Bohr magneton (& gyromagnetic ratio)

because atoms have electrons going around which forms current loops they behave like tiny magnets and as a result they have magnetic moments but here's the question we want to try and answer in this video just like in charges we've seen they are quantized and they have a minimum value of e the charge of an electron and charges have to be an integral multiple any charge you think of has to be an integral multiple of the charge on an electron the question is is there something similar for the magnetic moment of an atom does it also have some fundamental minimum value do these magnetic moments also obey the quantization principle it has to be some integral multiple of that minimum value well that's exactly what you want to find out in this video in a previous video we saw that if you take the simplest atom an electron going around a proton it behaves like a tiny magnet and the strength of that magnet the magnetic dipole moment is given by this expression it basically says that magnetic moments come from angular momentum in fact they're directly proportional more angular momentum gives you more magnetic moment and the reason why we i'm putting an over here is to remind us that this is due to the orbital motion because along with that we also saw electrons tend to have a spinning motion spin angular momentum although they're not really spinning around their own axis we can think of it that way and that generates an additional spin magnetic moment which we're not going to talk about too much in this video and so our main question is does this magnetic moment have a minimum value and if it is what is that now before we get to that we'll get to that in a second but before we do that i just want to focus on this number over here what does this number represent if you think mathematically this number is just the ratio of magnetic moment and the angular momentum right and we give a name to this number so this number is often called the gyro i'm just mentioning because you may hear it somewhere it's called the gyro magnetic magnetic ratio all it is saying is that it's the ratio of magnetic moment and angular momentum in fact you know it should have been called magnetogyric ratio because you know moment is magnetic moment is coming in the numerator and the angular momentum is coming to denominator but never mind it's just a name that we give and what's the significance of that it's basically this number tells us how much magnetic moment you get for a given angular momentum for high gyromagnetic ratios you get more magnetic moment for a given angular momentum and this value would be different for spin magnetic moment of electrons and we have a different value for spin magnetic moments of protons and neutrons even they have magnetic moments but you can immediately see that because protons have a higher mass protons and neutrons have a higher mass their gyromatic ratio would be much smaller than that of electrons meaning we can immediately see that the protons and neutrons will have much tinier but smaller values of magnetic moments compared to that of electrons and it's for that reason most of the magnetism that you see in nature is due to electron spin not the spin of the protons and the neutrons all right with that out of the way we can now go back come back to the main question does this have a minimum value and for that we have to ask does the angular momentum have a minimum value well how do you calculate angular momentum of electrons going in circles well angular moment of any particle going in circle is given by m v r where m is the mass mass of the electron here v is the speed of the electron and r is the radius of that circular path of the orbit so here's the question does should this have a minimum value well no right i mean our our electron should be allowed to orbit as close as possible r should be able to have any small value it want any arbitrarily small values it wants so there's there is no reason why they should have a minimum value at least when you think using newtonian physics right but the problem is atoms do not be atoms do not obey newtonian mechanics neutral mechanics is an approximation that works at the large scale so to really deal with atoms we need quantum mechanics and i know quantum mechanics has a reputation for being all complex and vague and abstract and whatnot but what we'll do in in this video is we'll limit ourselves to bohr's theory see niels bohr was one of the pioneers of quantum mechanics and he came up with you know the earlier versions of quantum mechanics and we'll only restrict ourselves to that so let's go to mr bohr neil's board and ask him what he has to say about it bohr says in his boss theory and we'll learn more about this in when we learn about atoms so bohr's theory says that in an atom he only talks about hydrogen-like atoms atoms which have only one electron so it's perfect for our atom over here so his theory says he postulates that the angular momentum of these electrons should have a minimum value they're not going to ask why we'll not get into all of that we'll just hear him out so he says should have a minimum value and he gives us what that minimum value is he says that the angular momentum of this electron cannot be smaller than h over 2 pi where h is the planck's constant and there's more to this theory see this is the minimum value right but electrons can have more than this if they want they can be in higher orbits that will give them higher angular momentum but again there are conditions you cannot have any higher values you want bohr says you can the next higher value available for electrons is twice this number the next higher value would be three times this number so in general any angular momentum that the electrons have due to their orbital motion has to be an integral multiple n times h over 2 pi so where n is an integer so this has to be an integer and so does this answer our question so do you think that magnetic moment should have a minimum value according to bohr's theory yes according to quantum theory yes and we can now figure out what that is by substituting it over here and i can do that and you can do that as well so why don't you pause the video and see substitute and see what that minimum value turns out to be all right let's substitute so i get magnetic moment mu zero and by the way now that i'm seeing this we have used mu naught before as my permeability here we're not talking about permeability we're using the same symbols i know but here we are talking about magnetic moment and o stands for orbital okay sorry for the confusion but anyways that equals and i'm not going to look at the vector signs anymore we only worried about the magnitudes right so that's going to be e divided by 2 m e that's the gyromagnetic ratio times the angular momentum we'll use the quantization condition n h over 2 pi and so we put it all together we get the magnetic moment has to equal i'll keep that n out because that's the integer so it has to be an integral multiple of what an integral multiple of e in times h divided by 4 m e pi so let's write that 4 m e times pi which means the smallest value you can have is when n equals 1 and that is this number so that is the smallest this is the smallest magnetic moment we have we can have this is very similar to how in in electric charges we have q equals n e where e represents the smallest charge and q has to be an integral multiple of that same same as the case over here so the question now is what is this value well you can substitute the value of e h when m e and pi all these are known constants and i'll not do that i'll just tell you what the value turns out to be if you substitute this turns out to be nine point roughly three times ten to the power minus twenty-four a very tiny value and what's the units now at first you might think oh my god so many units have to substitute but don't worry it's magnetic moment we've seen magnetic moment is current times m current times area so current is amperes times area is meter square so this is the smallest magnetic moment ever according to bohr's theory and we give a symbol for that we call that mu b we call that bohr magneton and just like when we're dealing with very tiny charges we talk in terms of ease if we say 1e or plus 3 or minus 40 or something like that when we're dealing with magnetic moments we talk in terms of more magnetons we say it has one more magnet on or it has two more magnetons and according to bohr's theory we cannot have 3.5 bore magnetons that's not allowed but remember this is only orbital magnetic moment similarly we have spin magnetic moment but the beautiful thing is it turns out that even spin magnetic moment for electrons happens to be you know pretty much one bore magneton so if you calculate the spin magnetic moment of electrons you get pretty much the same number one more magneton so long story short because angular momentum is quantized and has a minimum value magnetic moment also automatically becomes quantized you can only have integral multiples of this minimum value and that fundamental minimum value happens to be this number which we call a bohr magneton