Magnetic susceptibility & permeability

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 provide a number 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's 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 how what will happen to the field lines once it goes through aluminium what happens to it when it goes through you know 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 you know 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 a speciality of them all right what about paramagnets we know that these have tiny permanent dipoles they have tiny magnets inside but they are all 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 are 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 that look like well it looks 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 gonna put a thick arrow mark over there okay 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 feed lines would be very very close to each other and so you know this is how we're gonna draw again this is they need to be really really close uh compared to the field outside because it's incredibly strong we'll see how strong it is in a 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 it as the induced field be 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 some 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 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 there 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 whenever there's a proportionality sign we can always replace it with an equal to and a constant and that constant is often called zai 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 0 then that means that the induced field would always be zero there'll be no there'll 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 it 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 we'll make all the values will make a lot of sense then 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 they should should they should be a unit-less number dimensionless number okay what about its value for diamagnets first of all would be a high value or low value well we know diagonal magnesium 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 you would expect this number to be very very tiny lastly would it be a positive number or a negative number well for that think about the direction of the induced and the and the external field well if the external field is to the right then the induced wheel is to the left because diamagnets you know they tend to repel the field right oh this means the susceptibility value of our 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 it turns out susceptibility values for copper and water is very similar about minus 10 to the power minus phi look at how small it is exactly what we predicted negative exactly what we 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 g 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'd expect you know t value or susceptibility value to be very tiny something similar and it's 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 so positive small numbers for paramagnet 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 aluminium 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 oh that's because parameterism 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 but when it comes to diamagnetism it really really doesn't depend too much on the temperature so you get pretty much the same value or similar value for nitrogen liquid okay what about ferromagnets now to be very precise we there is no direct relationship between the induced field and the external field and you know we will take us we will 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 iron for example i just looked up turns out to have about susceptibility about about two hundred thousand 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 fields that we are familiar with maybe in future videos we'll introduce those new terms as well but the last thing we 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 in 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 you know chi another name we call that mu r and that's called the relative permeability me ability and that is just one plus this number and because diamagnets and paramagnets have very tiny values of g that basically means this number is pretty much going to be one for diamagnets it's gonna be slightly smaller than one it's gonna be one minus ten to the power minus five slightly gonna be 0.99 something for paramagnets it's going to be slightly higher than one one plus two times ten power minus five is one point zero zero zero something but for ferromagnets oh it's gonna be these values are pretty much comparable you know mu value is also gonna be ten thousand hundred thousand 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 g value to be negative one they're called perfect diamagnets i want you to ponder upon that why do we 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 you know and which materials are these i want you to do some research around that