 We know that changing magnetic flux induces some emf in a coil. The goal of this video is to figure out the magnitude or the amount of that emf generated. Let's start off by bringing back Faraday's experiment setup. Now Faraday used an ammeter in place of a lamp here. Ammeter is an instrument used for measuring current. For our purposes let's use a lamp, and whenever there is induced emf in a coil we should see the lamp glow. Now it was observed that when this magnet is moved towards or away from the coil the lamp started glowing, which meant that there was some induced emf in the coil. Let's see the magnetic field lines. Now when we move the magnet towards and away, see how the lamp glows. And this led Faraday to conclude that whenever magnetic flux through the coil changed there was some emf induced in the coil. It was also observed that the bulb glowed brighter when the magnet was moved faster towards or away from the coil. So for example if we take four seconds to move this magnet from this point right here to this point the bulb glows and there is some induced emf and it could look like this. Now if we move the magnet between the same points in two seconds if we reduce the time by half notice how the brightness of the lamp changes. I'm sure you must have seen that the brightness of the lamp increases and it turns out that the emf induced in the coil in two seconds was double the amount of emf that was induced when we took four seconds to move the magnet between the same points. Now if we reduce the time by one fourth if we take one second to move the magnet between the same points again notice how the brightness of the lamp changes. And there you go. Again we saw that the brightness increased and it turns out that the emf induced in this case is four times the emf induced when we took four seconds to move the magnet between the same points. This led Faraday to conclude that the magnitude or the amount of emf that is induced is equal to the rate at which the magnetic flux through the coil changed. And this can be expressed mathematically just like this. This right here is called Faraday's law of electromagnetic induction which says that the emf induced, the amount of emf induced, the magnitude of emf induced is equal to the rate of change of magnetic flux through the coil. This negative sign over here is called Lenz's law and this law helps us predict the direction of current in the coil. We will not be focusing on Lenz's law in this video. We will talk about Lenz's law in depth in one of other videos. For this video let's only focus on the magnitude of the emf induced in the coil. Let me remove the field line so that we can see this clearly. There you go. Now if we have only one loop and the magnetic flux through this loop changes, then the emf induced is directly equal to the rate of change of magnetic flux through this loop. If we have three loops then it turns out that emf is induced in each and every loop and the total emf induced is equal to the sum of the emf induced in each and every loop. So we add the rate of change of flux three times which gives us a total emf induced. If we have six loops then again emf is induced in each and every loop and the total emf is equal to the rate of change of flux in each and every loop. And if we continue increasing the number of loops that means you have to add the emf induced that many times. So if we have n number of loops then you multiply the rate of change of flux n times to get the total emf induced. Now this equation right here is used for figuring out the average induced emf. Average because we are figuring out the emf induced over a period of time delta t over some interval of time. This could be two seconds, three seconds, four seconds. There is also instantaneous emf which is given by d5 by dt. This is calculated when we are interested in figuring out the emf induced at any one particular instant in time. In this video we will only be focusing on average induced emf. So let's do that with a couple of examples. Now for our first case we have a region of uniform magnetic field and the strength of this magnetic field is two tesla. And let's say we have a square loop which is which has a side of six meters and it is moving to the right with a speed of three meters per second. Because it is moving to the right after some time this square loop will completely enter the region of uniform magnetic field and will look just like this. The question is to figure out the average the magnitude of the average emf induced when the loop has fully entered the region of uniform magnetic field. Why don't you pause the video and try this one on your own first. All right so we know that the magnitude of average emf induced is given by this relation right here. It's equal to the rate of change of magnetic flux. Now over here n is just one because this is just one square wire. There's no coil over here so n is just one. We need to think about the change in magnetic flux and also the time duration for which the magnetic flux is changing. So let's begin by thinking about delta t first. Delta t is the time duration for which the magnetic flux changes and the magnetic flux through the coil starts changing when the loop starts entering the region of uniform magnetic field. So kind of like this when the loop is moving to the right and at this very instant right when it starts entering the region that is when we should start recording our time. That is when we should start the timer for delta t. So we know the speed that is three meters per second. That means that in one second the loop travels three meters to the right. So that would look like this. It has travelled three meters to the right and the distance over here is this distance right here. It has travelled three meters to the right. Now if we take one more second then in one more second it will take further three meters to the right and the loop will move like this further three meters to the right. So now over here three meters is this distance right here. That is the point when we should stop the timer because now the loop has fully entered the