Lecture 23 Ampere's Law

all right welcome back we've been discussing Gauss's law so last time we started Gauss's law today we're going to continue Gauss's law and I'll show you an example of Gauss's law and then we'll look at what's the magnetic analogue of Gauss's law and also get into amperes law amperes law is another way to look at the current and magnetic field relationships so we had discussed last time Gauss's law Gauss's law is this statement it says electric flux through a closed surface equals 1 over epsilon knot kinds of some of the charges that are enclosed an electric flux so that the little circle on the integral means a closed surface okay and electric flux means this if I have some geometry of electric field going in a particular direction electric flux says okay if I pick an area through which I'm going to calculate the electric flux so then I the flux it counts okay the part that gives you flux is the part of the electric field that's perpendicular to that area right so if you have a case where the flux lines are hitting smack on the area you're calculating it's easy alright if you have a case where they're skewed away from each other then what counts is the part that the perpendicular component so if I think of these electric field lines coming into this slanted area I can break up the electric field into pieces that are parallel and perpendicular to this little area now am I only going to have a flat area no i might have a really complicated area but you know how calculus goes you take your complicated area you tile it with boxes and in every little box you're going to say that it's flat there and then later you'll take the calculus limit of very small boxes so thanks for now of a very small box and it's got a particular area to the box I've got an electric field coming in but I'm going to think of the box being tilted so what do I do in that case well it's like a net catching fish all right only the fish that swims toward the net get caught fish that swim this direction don't get caught so I think of the electric field coming in and I'm going to break up the electric field into components that are parallel and perpendicular to this area so there's a piece of this electric field that's perpendicular to the area those are like the fish that goes the net and get caught there's a piece of this electric field that's parallel to the area that component doesn't contribute to the flux so only the components of the electric field that are perpendicular to this area so you can either think of it there's two ways to think of that mathematically the way it's written down up there is y dot N Da where n is the normal to that area so here's that area again here's the electric field and I imagine a normal to that plane I'm going to take this electric field and dot it into the normal that's the part that counts okay so that's literally what that means you can do it a different way though okay which and we discussed this last time you can think about well what I really need is these guys to be smack heading into each other so rather than breaking up the electric field lines and the components perpendicular and parallel I could take the area and project it back onto the perpendicular component does that make sense I could take this tilted area and say well I just carry about the part that's perpendicular to the electric field and I could take that projection back so either way you like to think about it is correct geometrically and physically the way the math is written down is with the perspective of taking the electric field and dotting it into the normal you have any questions about that okay either way it's the component of the net that's perpendicular to the flow of fish that catches fish nets that are laid sideways don't catch fish don't know if you've been fishing lately but standard principle fishing make sure the nut is perpendicular to fish flow all right any questions about what the flux means okay so Matt's Gauss's law we were working it out last time for a point chart so this is correct all the time for any shape and for any number of charges in there they don't even have to be point charges they could be any shape of charges you like we were working it out specifically for the point charge case just to illustrate what goes on so in the point charge case so let me think of a point charge there in the middle and the electric field that radiates out of a point charge is kind of like a dandelion or starburst and it has this shape and this this magnitude to it one over four PI epsilon naught times Q over R squared times R hat where our hats pointing away from the point charge so in the case of the point charge if we wanted to think of applying Gauss's law need to find some net to put around it okay some area to enclose it by and in this geometry it makes sense to enclose the biosphere you can do any shape you like for Gauss's law but typically in a simple physical situation you want to find what's called a Gaussian box a Gaussian box that encloses your charge in the way that's going to make your life the easiest so we enclose this guy with a sphere I don't think about a sphere surrounding a point charge and the sphere is centered on the point charge so I've got this excellent geometric situation to where the electric field lines come straight out of the point charge and everywhere they poke this sphere they're automatically perpendicular to the sphere they poke up the top and they're automatically perpendicular