 A current carrying solenoid generates a magnetic field. Now let me call that magnetic field B naught and let's address that as a vacuum field because there's nothing kept inside the solenoid, it's vacuum inside the turns. But what if I keep say a piece of ferromagnet, say a piece of iron, then we've seen before that magnetic field inside the iron is going to be way stronger than the field outside because iron is a ferromagnet and gets magnetized. And we wrote a relationship between the two, we saw that the magnetic field inside any material can be written to be equal to mu R called the relative permeability times the magnetic field outside. Now what I want to show you in this video is that although this relationship works for paramagnets and diamagnets, it does not work for ferromagnets. For ferromagnets, the relationship is actually quite more complicated. To truly understand this relationship, we need to study something called the hysteresis graph and that's the goal of this video, hysteresis graph. So hysteresis graph is basically the graph between these two things, what we want to study. It's a graph between the vacuum field and the magnetic field inside the ferromagnet. And it's called hysteresis because the name basically says something is lagging behind something. We will see once we understand the graph that this magnetic field inside the ferromagnet actually lags behind the vacuum field. That's where the name comes from. All right, so let's do that. Let's try to study that. And for that, we're going to draw a graph. So here's our x-axis. On the x-axis, we're going to draw b0. And we know how to change b0 just by changing the current through the solenoid. Increase it, decrease it. You can even flip the battery and you can reverse the current. So we can do all of that. And then we will see how does that change the magnetic field inside the ferromagnet. And we'll call that b. And just to keep our attention on the ferromagnet, imagine our solenoid is invisible. It's just there, but let's imagine we made it invisible. OK, so how can we understand what's going to happen inside the ferromagnet? How do we understand that? For that, we need to recall the domain theory. Remember, we said that ferromagnets have magnetic domains, which are groups of atoms that are all completely aligned in one direction. So what we'll do is we'll keep track of what happens to these magnetic domains as we change the vacuum field. And that's going to help us understand what happens to the magnetic field inside the ferromagnet. And just to keep things simple, instead of drawing all these gajillion atoms, I'm just going to draw one arrow mark representing the direction in which the atoms inside the domains are all oriented. All right, so let's get started. So how do we start? Well, let's imagine right now there's no current running through our invisible solenoid. So we are at zero. And let's imagine we are going to slowly increase, increase, increase that current so the vacuum field starts increasing. What's going to happen inside? Well, we have seen before that when you expose a ferromagnet to the magnetic field, the domains tend to get, domains tend to turn and get aligned in the direction of the field. And so as we make the magnetic field stronger and stronger, more and more domains get lined up. So in essence, the magnetic domain is becoming bigger and bigger. These three have now become one giant domain. And as a result, the magnetic field inside the ferromagnet starts increasing. It becomes orders of magnitude larger than the field outside because there are billions of atoms, all of their magnetic domains align in the magnetic dipole line in the same direction. So as we make the magnetic field stronger and stronger, more and more domains get aligned eventually, eventually all the domains get aligned in the same direction, all of the magnetic domains. There are millions of domains, billions of them actually. And I've only drawn six for simplicity. But eventually we reach this point. Now what happens if I make my magnetic field even stronger? The vacuum field, I increase the current more in my solenoid. I increase it even more. What's going to happen? Well, now since all the domains are already lined up, you can't have a stronger field inside. This means there is a saturation point. So let's say this is that point. This means eventually we hit a saturation. Basically it means maximum. And so now based on this, I want you to pause the video and think about, can you draw a graph for this of what happens as you go from zero, the vacuum field, and keep increasing? You know eventually you're going to hit a saturation. Can you guess what that graph is going to look like? All right. So let's say because I know it's going to hit a saturation, after this point it's going to be straight, I can make a guess that the magnetic field inside should keep increasing, but not linear. Eventually it's going to saturate. And this means increasing the vacuum field further will not increase the magnetic field inside. It's saturated, it's reached the maximum. And at this point you might say, okay, the relationship is not that bad. It's not a straight line, but it's a curve. And that's not a big deal. We've reached this far in physics. Curves don't, you know, we're not afraid of curves, right? Wait for it. What if now we decrease the magnetic field? Say the current in the solenoid, we reduce it, and we take it all the way to zero. What do you think is going to happen to the magnetic field inside? What do you think is going to happen to all these domains? I want you to pause the video, think about that, and think about what the graph is going to look like as I take it back. Do you think it's going to follow back? Okay. If it has to follow back, reverse, that means all the domains have to go back and become random. And that's not what's going to happen because ferromagnets have this property of retention, which basically means that the domains, once they're aligned, they tend to stay aligned. So even when I reduce my magnetic field, most of the domains don't flip back. Maybe some of them do. So I'm going to over here, let's say one of these domains, they flip back. So some of them say flip and become random, but a lot of domains still stay aligned. That means the magnetic field inside is still very strong, very strong in towards the right. So the field will not go all the way to zero, it might come somewhere over here as the vacuum field goes to zero. And so now we can predict what the graph is going to look like. So our graph from here, as I come back, will not follow this, it'll not go reverse, but it'll go somewhere over here. And this point shows the property of ferromagnets. Even when the magnetic field outside, the vacuum field is zero. I can take this outside the solenoid now, we have permanent magnetism. Okay. So you can immediately see things that are starting to get a little bit interesting. But wait for it. What if now I start increasing the magnetic field in the leftward direction? I'm gonna call it as the