Ampere's circuital law (with examples)

coulomb's law helps us calculate electric field due to point charges and similarly in magnetism bo sawar law helps us calculate magnetic fields due to point current elements but we've also explored gauss's law which helps us calculate electric fields in symmetric situations and we've seen that one can be obtained from the other they are equivalent and so now the question is do we have something similar in magnetism something that can help us calculate magnetic fields for symmetric situations the answer is yes we have something called ampere's circuital law and in this video we're going to ask mr ampere to help us understand his law so mr amp here what's your law tell us ampere says let's take an example imagine we have three wires that carry some current i1 i2 i3 now ampere says draw a closed loop and i ask what do you mean he says anywhere in space any shape you want draw a closed loop i say okay cool let me do that so let's say i draw a closed loop that goes somewhat like this i am excluding i3 because why not he asked us to draw anywhere i want fine what next amp here next he says walk around this loop and i ask well there are two ways to walk either this way or that way which way should i walk he says any direction you want and i love that i love the freedom that he's giving us so let me walk this way i'm walking amp here what should i do now and now comes the important part he says at every point i'll write this at every point find the dot product of the magnetic field and a tiny length dl vector okay what does that mean so here's what ampere is saying at every point in space the three currents are together producing a magnetic field right we know that current produces magnetic field so maybe at this point just drawing random directions now maybe the magnetic field is this way maybe there's a point over here where the magnetic field is i don't know maybe this way and maybe there's a point over here where the magnetic field is this way now what amp here says is that as i'm walking at every point take a tiny step length which is dl and we'll have a direction of this you know tangential to this path so over here dl will be this way over here dl would be this way and over here because i'm walking like this dl would be this way you can imagine dl to be a very tiny step like a nanometer or something it's an infinitesimally small step and take a dot product of them scalar product of them so bdl cos theta you might know how to take the dot product by now and do that everywhere and he asks us to then take a summation of that so add all of that up over the entire loop and since we're dealing with infinitesimals we're dealing with calculus addition in calculus summation in calculus is what we call integral and so this is what ampere wants us to do in that closed loop take the integral of b dot d l everywhere now ampere is warning us not warning sorry reminding us that this will only work for closed loops so for example if i had chosen say a loop which looked like this then it will not work ampere says don't take this even here i would take i can take b dot dl right but ampere says no no only for closed loops and so to remind us he's going to put a circle over here oops let's use the same color circle over here and says closed loop all right all right what what happens if i do that ampere asks me mahesh what do you think will happen if you took this uh integral over the closed loop and i say i have no idea you tell us amp here this is where ampere smiles and laughs and says ha the answer is going to be and this is the circ this is the ampere circular law the answer is going to be always mu naught times i enclosed not available now before we continue i'm sure a lot of questions are brewing up in your mind like why do we need this law well as we will see in future videos we can use this to figure out the strength of the magnetic fields in certain symmetric situations we'll do that in the future videos okay but for now let's concentrate on the right hand side and ask ourselves what does this enclosed current mean well let's ask ampere up here what is this enclosed current well one way to think about this is basically how much current is enclosed the total current enclosed by the loop but ampere is a little bit more specific ampere says look to figure this out first step you have to do is attach a surface to this loop and i don't understand that as ampere what do you mean so ampere says okay imagine this imagine you took this and dipped in a soap solution what would happen there will be some soap film attached over here right here you go let's imagine that's the soap solution now ampere says enclosed current is the current that punches through this surface whatever is punching to that surface is the enclosed current so the enclosed current is basically the total current that is passing through an attached surface to the loop the attach surface is our soap solution so in our example what would be the value of b dot dl according to ampere's law well that's going to be mu naught times what is i n closed only i1 and i2 are passing through the attached surface they are the only ones enclosed i3 is not so i3 will not be in the picture so the total will be total will be i1 plus i2 but the moment i write that i feel uncomfortable because i know that one current is going up another current is going down so one must be positive and one must be negative right ampere says yes one must be positive one must be negative but how do i figure out which one is positive and which one is negative what do i do so ampere says we use the same thing that we've used so far in magnetism right hand rule he says take your right hand and curl it in such a way that the curved fingers are in the direction of your travel and then the thumb represents the positive direction so in our example since i'm traveling this way if i take my right hand and if i curl my fingers in that direction my thumb will point downwards and so this means according to my right hand thumb rule downward direction is positive for this loop so i2 would be positive i1 would be negative so this is now the correct application of ampere's law for this case why don't you quickly try one so let's say let's let's take another loop which is over here i'm going to take a rectangular loop because shape doesn't matter so let's say we take a rectangular loop somewhat like this and this time let's say we walk this way and calculate b dot dl can you pause the video and think about what will be the closed loop integral of b dot dl over here is going to be mu naught times something what will that be can you pause and think about this all right so the first step would be to dip this in a soap solution and attach a flat surface to it and now the current that penetrates through this surface will be our enclosed surface and you may be wondering why should we attach a surface we'll talk a little bit about that towards the end of the video but the current that penetrates is i2 and i3 and now we need to know which direction is positive for that we use our right hand thumb rule in this case we are moving in this direction and so if i use my right hand now let me keep it over here somewhere okay if i move the right hand now now notice the thumb points upwards so upwards is my positive so this is now positive so what i end up getting is plus i3 so i3 becomes my positive current i2 becomes my negative current so minus i2 and i1 is not in the picture because i1 is not penetrating to that surface and there you go this is how we use ampere's circular law now before we wind up i want to talk about some important characteristics of this law first of all this law can be derived from a bo savar law and you can derive bo sawar law from this law so they're both equivalent and we use whichever one is more convenient in our given situations in some in sometimes when things are very symmetric we go for ampere's circular law because it makes our calculations simpler again something we'll see in future videos secondly on the right hand side we only consider currents that are enclosed by the loop right so for in this example only i1 and i2 but not i3 but what about the magnetic field on the left hand side is that only due to the enclosed currents no that is the total magnetic field so the magnetic field which we are considering is due to all the currents enclosed and non-enclosed so how does that work why is it on one side we have total field but on the other side only the enclosed one matters well that's because again this is like mathematically we will not get into the details but what happens is what this means is that the contribution of bdl provided by the non-enclosed currents they add up and become zero so they end up be giving zero contributions so you can imagine as you walk around this loop when you in some cases the contribution of this is positive uh in some places the contribution is negative and so the total contribution of them is zero this is very similar to what we saw in gauss's law the char uh you know the total electric flux only depends upon the charges that are enclosed by the surface over there right the charges which are outside they will contribute to zero flux same thing very similar happening over here finally when it comes to the surface we said imagine a soap solution attached to it right but here's the thing about soap solutions you don't have to have them flat if you blow on a soap solution you end up having an open surface attached to our loop so the loop becoming the opening to that surface we can also attach such surfaces over here so imagine somebody is blowing from the top what will happen to that surface we might get something like this and now the iron closed becomes the current that is penetrating through this surface and so you can attach any open surface you want to your loop flat being the simplest one but you will always end up with the same value of iron closed now finally finally you may ask why should we even attach a surface i mean what's this business of attaching surface can't i just look at this loop and tell what is the enclosed current sure in most simple situations yes but in general we can have very complex situations in which it may not be so obvious and i'll not dig too much into it we will look at one such situation sometime in the future where we deal with calculating the magnetic field when there is a capacitor things will become much more interesting over there and this attachment of surface will make a lot more sense over there but for now it's completely fine in most of our examples it's fine if you don't attach it but ampere suggests you attach a surface to our loop and find the current that is punching through that surface that becomes our ion closed