to that sphere and so on so that's a great geometry to be calculating in this case so in that case then I think about well what's the magnitude of the electric field on this sphere the magnitude all over that sphere is constant because the sphere is a constant distance away from from the point charge so the magnitude of that is 1 over 4 PI epsilon naught times Q over R squared and I pull that out of the integral ok because as I integrate over the surface area that magnitudes not changing so I pull that out of the integral and now the interval I'm left with is the surface area integral and what's the surface area of a sphere 4 PI R squared so that's what goes here the 4 PI's cancel the R squared cancel and I get Q over epsilon naught ok and we're thinking of this for a particular sphere but it works for any sized sphere would work for the small sphere the small the smaller strip or the larger sphere so the reason it works is because as the field falls off like 1 over R squared the surface area grows like r-squared and they exactly cancel and that got a star then we thought of expanding this to to a general case so rather than just surrounding this with a simple sphere I'd like to be able to put it with any shape I like we said all right think of breaking up the sphere into segments like orange wedges or something like that and in any segment the contribution so I have to think all along this blue segment here the contribution to this shell is the same as the flux through that shell is the same through that part of the show so forth all right just in the same way we made an analogy between flashlight shining on a piece of paper so if I have a flashlight shining on a piece of paper let me put the piece of paper where it's going to catch all the photon it catches all the photons right so if the piece of paper is close to the flashlight I get a small circle of light but it's bright the piece of paper is far from the flashlight I get a large circle of light but it's dim catching the same number of photons same thing along here it's got the same flux through any of those shelves and then I wanted to think of surrounding now this charge with any shape I like by following different spheres in different places so we draw a line around this now and I think of following the outer sphere okay and then drop back into an inner sphere and follow that sphere and then drop back to another sphere and follow that one and keeping in mind that I find on up as long as I stay on one spherical shell it's got that constant contribution of the electric field on that shell and then when I drop back down to another spherical shell I'm going to do it with an area I'll take that slice of area and make sure that it's parallel to the electric field lines and how much fish there's a net like this catch nothing right so that little segment that drops down to another shell catches no flux okay and then I get on to another shell and I dropped on to another shell in a way that catches no flux so then the flux through this shape is exactly the same as the flux through that sphere okay so the flux through the outer surface is always the same no matter what and now I can in a standard calculus way of reasoning to make any complicated shape I like by thinking of tiling it into those little segments and then take the limit that those segments become finer and finer and finer until I recover the smooth shape and it all works out so now this works for any smooth shape I can take any shape I like for Gauss's law you can make it a funny shaped bag you can make it a box you can make it a prism a sphere whatever you like do you have any questions about how that worked okay all right let's apply it to a completely different geometry okay so Gauss's law works in any geometry okay we just proved that it works in any geometry now I want to think about what happens if I have a K we're not just using point charges so it also works for any shape of charge you put in there as well so let me show you how it works out for a plane so let me have an infinite plane of charge okay with a surface charge density Sigma equals Q over a so the way this goes is that Sigma is the charge density it's charge per unit area and it's the same all over the plane so if I think about this plane and then I take any particular piece of the plane and I take an area of it take the area and I could think of what's the charge on that little piece of area take that charge divided by the area that's Sigma go anywhere else on the plane take it any size shape area you like count up the charge in that area charge divided by area is the same so that's Sigma and so given that given infinite plane with uniform surface charge though to the Q per a I want to find the electric field due to the plane have you seen this problem before we did this before right so we did this in a different context and we derived it from thinking about what the point charge contributions were like and then we integrated it and found the total electric field so you've already done this problem once but now we're going to do it in the Gauss's law way and to illustrate Gauss's law so I have have a plane due to the geometry of the plane the electric field since this infinite plane the electric field has to simply point away from the plane but it's positively charged it points away it's negatively charged it points towards so I like to think of the positive charge case first so think of this positive plane electric field just