Emf induced in a rotating rod (& disc) - Motional EMF

suppose we have a magnetic field directed into the screen maybe due to some giant magnets and let's say somebody comes along takes a conducting rod of length l and starts spinning it this way about one of its ends with some speed our goal is to figure out what is the induced emf or what is the potential difference induced between the ends of this rod due to the spinning motion now you may be wondering why are we spinning a rod in a magnetic field well that's what physics is all about we come up with some unrealistic situations but to help you strengthen your concepts all right so the rod is spinning let's say it's spinning with some angular speed omega the immediate question i have is why would there be a potential difference induced what's going on over here this is something we've talked about in previous videos called motional emf and the whole idea is let's take let's quickly recap if you have a rod which is let's say not spinning but let's say it's moving in a magnetic field then we this conducting rod has electrons inside of it and now these electrons are moving inside a magnetic field and we know that moving charges experience a lawrence force given by f equals q times v cross b and so these negatively charged electrons inside the conductors are all going to experience a lawrence force can you quickly pause the video and think about what direction that lawrence force would be in okay so the force would be in the direction of v cross b so v is upwards and b is into the screen so if i were to draw that to show you from some angle upwards into the screen then when you do v cross b you have to take your right hand and and you use your four fingers to cross it in this direction v cross b and when you do that you get your thumb points towards the left which means the force is towards the left but that's for a positive charge because electrons are negative charge this means the force would be towards the right and so the lorentz force acting on them would be towards the right and as a result of that all these electrons would so all these electrons will start migrating to the left side giving you a negative charge and leaving behind positive charges on the left side and there you go because there's a charge separation there's an electric field generated induced and that produces a potential difference and in the previous video we derived the expression for that induced potential difference we called it the motional emf but it's the same thing it's actually the potential difference due to charge separation and we derived it to be the product of b l l the length of the rod times v where v is the speed of the rod this simple expression only works when the velocity is perpendicular to the magnetic field which is the case that we are dealing with right now and if you need clarity on where this comes from feel free to go back and check out our videos on emotional emf and video videos on induced current insider wire moving inside a magnetic field anyways the same concept applies over here even here the rod is moving in a magnetic field charge separation will take place and as a result there will be a potential difference generated the problem over here is instead of moving linearly this char this rod is rotating so the problem the complication over here is when things are rotating different parts are moving with different speeds so again let me show you this let's look at the animation so as we spin this notice the center hardly moves it's at rest look at that center hardly moves however if you look at the tip the tip is the one that's moving the most distance in a given time and as a result has the highest velocity which means as you go away from the center particles are moving faster and faster so the way to think about this is different parts of this rod are moving with different velocities and so what do we do now how do we figure out what's the what's the emf what what do i substitute for the velocity well the way to think about this at least one of the ways to think about this is instead of trying to figure out what the emf or what the voltage is potential differences for the entire rod we can divide this rod into tiny tiny pieces here we go and because each piece is having its own velocity we will see that each piece will generate its own emf or the potential difference across each piece will be different it will be very low over here and as i go towards the right the potential difference keeps increasing increasing more more more more maximum over here but what i want to calculate is the total potential difference bet from here to here which means i have to add up all these potential differences in other words i need to set up an integral and it'll be a great idea now to pause again and see if you can set up that integral yourself it's okay if you go wrong don't worry about it but try setting up that integral yourself first see how you would do it okay let's set up the integral so the whole idea behind setting up the integral is because different pieces are having different potential differences generated let's just concentrate on one of these piece find out how much potential difference that tiny infinitesimal piece generates and then integrated so let's pick one piece at random so let's say i want to concentrate on this piece all right this one let's say okay at some distance so this piece is at some distance from the center let's call that distance x the most favorite variable okay and now the length of this piece matters in our formula that length would be the length of this piece we're only concentrating so since this is this is an increment in x we'll call the length to be dx so if you use the variable the variable to be say r over here this increment would be dr okay you have to use the same variable all right so now let's only focus on this piece and think about what's the potential difference generated over there that potential difference it's the same thing as the emf different words but same thing that potential difference is going to be the magnetic field b times the speed at which this thing is moving and we don't know what that speed is but let's just put that times the length of that and the length of that is just dx and now since we want the total potential difference we integrate this so the total