region of uniform magnetic field and now when it continues to move forward there will be no change in magnetic flux because the number of magnetic field lines through this loop aren't really changing. So the time duration for which the magnetic flux changes that comes out to be as two seconds. Now let's think about the change in magnetic flux. That is delta phi. So delta phi is the final magnetic flux minus the initial one. We can think about the initial to begin with. Initially the loop is completely outside the region of uniform magnetic field and when that is the case there are no magnetic field lines passing through the loop so the magnetic flux is just zero. So that leaves us with the final flux. Now flux is given by Ba cos theta. We already know the strength of the magnetic field and we can figure out the area of this square loop which would be 6 x 6, 36 meter square. But what about this cos theta over here? Now theta is the angle between the magnetic field strength and the area vector. So in order to figure out what would theta be let's try and look at this setup from a certain angle. So let's say these are our coordinate axis and here is a loop. So we are looking at this setup from a certain angle. Now if we try and draw the magnetic field vector that will always be pointing down because these magnetic field lines are going away from you they are away from the plane of the screen. So the magnetic field vector could look like this. This right here is your magnetic field vector. And the area vector could either be along the magnetic field vector or directly opposite to it. So if it is along the magnetic field vector it would look like this. Now over here the angle between the area vector and the magnetic field vector is zero degrees. So that would be zero and cos zero is just one. It is also possible that the area vector instead of vertically down it's pointing in a direction completely opposite to the magnetic field direction just like this. Now in this case the angle between the magnetic field and the area vector is 180 degrees. So that is minus one. Now it turns out that the direction of the induced current determines the direction of the area vector and we can predict the direction of the current using Lenz's law. Again that is something that we will not go into this video. We are only interested in the magnitude of cos theta and in either case the magnitude of cos theta comes out to be as one. So now let's plug in the values and if we do that here is your magnetic field strength. Here is your area and the cos theta is just one. So this will give us the final flux and rather the change in flux to be as 72 vectors. Now when we put all of this the change in flux and the time duration in the first equation that will give us 72 divided by 2 seconds. And finally the EMF induced comes out to be as 36 volts. Now in our next example we have a coil between the two poles of a magnet and the magnetic field lines look like this. They go from north to south pole and this coil can be rotated in a clockwise manner just like this. The strength of these magnetic field lines is 2 tesla. The area of this coil is 6 meter square and we can say that this coil is made up of 10 loops of wire. So 10 loops of wire they make up to form a coil. So N capital N becomes 10 in this one. Now this coil rotates from an angle of 60 degrees to an angle of 180 degrees in 3 seconds. This theta is the angle between the magnetic field vector and the area vector of the coil. The question is to figure out what is the average induced EMF when the coil rotates from 60 degrees to 180 degrees in 3 seconds. Why don't you pause the video and try this one on your own first. Alright so the average induced EMF is given by the same relation which is N into change in magnetic flux delta phi by delta t and we are interested in the magnitude so we see the mod signs over here. Now we already know the time interval delta t which is 3 seconds in this one. So we only need to figure out the change in magnetic flux. Change in flux is final minus initial and flux is B A cos theta. So we can write change in flux as B A cos theta F which is a final angle. Minus B A cos theta i which is the initial angle. Let's try and understand how do these angles really look like. So for that let's place ourselves on top of this coil. Let's say we are on the top this is our i and we are looking down on the coil. So the magnetic field lines would look just like this. They go from the north end to the south end just in this manner. And to begin with the angle between the area vector of the coil and the magnetic field lines is 60 degrees so the coil would be in an orientation like this. If we show the area vector that would be right over here. Now theta that is 60 degrees is this angle right here. That is the angle between the area vector and the magnetic field. Once it has rotated by some degrees and finally the angle between the area vector and the magnetic field is 180 degrees. So in that case the coil would look just like this along with the area vector. You can see how the angle between the magnetic field vector and the area vector is 180 degrees. So theta 60 degrees to begin with. So that makes cos theta as 1 by 2 and theta 180 degrees after 3 seconds. So that makes cos 180 degrees to be as minus 1. Now if you plug in these values right over here. We should get change in flux to be as 2 multiplied by 6 which is the area multiplied by minus 1. Minus 1 because we are starting with the final flux. So it has the final angle theta f. So that is 180 degrees. Therefore this is minus 1. Minus the initial flux and that would be 2 into 6 into 1 by 2 cos 60 degrees. Now when we work this out this comes out to be as minus 12 that is the final flux minus the initial flux which is 6. This comes out to be as minus 18. So minus 18 if you take the mod it becomes 18 vebbers. Now if you place these values into the first equation that will become 10 into 18 divided by 3. 10 is the number of turns. Tie interval is 3 seconds and the change in flux came out to be as 18 vebbers. So the emf induced in this one is 60 volts.