points away anywhere you are it just always points away from the plane and I want to use Gauss's law now to find the magnitude of the electric field we know the electric field is going to have the same magnitude everywhere okay we've calculated that before but in fact we can we can also use Gauss's law to show that that's the case so let's use Gauss's law now the key to Gauss's law is to choose what's called your Gaussian box wisely all right the flux you can calculate this flux and it will work for any shape you choose so given the geometry of your problem choose a geometry for your box that makes your life easier okay so you get to choose the box make your life easier in this case I'm going to choose a cylinder that's oriented perpendicular to the plane okay I could have chosen a box that could have chosen a cube you know whatever but the key is that I want to choose it in the same way okay in the same way to build up this shape I thought of taking pieces of the area there were perpendicular to the electric field and other pieces of the area that were parallel because the perpendicular ones are easy to calculate and the parallel ones give me zero contribution when when coming up in your gaussian box use the same principle find something so that you get pieces of the area that are perpendicular to the electric field and it easy to calculate and other pieces of the area that are parallel and give you no contribution so here I did a cylinder and so I have then this cylinder and the face of it is the perpendicular part that'll be easy to calculate because it's perpendicular and then the part of the cylinder that's the smooth curvy part is parallel to the electric field parallel nuts catch no fish okay so there's no flux through that part that'll be easy to calculate the only other thing I need to figure out well what does the Q inside mean okay it means the enclosed charge the charge that's enclosed in my box and in this geometry the charge that's enclosed in my box is I just need to count up based on the charge density I just need to count up the area in there and that will give me the charge enclosed I don't care about what's outside that's the other beauty of Gauss's law who cares what's outside okay just what's inside that matters you have any questions about the setup okay so always choose it in a way that works easiest for you so here the electric field is constant and perpendicular to the end area which is a so when I take the flux then through this box on the right hand side okay I'm getting a contribution out here that's the electric field times the surface area so I get e times a but then I have the other one to worry about as well right I have to count up over the whole box so there's this other face which also has end area a so e times a over there plus e times a over there gives me e times 2a or two times e a however you want to think about it then the contribution from the parallel parts is zero so this is the flux the electric flux through that gaussian box e times 2 times a you have any questions about how I got the flux alright so now I just need to think about well what's the charge inside there's always this one of our epsilon naught and now I need to charge inside so how do I get the charge inside that's going to have units of charge okay so how do I go from charge density surface charge density which is Q over a to Q I need to multiply by a ok so 2 over a times a where this area a is the area the cross-sectional area of that cylinder so taking that together I get Sigma times a and that's the charge inside does that make sense that that's Sigma times a guess it all right ok all right now that a is are going to cancel so cancel off the eighth and I can solve for the electric field and then electric field is Sigma over 2 epsilon-not electric field of a of a plane do you have any questions about how that worked ok all right now we did it for a simple geometry ok it'll work for any geometry you choose right you could choose any shape box I could choose rather than choosing a plane I could choose another charge configuration I could choose you know 15 point charge is something like that but the key is if you have a simple geometry of charges see if you can find a box that makes your life easy we're part of it's going to have a constant perpendicular contribution to the flux and part of it will have zero contribution to the flux so so for example if I go to a case let's say that I have a line charge take this apart for now ok let's say that I have a line charge alright and let lets us see like an infinite line of charge and the charge density on a line you would count as charge per unit length ok we've got different links different units from up there but a uniform charge density all on an infinite wire ok what's the shape of the electric field coming off this guy think of a positive charge yeah I just spread now ok so just come straight off the wire in older straight off the wire in all directions kind of like a pipe cleaner think of a pipe cleaner right so what shape box would you want to choose in this case cylinder okay and how do you want to orient the cylinder yes like if this is my coke can which is cylinder not going to do it this way I'm going to orient it this way and enclose it that way okay so think of enclosing this guy with a big you know soda pop can and then I'll get contributions where now the end phases have how much flux on the end phases zero right because the electric fields coming straight