Displacement current & Ampere Maxwell's law

we've already explored ampere's law which says the circulation of magnetic field through a closed loop equals mu naught times the current enclosed by the loop but in this video we'll see that this equation is not complete it's not applicable to all scenarios and we'll see how maxwell steps in and completes it by introducing this idea of displacement current so let's begin now before we start if this equation looks unfamiliar to you then we've talked a lot about ampere's law in our previous videos if you've solved numericals so it'll be a great idea to go back and watch those videos so let's start with the question of what are some scenarios in which this ampere's law does not work and why so one of the most common scenarios is when we're dealing with a charging or discharging capacitor and let's zoom in a little bit so we can see better all right so we have imagine two circular plates of a capacitor and let's say that our current over here is increasing just for the sake of example so the capacitor is charging and now what we'll do is we'll focus on two points one point over here let's call that point p and another point somewhere in between the capacitor but at the same height let's call it point q i want us to think about what is the magnetic field at point p and point q according to ampere's law we don't have to calculate it but let's just compare them so if i were to calculate the magnetic field at point p using ampere's law i would get some answer some non-zero answer that's important right because it's enclosing some current what would i get if i were to use ampere's law here well let's see i'll again choose a circular loop because everything is nice and symmetric so if i go ahead and draw a circle a closed circular loop attach a flat surface and if i apply ampere's law over here the one thing that i see is that there is no current passing through this surface why not because this loop is right in between the two plates of a capacitor and there is no current over here this vacuum which means if i apply ampere's law the right hand side becomes zero which means ampere is saying the magnetic field over here is zero magnetic field according to ampere's law at point q is zero and that's the problem because experiment says that's wrong according to the experiment magnetic field at point q is pretty much the same as magnetic field at point p that's what experiments tell us and that is the problem with ampere's law so you can see there is some problem with ampere's law because it's not giving us the right answer in this situation so there is some fundamental problem with it but what is it for that let's you know step one you know let's go one step deeper beyond mathematics and see what this equation is telling us you see if you asked ampere let's go ahead and ask emperor hey ampere if you ask him what generates magnetic field ampere says currents and that's why in his equation is saying look your loop should enclose some current only then there will be magnetic fields if you don't enclose you will not get magnetic field and it's for that reason amp here or for that matter any other scientist back then would say that you get zero magnetic field predicting zero magnetic field over here because it's not enclosing a current because no current no magnetic field but we do get a magnetic field experimentally so the question is why is that happening this is where maxwell steps in maxwell says maybe current is not the only thing that generates magnetic field maybe there's something else that's generating magnetic fields which is missing in this equation and i'm sure at this point you would get up and you would ask maxwell what else can generate magnetic field besides current and maxwell says i won't tell you the answer right away i actually want to walk you through the thought process that max maybe max ever had to give you an opportunity to guess what that other way could be so let me zoom out and you know let me just walk you through the process so maxwell is saying let's zoom out let's recap a little bit if we go back to electric fields electric fields and we ask ourselves what generates electric fields what creates electric fields what's the answer for that well we have studied electric fields come from charges positive and negative charges right charges generate electric fields but we saw there is another way to generate electric field even without charges remember faraday showed us that faraday showed us that you can generate electric field by changing magnetic flux so let me write that you can also generate electric fields by changing magnetic flux and we call this electromagnetic induction we talked about all the cool applications of that so there are two ways of generating electric fields now similarly we come to magnetic fields magnetic fields and we ask ourselves hey what generates magnetic fields what's the source for that and so far we had only one answer for that currents currents generate magnetic field currents or moving charges and that's why you know the the unit name was called moving charges and magnetism but now maxwell is coming in and thinking maybe there's another way to generate magnetic field which is similar to this now can you pause and think about what that other way could be i want you to guess this all right so changing magnetic flux can produce electric fields so maxwell is thinking maybe changing electric flux changing electric flux can produce magnetic fields maybe that's another way of generating magnetic fields how does that answer this question well you can see as the capacitor is getting charged there is an electric field that's being generated over here and as the charge is increasing that electric field is increasing and because the electric field is increasing the flux through this loop is increasing and maxwell is arguing and saying because the flux is increasing that's why there is a magnetic field generated over here which is missing over here the term is missing and so what maxwell does is he adds that other term he says there needs to be another term which takes care of the changing electric flux and we're not going to go through the derivation but it turns out that other term the other term happens to be epsilon naught times d over dt of electric flux and this now is what maxwell says is the complete equation so this equation takes care of both the sources you get magnetic fields due to currents and you get magnetic fields due to changing electric fluxes this complete equation is now called ampere maxwell's law for obvious reason because maxwell completed it and finally what do you think is the unit of this term that maxwell added well at first i would say think that oh my god so much calculations unit of epsilon naught multiplied by units of phi but another way to think about it is remember you can only add two physical quantities which have the same units which means this should have the same units as current and therefore this should also have the units of ampere and you can go ahead and check that it indeed does have the unit of ampere and it's for this reason maxwell gave this term a name he thought this is some kind of current flowing between the capacitor plates and he called it the displacement current displacement current and i always wondered why did he call it that i read up some articles i'm not sure how credible this is but some articles said that back then maxwell believed that there is ether medium everywhere in the universe even in between the capacitor and he thought that as the electric flux is changing it's the particles of ether that is getting displaced and maybe that's the reason he thought that there's some kind of current and so he called it the displacement current but of course today we know there is no ether medium and there is truly vacuum in between these two plates and so there is no current so this is not really any current but the name is stuck and just to differentiate this term from the actual current we also give we also call this the real current or the actual current conduction current so conduction current is you know the actual current the good old familiar current and displacement current is just the term that takes care of the changing electric flux that gives us that produces magnetic fields and now if we go ahead and use this ampere maxwell's law and apply it at point p and point q we will indeed see the magnetic fields will be exactly the same and we'll do that calculation in another video but long story short what did we learn we learned that back then ampere and many other people thought that there's only one source of magnetic field electric current but today we know that's not true there is another way to generate magnetic field by changing electric flux and so this quantity that represents the magnetic field generated due to changing electric flux is what he called the displacement current and of course today we know that it's not really a current