potential difference will be an integral integral from where to where so we are integrating with respect to x so i ask myself what should be the minimum and the max value of x well i want to start from here so x equals 0 and i want to end up over here where x equals l i need to integrate this b times v times dx and we can remove all the constants out so b is a constant the magnetic field everywhere is the same so i can pull that out what about the velocity is that a constant is it the same for each and every piece not at all the whole reason we are integrating is because the speed is not the same so the speed stays this stays inside the integral and now i need to figure out i need to figure out an expression for speed in terms of x because you're integrating with respect to x clearly v is a function of x we depends on x right if x is very small v is very small if x is very large v is very large so we depends on x i need to figure out what that connection is can you pause the video and think about what that connection would be and while you're at it once you get that try integrating and see what you end up with yourself okay so the connection comes from mechanics and circular motion we've already seen before a long time back probably that if an object moves some distance s in a circular path and the angle covered is theta in radians then the connection between them s equals r theta so if you divide now both sides by time this becomes the linear velocity or the speed and this now is the angle squared per second becomes omega and so the connection becomes v equals r omega in our case r becomes x and so clearly you see as r increases or as x increases the speed increases exactly what we predict so we can plug that in over here now so for our piece that velocity v is going to be r times omega so it's going to be x times omega times dx so omega is a constant that comes out and now we can integrate we're integrating from 0 to l so if i pull that omega out we have to integrate x dx and integral of x dx is x squared by 2. so what we end up with now let me move this a little bit up so v equals b that omega gets pulled out times x squared by 2 so x squared by 2 and we put the limits from 0 to l so when we substitute the upper and the lower limits and we subtract we will end up with b omega l square by 2 minus 0 we'll end up with l square over 2. and so that is the expression for the voltage that gets induced or the emf that gets induced or the potential difference basically this is the potential difference that gets induced okay shall we quickly try one more problem and i want you to give it a shot very similar problem magnetic field constant uniform field and instead of a wire this time we have a disk of radius r and that disk is spinning and the question is what's going to be the potential difference between the center of that disk and the edge of that disk can you pause and think about what that would be all right we can redo all the integration if we need but there's another way to think about this if i just consider a tiny radial piece like imagine i just take concentrate on a radial piece of this disc then that particular piece is doing exactly the same thing what that rod was doing in a previous case it's going in a circular path about this point which means the potential difference generated or the emf induced in that piece is going to be exactly the same it's going to be b omega and the l is going to be r now so it's going to be b omega r square over 2. and the beauty is if i choose another radial piece like this over here that piece is also doing the exact same thing it's also spinning that should also have the same potential difference which means if i choose various pieces if i divide this disc into tiny tiny radial pieces each piece should have the exact same potential difference so what's the answer to our original question what's the potential difference between the center and the edge of the disk well it's going to be the same b omega r square over 2. if you want to choose between this point the center and this point on the edge you think of the potential difference between this radial piece if you want to consider the potential difference between center and this point on the edge you imagine that's the potential difference across this radial piece and so if you think of it that way anywhere you consider between the center and the edge the potential difference should stay the same but a couple of questions that comes to my mind sometimes is wait a second all these wires are like you know in this case all these pieces are touching each other so wouldn't that cause some charges to move and wouldn't that change things well remember charges only move if there is a potential difference right and over here if i go at some radius over here let's see if i go to some radius let me choose the same color okay if i go to some radius over here the potential here if it's say i don't know maybe 5 volt then at this point the potential would be the same 5 volt at this point the potential will be the same so at a particular radius the potential everywhere is going to stay the same and so charges will not move from here to here or here to here there is no reason for them to move from here to here at all and so the fact that they're touching each other is not going to make any difference another question that comes to my mind is wait a second shouldn't we like i don't know maybe add all the potential differences and i'll ask why why would why should we do that um and just to convince myself there's another way to think about this you see since each piece is having a potential difference you can sort of kind of think of it as a cell as a battery all right so if you imagine this to be it's not really a battery okay the potential difference is coming because it's spinning but you can imagine it's like acting like a battery and so if you imagine these things were tiny tiny batteries and see how all these batteries are connected they're connected in parallel with each other can you see that right n's ends are all connected and because they're all connected in parallel in parallel voltages don't add up they stay the same okay and it's for that reason you don't add up the voltages and so the potential difference in the center and the edge anywhere you take is going to be b omega r square over 2.