out and it doesn't pierce through that part so anywhere the electric field pierces the can though I get a contribution and so now all along the curved part I'll get a contribution and it'll be constant if I choose that cylinder I Center it on there correctly so in in simple geometries like that you're going to be able to find simple galaxies and boxes that will make your life easier any questions yes well okay so that's a good question I so I could view it on the whole wire alright the physics equation is correct and will always work so what will happen is if I take an infinite wire and then I make my brows clean box infinite it'll be difficult to calculate because what'll happen is that I'll get an infinite flux and then I won't really know what the proportionality constant is anymore right once infinity happens it's hard to track so it's better to take a finite box yeah good question it'll work just probably won't won't help you calculate the electric field yeah good point other questions okay all right so okay so that was Gauss's law for charge so this one related this related the flux of the electric field okay to the enclosed charge now we could write down the same kind of integral for a magnetic field right this integral was just about taking electric field lines and seeing where they poked through and pierced the area that you're that you're studying okay we could think about magnetic flux as well we could think about well what if I take an enclosed surface and I calculate what's the magnetic flux poking through that thing so so here's what you would expect all right you expect okay maybe you can generalize this okay this is the case for charges what if I tried to generalize it to the magnetic field case I can write down the left hand side for for the magnetic field the only question is what kind of charge should I put inside right so it turns out that if I try to write down Gauss's law for magnetism it's much simpler right I can think about the same kind of geometric consider considerations that I'll take a flux of the magnetic field and you do it in the exact same way you calculated flux of an electric field you just look for the magnetic field lines try to put your area perpendicular to it if you can't mean you're going to have to take some projections all right to get the perpendicular component that's your flux it turns out though that when I take a closed surface it's a little surface on the integral symbol sorry the circle on the integral symbol means closed surface when I take a closed surface for magnetic field I always get zero flux every time no one's ever found a contradiction to this so how can that be so think of a magnet to a magnetic think of the magnetic field coming out of a bar magnet so mark bar magnet has a plus a North Pole and a South Pole the magnetic field lines come out of the North Pole and they wrap back around and come back into the South Pole right so now I'll surround that guy with a with essentially a Gaussian box and what you'll find is that you'll find that a magnetic field lines are poking out on one side but they move back around and peer speck on the other side so you always get this net zero contribution it's very much like that situation of thinking of a water fountain where water's flowing in and water is flowing out okay so you never catch anything with Gauss's law for magnetism no fish okay zero fish always the fish swim in the fish to a mount so no magnetic monopoles is what that means when I have a bar magnet a bar magnet is what we call a magnetic dipole it has a North Pole and a South Pole wouldn't it be kind of a cool Universal if I could break that apart and have hand you the North Pole and I keep the South Pole okay if we could do that that would be kind of fun we'd call that a magnetic monopole no one's found it yet okay people look they're clearly not common because no one's found one I have a friend dr. Bob my friend dr. Bob who his he did his PhD at Caltech in physics and his dissertation was looking for a magnetic monopole and you might think why bother no one's found one yet okay there are some interesting theoretical physics reasons to expect that there might be one somewhere in the universe okay so he went looking for the thing with really sophisticated techniques and his PhD was taken awhile because he didn't find one and and at some point he decided okay now I really don't want to find one because you know if I find one now it's going to take me forever to write up my dissertation and so anyway he is true he never did find that magnetic monopole but people keep looking maybe someday we'll find one and then we'll update that equation to have a magnetic monopole contribution do you have any questions about that okay all right now next up something called amperes law ok ampere mmm-hmm do you know what it answers right okay amps named after ampere when you see current expressed in amp it's after this guy so amperes law is going to involve some current and it's going to involve magnetic field let me first remind you of the biot-savart law and then we'll build up to what's called amperes law okay so the biot-savart law was the following geometry right the abuse of our law was about what happens when have a current carrying wire so here's current and what does the magnetic field look like around a current carrying wire it's this geometry okay so that if you have current running this direction the magnetic field circles this way and the analogy we made in