense. And the same thing continues over here. You get 0 slope. And so you get 0 EMF.