 In this video we want to put a number to how diamagnetic or paramagnetic or ferromagnetic things are. For example, if somebody asks how diamagnetic is wood or how paramagnetic is aluminium, how do we put a number to that? That's what we want to explore over here. And we'll start by introducing a uniform magnetic field through all these materials. Imagine there is a giant bar magnet somewhere. Now we've already seen earlier how different materials react to magnetic fields. So based on that and based on your knowledge of magnetic field lines, can you predict how the field lines would change inside these materials? How the field lines would be inside wood? What will happen to the field lines once it goes through aluminium? What happens to it when it goes through iron? Can you pause and try to predict how the field lines would change inside? Okay, let's start with diamagnets. What is their specialty? They tend to repel magnets, right? And how do they do that? Well, they do that by inducing very tiny magnetic moments in the opposite direction. Tiny magnets get induced in the opposite direction. And as a result, there is a very tiny magnetic field induced inside wood in the opposite direction. So let me put that this way. As a result, the total field inside becomes slightly less than the field outside. And we've seen in magnetic field lines, if the field value is less, we like to draw the lines farther away. Okay, so from this, can you pause and now think about how the field line should look inside a diamagnet? Okay, so the field lines inside should go farther away from each other. So the field lines would look somewhat like this. So you see, diamagnets tend to expel the field lines out of its body. That's the specialty of them. All right, what about paramagnets? We know that these have tiny permanent dipoles. They have tiny magnets inside, but they are all in random direction. And when you put a field, they all get aligned pretty much in the same direction. And as a result of that, there is a very tiny magnetic field produced in the same direction. Which means the total field inside becomes slightly higher than the field outside because they're induced field and this is in the same direction. Which means the magnetic field lines are slightly closer to each other compared to the outside. So what would that look like? Well, it would look somewhat like this. So you can see paramagnets tend to kind of suck the magnetic fields towards each other, but very slightly. Both these effects are very slight, very tiny, I've exaggerated over here. What about ferromagnets? Well, we've seen that these have domains inside. And when you put an external field, if it's strong enough, the domains turn. You have superb alignment and as a result, you have an extremely strong magnetic field generated. Or induced inside. I'm going to put a thick arrow over there. Very strong magnetic field get induced. And as a result, the field inside is incredibly strong compared to the outside. So what would it look like? Well, the field lines would be very, very close to each other. And so, you know, this is how we're going to draw. Again, they need to be really, really close compared to the field outside because it's incredibly strong. We'll see how strong it is in a while. All right, so to start putting a number to this, we have to build some equations. And for that, let's try to label these magnetic field. Now, there are multiple fields over here. First of all, we have the field which is in vacuum. The field in the absence of say the wood or this aluminum. Let's call the vacuum field B naught. So I'm just going to call that B naught. The vacuum field is B naught. The vacuum field is B naught. Then we have the field that gets induced as the materials react to it. Let's call that as the induced field. B in, B induced. Let's call that as B induced. And then because of this, the total field inside is going to be a sum of these two, right? It's just going to be a sum, a vector sum, you can say. And so that total field, the net field inside is going to be just B. I'm just going to call that as B. So now the main question I have is what does this induced magnetic field depend on? Well, clearly they depend on the material, whether it's a diamagnet or a ferromagnet and, you know, whether we're dealing with copper or we're dealing with nitrogen, definitely depends on the material. But it also depends upon the strength of the external field, right? I mean if the external field goes to zero, we've seen that the induced fields go to zero in these cases. They are temporary effects. On the other hand, if the external field gets stronger, we would expect the induced field also to become stronger. So it turns out to some degree of accuracy, we can write that the induced magnetic field, and I'm writing it as a vector, is proportional to the external or the vacuum field. And whenever there's a proportionality sign, we can always replace it with an equal to and a constant. And that constant is often called xi or chi, I think, one of them. And it's called susceptibility. Magnetic susceptibility. And what does this constant tell us? Well, let's see. Let's put some number. If we put this zero, then that means that the induced field would always be zero. There will be no magnetization at all. It wouldn't react to the magnetic field at all. On the other hand, if the susceptibility value is very high, that means you have a strong induced field. Very nicely, very vigorously, it reacts to the external field. So, hey, this number directly tells us how readily materials can get magnetized when you have an external field. So this is the number that we're looking for. And that's why it's called magnetic susceptibility, because it tells you how susceptible or how readily materials can get magnetized. Now, before we start looking at the values of susceptibility of different common materials, I want you to make a prediction again. I want you to think about whether the susceptibility values would be very high or low for these materials. And also think about, would they be a positive number or a negative number for these materials? What would be the unit of this? Think about, can you ponder upon all of this before we start looking at the values? Then