class was if you want to remember which way this goes think of your think of a clock and think of how you feel about your alarm clock in the morning and think of stabbing okay the direction of your knife is the direction of the current and the direction of the clock space is the direction of the magnetic field or you can use the right-hand rule or ain't your thumb along the current and the magnetic fields curls around it there is the equation okay and what we found before was that for a current carrying wire we said that the magnetic field is nu naught over 4 pi times il over R times the square root of R squared plus L over 2 squared all that in the theta hat direction theta hat just means because it's a geometry that lends itself to cylindrical coordinates will work in cylindrical coordinates okay where Stata hat is this direction and R is just the distance from the wire could be here could be there but it's always that the shortest distance to the wire and now I want to think about very close to the wire so let's get really close to the wire R is going to be small much much smaller than L and you remember how these approximations go when you're doing approximations where I'm going to think of are much much smaller than L I turn it into money and think about that so I think about L being billions of dollars in our being pennies and then I look for places where those terms are added together in the equations if they're multiplied I can't approximate okay because a billion dollars times point zero zero zero one you would care if someone did that's your bank account right all right but just think about things you wouldn't really care if someone did your bank account if you have a billion dollars and you know there's a penny zero by the bank you don't really care so think of billions of dollars plus pennies all right and you're Bill Gates it's not worth your time to pick up the pennies so you cross off the R and you're left with that square root of R squared plus L over two squared is approximately equal to L over two so this gives you in the equation U naught over 4 pi times il over R remember I didn't get to approximate the R because it's multiplied not added can only approximate things that are added and then I have times L over two and all together that gave me Mew not over 4 PI times 2i over R theta hat okay and this is close to the wire so that's basically this geometry is what that's telling you seda hat gives you that direction now watch the slide closely ok this slide was about very close to the wire R is small compared to L okay the new thing on this slide all the same slide I just changed the words to be about a very long wire all right R is much smaller than L is the same thing as making the wire very long so for a very long wire I get the same thing but it's that the magnetic field is Mew naught over 4 pi times 2i over R in the theta hat direction oh that's from a review from lecture 13 so now think of this very long wire okay and I'm going to have the very long wire oriented current is coming out towards you the currents coming out towards you and the magnetic field lines go all the magnetic field lines you see are circles okay you see the cross-section here as circles so this is viewed from the end the red dot means currents coming out of the board towards you and the pattern of the magnetic field is cylindrical and I want to take what's called a line integral okay so I'd like to take a line integral of the magnetic field along one of these circles so you've seen a line integral before where have you seen a line integral before electric field right you used a line integral before to tell you the potential difference between two points is - the electric field dotted into DL and the way you did that was that you know the way you took any line integral so if you were thinking about potential difference you would say okay take a step that's the DL dotted into the electric field where you are take the contribution put it in your pocket and step again multiply the distance you step two times the parallel electric field put that in your pocket and keep going that's a line integral here we're going to do the same thing for magnetic field so I have this magnetic field line integral the circle on the integral means make a closed loop so come back to where you started and tonight that this this line up front is is a curved so we can think of this as being like along a really big circle where there's some flagpole in the next building that has current running up okay but a really tall flagpole so again to take the line integral I'm going to think about B dot DL so DL means take a step DL okay take that links multiply the links I just stepped by the parallel field component put that in my pocket step again take that DL length times the parallel field component put it in my pocket and add it all the way up and as I walk around this circle you might imagine okay magnetic fields going in this direction if I walk along that circle and take that line integral scooping up magnetic field times the distance I walk at every step I get a contribution okay so let's add that up I can take the equation here for magnetic field that's going to be line integral of MU naught over 4 pi times 2i over our theta hat dotted into DL okay Mew naught over 4 pi is constant I can pull that out too constant I can pull that out of the integral current should I pull the current out of the integral current constant just depends how I set it up right