Magnetic moment of electron around a proton

we've seen current carrying loops generate magnetic fields and behave like tiny magnets and since atoms contain a lot of electrons going around and electrons are charged particles meaning atoms also have current loops and these current loops should also generate magnetic fields meaning atoms also tend to behave like magnets so the question we want to answer in this video is what does what decides the strength of this atomic magnet and as we answer this we'll also uncover mysteries like how did people realize that electrons have a spin what's that going to do with magnetism and finally if everything is made up of atoms why isn't everything magnetic so imagine the simplest atom we can think of one proton and one electron going around it in perfect circle okay so we are looking at it from a side so we can't see the circle properly it looks like an old but imagine this is a perfect circle now we know electrons moving around since electrons have have charge moving electron produces a current and because it produces a current we know that current loops behave like tiny magnets and so this atom now immediately you can see should behave like a magnet but one question that immediately comes to my mind is what would be the direction of that magnet would this behave like a magnet with the north pole pointing up or would it behave like a magnet with the south pole pointing up how do we figure that out well to quickly recall we use our right hand thumb rule so if you clasp your right hand in such a way that the the four fingers give you the direction of the current then the thumb points in the direction of the north pole so can you use this and pause the video and think about what the direction of the current is and then use your right hand thumb rule to figure out whether the north pole or the south pole points upwards so can you pause and give it a shot all right so in this case the electron is going in this direction so immediately what comes to my mind is hey if i use my right hand thumb rule so that the encircling finger goes in this direction then my thumb would point downwards so maybe the north pole is pointing downwards right we need to be careful the encircling fingers should represent the direction of the current not the direction of the motion of the charge why that distinction because remember electrons are negatively charged particles which means if the electrons are going in this direction then that means the current is in the opposite direction so the current direction is actually in this way this is the direction of the current and therefore if you use your encircling you should encircle your fingers in this direction which means the thumb points upwards which means the not pole should be pointing upwards so indeed this behaves like a magnet but it's not pole pointing upwards does that make sense okay this now brings us to the main question of the video what is the strength of this tiny atomic magnet and just think about what we are doing over here we're trying to figure out how strongly an atom which are invisible to our eyes how strongly it behaves like a magnet using a pen and paper i mean come on isn't that mind-blowing just think about how far we have come all right so how do we do this well in physics when we talk about strength of a magnet we remember we're talking about the quantity magnetic dipole moment and we've seen the expression for this uh it is the product of the current current and the area of the current loop and if there are multiple loops then we can multiply by n but since here is only one loop we'll not do that and this basically means if you have more current it behaves like a stronger magnet if it's a larger area again it behaves like a stronger magnet and if you're wondering where this is coming from if you've derived this in our previous videos feel free to go back and check that out so let's go ahead and calculate it let's first do the area i think that's a little simpler so current times what's the area of this loop well since we're assuming this to be a perfect circle if we call the radius of this circle to be r okay let's write that over here r then the area of the circle is going to be pi r squared all right that leaves us with the question what's the current and i think that's a little tricky because we just have one electron going around so it does produce a current but i think this is you know you have to think a little bit about it so i go back to my basics what is the definition of current well we can say mathematically current is charge by time but that's not enough you have to understand the meaning of this the way i like to think about it is this basically means at any point you wait for some time t and you figure out how much charge q is passing by in that time t and then that ratio represents the current how much charge is passing per second and so we can do the same thing over here since the electron is whizzing around what we can do is we can take some point on this loop wait for some time t and figure out how many times the electron visits around it and that will give us the current for example if we find that the electron is missing around 10 times then we know the charge is 10 e 10 e divided by the time t will give us the current the question now is how how long do i have to wait how should i wait how do i still do this so the trick that you know we like to do over here is instead of waiting for any time wait for exactly one time period a time period is the time it takes for the electrons to make one complete circle so if you wait for exactly one time period then the electron will pass through that point just once think about it if the time period was say 10 seconds if it took 10 seconds for the electrons to go around then if i were to start my timer right now then in that 10 seconds the electron will just complete one circle which means in that 10 seconds the electron went through this point once which means the charge that went through this point is e and if i can wait for another 10 seconds and i'll again find the charge that goes through this point is e and therefore can you see that if you wait for one time period then the charge that goes by at any point is e and therefore the current would then be e divided by the time period and so now all we have to figure out is what that time period is and i think we we can do that just by using speed distance and time relation we don't know the speed so let's throw in some variables so let's say the speed is v the speed is v so i want you to now pause the video and think about this can you given the speed and you probably can figure out what the distance is can you figure out the time period and then can you plug it and figure out what the current is going to be and you know while you're at it feel free to see what the expression for magnetic moment ends up being so pause and give this a shot okay so the current will be e divided by how do i calculate the