negative B naught. And we can easily do that by flipping the battery. What do you think is gonna happen? So I'm gonna increase it in the leftward direction. Again, pause and ponder upon this. All right. So as I increase the magnetic field in the left direction, now it's going to force the domains to flip in the leftward direction. So some of the domains, oops, sorry. So let's say this domain gets flipped to the left. Maybe this gets flipped to the left. And as a result, as more and more domains get flipped to the left, notice they start cancelling the domains which are flipped to the right. And as a result, the net magnetic field starts decreasing inside. So as I go towards the left, the magnetic field starts decreasing. So it sort of like goes here. And then comes a very interesting point somewhere when half the domains are to the right and half the domains are to the left. That's when all the domains cancel out pretty much. And the magnetic field inside is pretty much zero. So that means as I make it more and more negative, eventually there comes a point where the magnetic field goes to zero. Ooh, this is turning out to be really interesting. So now the graph continues and eventually goes to zero. So this is that point where pretty much half of the domains are to the left and half of them are right. And the magnetic field inside has gone to zero. But we won't stop there. We won't stop there. We're gonna increase our magnetic field even further. What if I make that field even stronger to the left? Well, then the story continues. More and more domains, oops, more and more domains are gonna start getting flipped to the left and eventually it'll saturate in the leftward direction. So what will the graph look like? Well, the graph will continue. Now the magnetic field inside will become negative. And so the graph will go like this and just like what we saw on the other side, it's gonna hit saturation. So now we are saturated. Pretty much the same value as we had earlier, but now in the leftward direction. So this will be again, B saturation. And now increasing the magnetic field even further, it's gonna be useless because we know it's gonna be straight. So we're gonna decrease that magnetic field and make it zero. And again, what do you think is gonna happen? Again, it would be a great idea for you to pause and see if you can now complete this graph. Well, just like before, we will find that not all of them will turn back. In fact, most of the domains will stay aligned in the leftward direction. It's kind of like they have inertia, right? They tend to stay aligned. Some of them will flip back. Maybe this is that notorious domain that tends to flip back once the magnetic field is gone. So maybe it flips back. And so the graph is gonna look pretty similar to what we got over here. It's gonna come back, but not all the way to zero. A lot of magnetization is there over here. Again, retention. That's what we see over here. And then what we can do, well, we can again flip the direction of the current in our solenoid and make the magnetic field stronger and stronger to the right. And something very similar is gonna happen. We will now find that more and more domains, more and more domains are gonna start flipping to the right. There'll come a point where half the domains are to the right, half the domains are to the left. And that's where we are over here. We can now speed up our explanation because very similar to what we saw earlier. This is the point. Half the domain are to the left and half are to the right. And eventually if I make my magnetic field even stronger, the vacuum field even stronger, finally I will find that all the domains, all the domains have turned to the right. And we now go back to that saturation. So eventually we will find this thing goes all the way back. And this means that the initial graph which we got when we had a fresh ferromagnet where its domains are all randomly aligned, that is no longer, we no longer get that. That's gone, that innocent looking ferromagnet you can see. Once you magnetize it, it's gonna go along this path. You will never get back that path. And so this graph is called a hysteresis graph and every ferromagnet will have its own graph and depending upon that graph, they can be used for different applications which we'll talk about in a separate video. But let's summarize a few things from this graph. First of all, can you see why the magnetic field inside is lagging behind the vacuum field? You can see when we reduce the vacuum field to zero, the magnetic field inside did not go to zero because of the retention property. It's only when I made it negative, then the field inside became zero. Then I make the vacuum field even more negative then the field inside becomes negative. So you can kind of see that the field is, yield inside is lagging behind the vacuum field. The second thing you can see immediately is not that this relationship doesn't work and I can show you in a very simple way, you take a specific point on this graph, let's say this point and I ask you for this amount of vacuum field, what is the magnetic field inside the ferromagnet? So here's what I'm saying. I'm gonna give you what is the current running inside the solenoid. From that you know what the vacuum field is and I ask you what the magnetic field inside this ferromagnet is going to be and you have two answers. You can either have this much positive value or you can have this negative value, both of them are possible. I'm neglecting this one because this, you only get initially, you don't get it later. And so this means that the magnetic field inside the ferromagnet not only depends upon the vacuum field, it also depends upon the history, how it went through cycles of magnetization. So just by knowing the field outside, the vacuum field, you can't tell what the magnetic field is inside and that's the whole idea behind the histories that's the whole idea behind this complication. And finally increasing or decreasing the magnetic field requires energy. And you know that energy is eventually dissipated as heat in the ferromagnet. What's happening is every time these dipoles are turning there's heat generated. And it turns out that the amount of heat generated is given by the area under this hysteresis curve. So this area represents hysteresis heat we can say or mostly we call that hysteresis loss because it's kind of an energy loss. So it's in the form of heat. So ferromagnets that have a very fat hysteresis curve, that means that it's harder to take them through that cycles of magnetization, it takes a lot of energy. And ferromagnets that have very slim hysteresis curve, it means that it's easier to take them through that cycles of magnetization because it takes less amount of energy. And in the future video, we are gonna use the hysteresis graph to understand retentivity and something called the cohesivity of ferromagnets based on which we're gonna decide whether to use the ferromagnet as a permanent magnet or electromagnet. We'll talk about those in future videos.