so if I set up this situation to be steady state constant current situation then it's constant and I'll pull it out of integral just like you were all saying our is our going to be constant okay now it depends how I set it up right so if I set it up so that I'm going to take this line and go on a circle that's centered on the wire then R is constant okay and again it's like Gauss's law I get to choose where I take this line integral so I'm going to choose it so that R is constant and I pull our out some left with a line integral of theta dot DL okay how should we think of this line integral of theta dotted into DL first of all what are the units on that it's the unit still not integral units of Newton's coulombs so what what you just you expect here there's at least a linked right okay because there's a DL sitting and this is a link what do I think about four units on an angle okay well we're writing these equations the angles pop up in in radians and that's unitless okay so we don't we don't count that as unit so this guy is going to be in units of length right and basically at this point to take that line integral I think of the same thing I take a step that's a DL okay I take the distance I just walked got it into theta hat which direction is say a hat point well it turns out that theta hat is always tangent to the circle just like DL is always tangent to the circle okay so theta has tangent to the circle because it's cylindrical coordinate DL is tangent to the circle because I set it up so that my line undergoes a circle okay so every step I take you know I take DL times what well times the magnitude of theta hat that's parallel to me theta hat the unit vector so I get a factor of one there all right so I can I can think then about how DL itself is Rd theta you've seen this be for how do I know how far I walked well I take the radius times the angle ice obtained it the reason that works is because if I walk around the entire circle I get two PI R back right so if I take integral say two hats dot DL all around the entire circle I'm going to get back at 2 pi R right if you have any questions about how to get the 2 pi R just walk it around the circle and pick it up the links all the way around so here then I have mu naught over 4 pi times 2i over R times 2 pi R okay that's what this line integral of magnetic fields around the circle ends up being and two times two times pi councils of 4pi the RS cancel on a must of you not I so there's amperes law the line integral of the magnetic field around a closed loop equals mu naught I okay and what we really mean for this is the enclosed current so it's the current enclosed in your loop you have any questions so far okay all right so to tell you a little bit more about what we mean by it should have labeled that I enclosed it means in closed loop so if I think of current see sometimes I show you visual aids all right let's show you just a segment of wire but if I have current running through the wire in any real physical situation if current runs through the wire got to be connected somewhere so there's always a connection somewhere in a real circuit that brings it back so now when I think about well is the is a circle I'm making going to enclose this or not I'm gonna look for a loop so let's say ah yes I'll unplug that all right so let's say that I take this loop alright and and okay so I have a loop pretend the loop stays the loop going to take the loop and put it here it's pretty clear that it's enclosing the wire right but if I could take the same loop and pull it off the wire not really right because in a real physical situation this wire cannot end got its current right the current has to make a complete circle so or complete circuit so this loop if it's stuck on the wire it's stuck on the wire it's stuck on the wire the whole time so it turns out I can deform the loop in the same way we did with Gauss's law I could deform the shape I can deform this loop I can twist it back and forth I can move it along the wire and then closed current is always the same right no matter the geometry that I have here all right so I'll show you that math okay so this is to remind you though it's going to look very much like what we did for Gauss's law so Gauss's law for the point charge okay there we go I want to remind you of the previous slide you saw okay so the left hand side here is going to be what we did earlier in lecture for Gauss's law the right hand side is going to be extending this now two amperes law okay so basically we already saw that in the gallop of law case for a point charge it worked for any sized sphere okay because the are canceled something similar is going to happen for the magnetic field of a wire all right so in the same way if I think of amperes law for the long wire there was this piece that had a cancellation in it the ars canceled all right so remember how back in the Gauss's law case the ars cancels when it came time to calculate the flux and I thought about any sphere so in the Gauss's law case it was a it was a spherical area over here it's a line integral but there's something similar going on so on each sphere the contribution of the flux was the same because if I think about a small sphere close to the point charge it's a high electric field with a small area high electric field alright controlled by one over R squared small surface area controlled by four PI R squared or if I think of going to a large sphere right now it's a weaker electric field