time period well i use p equals distance by time so time equals distance by speed what is the distance that it travels in one time period that's two pi r that's the circumference right so the distance traveled in one time period is two pi r and what's the speed so time equals distance by speed and speed you are taken as v there we have it that's the current i will just write that over here now so that gives you e times the v comes on the numerator divided by 2 pi r times pi r squared and we're done with all the substitution now we just have to simplify so r cancels we have a pi that cancels and that gives us m equals e times v e times v divided multiplied by r divided by two and so this means that the strength of this magnet depends upon two things how fast the electron is going and what the radius of that orbit is in fact it's actually another way of just saying the current and the area if you think about it right okay now one question i have is instead of thinking in terms of two things like the speed and the radius of the orbit can we club this together and come up with one single you know one single quantity and there is such a quantity that we can come up with remember we may have learned in mechanics before that when things are going in circular path we can say they have an angular momentum and just to jog your memory that angular momentum which we used to represent as l for particles going in circular path it would just be m v r so we could say the electron has an angular momentum and its angular momentum would be m e times v times r and so what i can do now is i can represent that v r the product of v and r as l divided by m e and so in doing so let's see what happens so we'll now get m equals e divided by 2 e divided by 2 and v r is l divided by m e l divided by m e where m is the mass and so in doing so what we have done now is we can now say that the magnetic moment depends on this is a constant depends on what depends on one thing about the electrons it depends upon its angular momentum and that's why we like this expression let me box that and so this means that the origin of the magnetism will now answer the question the origin of magnetism comes from angular momentum of electrons more angular momentum means stronger magnetic moments now before we wrap up i want to take a couple of steps further first of all i want to write this equation in a vector form because both magnetic moments and angular momentum they're both vector right so what's the direction of the magnetic moment well the direction of the magnetic moment over here is given by the thumb itself okay or you could say uh whichever direction is the north pole that direction is the magnetic moment so in our case the magnetic moment is pointing upwards what about the direction of the angular momentum that's also given by the right hand thumb rule but this time the encircling fingers give us the direction of you know the direction which the particle is moving not the current now current doesn't matter now notice since electron is going in the opposite direction if i were to use my right hand rule to represent the angular momentum that would be in the opposite direction does that make sense because the electron is going in the opposite direction so my encircling finger should be in the opposite direction this means that my angular momentum is pointing downwards and guess what this will be always true for electrons and this means that magnetic moment and angular momentum will always be in the opposite directions so i can now rewrite this vectorially magnetic moment will always be in the opposite direction and that's why i would write negative over here opposite direction of the angular momentum you can also imagine that the negative sign is purely coming because the electrons are negatively charged if this was a positively charged particle then i hope you agree that both of these would be pointing in the same direction finally here's an interesting question for you imagine we just took an electron which is not moving not inside an atom norm it's nothing do you think that would have a magnetic moment what do you think well if you ask me i would say like look magnetic moments require current loops forget about loop there is no current here at all so it shouldn't have any right that's what people thought too but guess what experiments showed otherwise experiment showed that even individual electrons themselves behave like tiny magnets what you may be wondering what kind of experiments are these how do you do this well just like how you know a magnetic needle experiences a deflection in a magnetic field in a similar manner we can check if these things experience a deflection if they do we can say that you know they have a magnetic moment of course things may not be that simple but let's take that for granted that you know you can experimentally figure this out now the question is what do you do of this why do electrons have magnetic moment it doesn't make any sense people were baffled and this is where some people hypothesize that maybe just maybe electrons are spinning around their own axis and so if you imagine electrons to be a ball of charge spinning around its own axis then you have an angular momentum and because of that angular momentum maybe there is another magnetic moment generated and this is how we say electrons have a spin right i used to always wonder how did people figure this out their electrons are spinning on their own axis right well i need to mention though that when today you know you know when we say electrons have a spin we don't really mean they're a ball of charge spinning around in one axis today we realized that the picture is a little bit more complex right the spin is a more quantum mechanical phenomena but we don't have to worry too much about it so what we wrote over here is orbital magnetic moment due to orbital angular momentum along with that we will have a spin magnetic moment generated due to spin angular momentum and it turns out that although we cannot derive it because it requires quantum mechanics you get a very very similar relationship and this means to find out the total magnetic moment of any atom you need to add up all its orbital magnetic moments and its spin magnetic moments and in most materials when you add up all these magnetic moments of all the atoms together because they're all in random direction they all cancel out and that's why most materials are not magnetic finally i want to leave you with one question just like how electrons have spin magnetic moments turns out that protons also have a spin magnetic moment which also obeys a very similar relationship you could also say that their magnetic moments is due to their spin angular momentum but it turns out that their magnetic moments of protons and neutrons are way smaller way way smaller compared to that of electrons and we can completely neglect it can you guess what could be the reason for that this equation has the clue for it