EMF & flux equation (& graph) of AC generator

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 its 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 then this changing flux pretty 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 okay 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 all right 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 of 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 okay so this is zero angle all right 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 okay 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 cannot 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 times cos of i can say that angle's curve 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 sort of find not 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 turned angle between the area vector and the field vector is 90 degrees so cos 90 is 0 so i get 0 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 1 cos 180 is negative 1 and so i get flux equals negative maximum and so 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 windows perspective the air is blowing 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 back 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 the 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 okay 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 dmf 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 individual and kind of predict what the graph is going to look like okay hopefully you made a prediction now let's check whether you know 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 d5 d phi over dt and so all you 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 nba are all constants we can pull them out so you get nba over here and then you get differentiation of cos omega t differentiation of cos is negative sign 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 nba and 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 mf 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 turns 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 okay now let's go ahead and plot this over here so and when omega t equal to zero when at t equal to zero when theta is zero sine zero is zero so you get induced emf at this position is zero interesting did you predict that at this point you get zero emf in this position okay then after after it has turned through 90 degrees sine 90 is one 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 zero so you get zero here and then at this point you would have turned by 270 degrees sine 270 is minus one and so you get now negative maximum so you get somewhere over here and finally by the time you come here you would have done by a whole 360 degrees 96 360 or 2 pi is a 0 and so emf becomes 0 again here 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 confluctuates between positive e naught and negative e naught 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 hundred times per second then our emf will alternate between plus e naught and minus e naught 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 zero emf and when the flux is zero 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 okay so at this point if you consider this like if you consider this as a mountain then the height represents the flux faraday's 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 zero you 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 slope is zero and that's why the emf at this point is zero okay what about this point at this point although the flux is zero if you stand over here notice it's very steep very very steep you put one step forward and you goes 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 lens is law if the flux is changing if the doctor is decreasing am is trying to increase the flux since 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 induced is zero 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 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 zero slope and so you get zero emf