that fell off like 1 over R squared but it's distributed over a larger surface area so catch the same number of fish okay so it works for it works for any size same thing here I take this line integral around this circle if I take a line under grow around a big circle the magnetic field is small the magnetic field a small bite by 1 over R the distance from it from the wire but I'll walk around a larger circle second same amount as if I'd walk throughout a small circle which had high magnetic field but a small circumference so the RS cancel I always get the same contribution because the circumference goes like R but the field falls off like 1 over R ok all right that gets a star the Stars I mean it's an important slide ok now I can also think in the same way we did for Gauss's law we thought about segments right the Gauss's law case we thought about the electric fields coming out here in this wedge as like light shining out of a flashlight okay and the flux through here was the same as the flux through here is the same as the flux through there similarly in any segment of the circles okay so take this wedge shape here all along this segment the contribution in any circle is the same so the contribution to the line integral here is the same as the line integral I would get for out there as long as I'm subtending the same angle you have any questions about that ok all right so in the same way that for Gauss's law I thought of okay let's divide this up into wedges and I'll surround the charged with any shape by following different spheres in different places I can do the same thing over here for for amperes law so and in in both cases I'll think about a boundary right so here I was thinking about boundaries that were on spherical shells that I stay on the spherical shell and then have an area that parallel to the electric field lines to drop back down from the most fearful shell similarly here let me think of walking along a circle and then if I want to change my radius I'll drop straight I'll just hop straight down to the next circle okay in a way and I'll hop down in a way that I'm not picking up any magnetic field lines as I do it as I go toward or away from the wire there's no contribution to the magnetic field that's parallel to my path so contribution to integral B dot DL is zero along these lines and then I'll walk along another circle if I want to change my radius I drop straight down to another Circle and then follow that radius by doing that I can build up a line integral okay it's any shape I want right it's the same kind of of calculus concept if I want to make any complicated shape I want I just think of making this gradation of which sphere I'm on finer and finer and finer and then the limit of very small changes of spheres I can get any smooth path I like but it's it's the same the same concept that we use for the Gauss's law case okay Gauss's law here amperes law here you have any questions about that generalization to the shape okay all right so it works for any smooth path also it's not just that I need to think of this loop of wire needle loop of wire it's not just that I need to think of this loop of wire as staying flat in the plane I mean what I've drawn up there on the board it's flat right I started with a circle and then I things for you and they're all kind of stayed in the plane but I could think of going out of the plane as well it's not a problem okay still works for amperes law so the reason it's still going to work is I could again think of walking perfectly walking parallel to the wire direction any time I walk parallel to the wire direction to get myself on another circle over here but well parallel to the wire direction there's no magnetic field parallel to my path integral B dot DL is zero along that segment okay so I can deform it this way or that way and I still get the same thing and I can move it anywhere I like right I can move move the loop I'm considering anywhere I like along the wire right twisted however I like and it's clear what's the current enclosed and what's not the current enclosed if you have any questions about amperes law yeah okay so good question all right so I did show you the specific case where the magnetic field will be used for the magnetic field the biot-savart law which assumed an infinitely long wire okay so yeah you're actually asking an excellent question this law here okay which I demonstrated to work for the views of our case of an infinitely long wire it also works for smaller wires so so this this line integral of B dot DL equals mu naught times the sum of the enclosed current always works for exactly the magnetic field right so even if I'm in a non infinite wire case which is the case in the lab right and no infinite wires in the lab so I have some strange geometry all right and I think of this circuit you know every piece of that wire is going to contribute something to the magnetic field over here it all works it's pretty pretty impressive all right but it all works so every little bit of a complicated geometry of the circuit contribute something to the magnetic field I measure and yet when I go around that line integral somehow it knows Sam this was the current going through it pretty cool and then as I you know as I change the shape of the wire it'll still work okay so yes so it works for any closed loop and you care about the current that's enclosed so you could have multiple wires inside and they'll still work you could have somewhere is inside