the values will make a lot of sense there. All right, let's see. First of all, let's look at the units. Both of these are magnetic fields, right? So the units should be nothing. It should be a unitless number, dimensionless number. Okay, what about its value for a diamagnet? First of all, would it be a high value or low value? Well, we know diamagnetism is a very, very weak phenomena. That means the induced field is going to be very tiny, very, very tiny compared to the external field. And so we'd expect this number to be very, very tiny. And lastly, would it be a positive number or a negative number? Well, for that, think about the direction of the induced and the external field. Well, if the external field is to the right, then the induced field is to the left because diamagnets, you know, they tend to repel the field, right? Ooh, this means the susceptibility value for diamagnets should be negative. It should be very tiny. And of course, it should be dimensionless. So let's look at some numbers and see if that makes sense. So turns out susceptibility values for copper and water is very similar about minus 10 to the power minus 5. Look at how small it is. It's predicted negative. Exactly what you predicted. No units. Oh, look at the susceptibility of nitrogen gas. It's way, way smaller, mainly because it's a gas. So less number of atoms, less induction. And as a result, smaller induced field, smaller value of chi. And by the way, we don't have to remember any of these numbers. What about paramagnets? Well, again, there's a very weak phenomena. So we would expect, you know, chi value or susceptibility value to be very tiny. Something similar. And do you think it's positive or negative? Well, over here, they induce field in the same direction as the external field. So we'd expect them to be positive. So tiny, positive small numbers for paramagnets. So let's see. Let's look at some values. And we get small numbers and we get positive numbers. Yay, that's nice. So for aluminum, it turns out to be about 2 times 10 to the power minus 5. Look at for oxygen gas. This is oxygen gas about 2 times 10 to the power minus 6. But look at liquid oxygen, 3.5 times 10 to the power minus 3, much higher. Why is it much higher for liquid oxygen? Something we've seen before. Ooh, that's because paramagnetism depends on temperature. Lower the temperature, easier it is to align these dipoles inside and as a result, stronger paramagnetism. And we've talked a lot about this in previous videos on, you know, videos on paramagnets and diamagnets. So feel free to go back and watch that if you need a refresher. When it comes to diamagnetism, it really, really doesn't depend too much on the temperature. So you would get pretty much the same value or similar value for nitrogen and liquid. Okay, what about ferromagnets? Now to be very precise, there is no direct relationship between the induced field and the external field. And, you know, we'll talk about that in a separate video when we talk about hysteresis. But we can give some ballpark number connecting these two. We know that the induced field inside is going to be much, much higher than the external field and it's going to be in the same direction. So you would expect susceptibility to be incredibly high number and it's going to be positive. So it turns out the numbers can be somewhere between 100 to about 10,000 or even a million. I and for example, I just looked up turns out to have about susceptibility about about 200,000. That's ridiculously high value. Now before we move on to the last part, this quick disclaimer, if you look into your textbooks or some online resources, you will find this is not how susceptibility is formally defined. It is usually defined in terms of a vector called magnetization and magnetic intensity. But to understand the meaning of susceptibility, we don't need to introduce those terms and that's why I like to at least introduce them in terms of normal magnetic field that we are familiar with. Maybe in future videos, we'll introduce those new terms as well. But the last thing you want to do is find an expression, write an expression for the total magnetic field, which is just the sum of the external field and the induced field. So again, can you pause and see if you can go ahead and write that equation? Okay, let's do this. So the total field inside is just going to be the sum of the vacuum field and the induced field. If there was no material, then the induced field would be zero and the total field would be just the vacuum field. Makes sense, right? And we now know that the induced field can be written as the susceptibility times the vacuum field. And therefore, the total field inside is just going to be one plus the susceptibility, one plus the susceptibility times the vacuum field. And because this relationship is pretty useful for us because most of the time we are interested in what the total field is inside a material. We like to give this one plus chi another name. We call that mu R. And that's called the relative permeability, permeability. And that is just one plus this number. And because diamagnets and paramagnets have very tiny values of chi, that basically means this number is pretty much going to be one. For diamagnets, it's going to be slightly smaller than one. It's going to be 1 minus 10 to the power of minus 5. Slightly is going to be 0.999 something. For paramagnets, it's going to be slightly higher than one. One plus two times 10 to the power of minus 5 is 1.000 or something. But for ferromagnets, these values are pretty much comparable. Mu R value is also going to be 10,000, 100,000 or a million. So the magnetic field inside any material is just going to be mu R, the relative permeability times the magnetic field, the vacuum field. All right, before we leave, I want to give you a fun fact. There are certain diamagnets and I won't tell you which one. I want you to do the research. They have chi value to be negative one. They're called perfect diamagnets. I want you to ponder upon that. What does it mean to have negative one? What happened to their permeability? What would be the magnetic field inside? What's the implication of that? And which materials are these? I want you to do some research around that.