and somewhere is outside and it only count the wires that are inside okay so reason the wires on the outside don't contribute is because if I think about what kind of shape magnetic field is the wire on the outside making I think well alright a wire on the outside of my loop so my loops here and have a wire over here giving me a magnetic field they just kind of come straight through and they'll give me some positive contributions and some negative contributions it'll just end up canceling if the wire is not going through the loop any questions other questions good question alright so we have built up a lot of what's called Maxwell's equations so Maxwell's equations tell us everything we need to know about how electric fields and magnetic fields and charges and magnetism all work and it's all contained in a set of four equations known as Maxwell's equations I've got the caveat up here that it's incomplete right now here's what we've learned so far we discussed today Gauss's law ok Gauss's law for charges is that the flux enclosed in a closed surface area electric flux enclosed in a closed surface area equals 1 over epsilon naught times the sum of the charges enclosed the magnetic extension of that says that the magnetic flux through any closed surface is zero because no one's found a magnetic monopole despite my friend dr. Bob trying really hard then there are these other equations we just got to this part which is called amperes law we saw that the line integral of the magnetic field around a closed a closed line path is Mew naught times the sum of the charges enclosed and we've also talked about how the line integral of the electric field around a closed path is zero in the steady state situation remember we made that analogy between potential differences like height and if I go around any closed loop I better come back to the same height so these equations all are correct in the steady state case steady state means the current that I've got is the same the whole time I'm measuring the system so steady current and it means the charges aren't moving around okay if I go to the case we're seeing they're changing in time some of these equations are going to get updated and that's what we'll be doing the next couple of chapters is talking about the non steady-state situation where I have fields that change with time K could be magnetic field could be electric field in those cases I'm going to have some updates to these equations alright so one of the updates is that we're going to update the line integral of the electric field around a closed path if I have a changing magnetic field around okay of a changing magnetic field around we're going to get a new term there that's going to well update that physics okay and it's and how would I get a changing magnetic field one of the ways to get a changing magnetic fields is to take a magnet and wave it around new your experiment changing magnetic field okay that's one way to do it that will update this equation all right that'll be in Chapter three and in chapter 24 we'll see that we should add a term here as well in case we have a changing electric field okay how would I make it an electric field that changes in time well whatever circuit you're measuring I'll take a point charge and wave it around near your circuit and now there's a changing electric field acting on your circuit and that will be chapter 24 once we have all that together now it's the full story okay just giving you a preview of what's coming in the next two chapters once we add in the time-dependent cases we've covered everything right and then that'll be it that'll be Maxwell's equations now this might look like a lot of math to you but this is remarkably simple for only four equations this is remarkably simple for something that describes so many different phenomena for something that ascribes for example photons are in here okay so the fact that that light can just travel through space okay that's in their lights in there these equations in table all any kind of telecommunications you like whether it's communications over a wire or whether it's communications in the airwaves okay it enables radio and enables your cell phones if you've got any kind of electronics device with a chip in it it's enabling all that so just a small set of equations right background can you imagine being a James Clerk Maxwell and figuring this stuff out right so so Maxwell I'm actually you'll see this is called the an for Maxwell law Maxwell is the one who noticed that this term should be there and once he did that he was able to unify all this stuff and say well because I've put all these things together I can now explain light as well as magnetism as well as electricity etc etc can you possibly have foreseen the impact on society of his four equations I don't think anybody could have foreseen the impact right these four equations because Maxwell figured that term out its unified light with magnetism with electricity and laid the foundation for global communications so never say physics didn't do something for you gave you cell phones right and the internet and all that good stuff so anyway Maxwell's equations the full story is coming up in chapters 23 and 24 and we're done for today

Channel: Prof. Carlson

Views: 1,404

Rating: 5 out of 5

Keywords: Physics, Phys 272, Matter and Interactions, Electricity, Magnetism, Ampere, Gauss, Maxwell, Equations

Id: xEYMxHOvTcU

Length: 44min 16sec (2656 seconds)

Published: Sun Jul 30 2017