Lecture 26 Maxwell Equations - The Full Story

alright guys welcome back and as you know as I know you know we have a final exam coming up right and a good way to practice is that I put a practice exam up on Blackboard Learn so there's a practice exam already up there the solutions to the practice exam will be posted tonight at midnight please please please before you look at the solution set give yourself every opportunity to make this into a good study experience so I highly recommend that you take the practice exam and you just give yourself the practice exam like regular testing conditions okay carve out a two-hour block where nobody's going to interrupt you shut the door tell your roommate not to disturb you get yourself some pizza so you're nice and set up and you don't need to be interrupted and for two solid hours just take that practice exam okay for real and then when you know what your answers really really really are after that check our solution set then you will know then you've used it as a diagnostic tool to tell you which pieces of the material you need to study more and which pieces you're pretty good on okay now so you shouldn't also study those all right but it gives you an opportunity to know which pieces you really need to focus on and then you can go into WebAssign and find those practice problems all right and practice the ones that you need extra help with you may even spot a pattern right oftentimes what we find if a student misses say five questions on the exam oftentimes it will find is that they're all kind of in the same category sometimes they're all about the same set of material sometimes there's something else that's biting you like maybe you're not drawing diagrams carefully so maybe sometimes you might figure out that oh if I had just drawn my diagrams carefully I would have gotten those five questions correct or maybe you'll find out that oh what what got me on those five problems is that I wasn't thinking very carefully about the vectors which again usually goes back to drawing your diagrams really carefully so take that practice exam okay give yourself a full two hours take it see how you personally do and then use it as a diagnostic to help you know how to spend your study time then you'll make the best use your study time the right way to study at this point okay is the book I'm assuming you're already read and if at this point in the class you haven't read those chapters yet the right way to recover from that is to just dive in and do problems okay solving the problems that's what's going to help you solve problems on the final exam yeah all right so study hard and good luck remember the better you do the better we feel about ourselves as teachers so we want you to do well because we want to have taught you well and so so do well okay study hard good luck okay so last time in class we discussed Faraday's law we learned all about Faraday's law this time we're going to get to the entire complete set of Maxwell's equations so we will finish out everything that's known about electricity and magnetism well not everything that's known but we'll we will finish out the equations that govern it okay working out the consequences is actually very hard but we'll finish out the equations that govern it and we'll have the complete set of equations that govern everything having to do with electricity and magnetism and if we have time for it I'll show you the wave solution of those guys so here's what we've had when we were talking about the time independent pieces of Maxwell's equations so what I mean by time independent is the steady state case okay so this was the case where we had charges and we could have charges moving as long as they were moving at constant velocity which we report in Maxwell's equations as a constant current so in the steady state case these are Maxwell's equations what we did last week though was we said okay these places where there's a big fat zero we're going to update okay last week in chapter 23 we updated this piece and said well look a changing magnetic flux with time can cause a curling electric field that was just that was Faraday's law and so we updated this piece that's not a steady state piece right that's got a time dependence to it so what it told us was that if there's a magnetic flux and that magnetic flux changes with time it causes electric fields to curl around itself what we're going to see today is that this part here amperes law also needs to be updated with a time-dependent piece so amperes law told us that I can get a curling magnetic field if I have a current so a current running causes a magnetic field to curl around but by analogy with Faraday's law you might think well look at changing magnetic flux just to changing magnetic flux cause the curling electric field may be a changing electric flux causes a curling magnetic field okay and this is exactly the insight that James Clerk Maxwell had so this was called amperes law but if you add in that time-dependent piece that we'll study today this was this was Maxwell's big idea that that now that's the complete set of equations for electricity and magnetism so this piece was added by James Clerk Maxwell and everyone agrees this was a great leap of genius to do this he was thinking about the symmetry of these things actually he also reports that he was thinking very deeply about the concept of Trinity so he was having deeply spiritual thoughts while he was trying to to solve these equations and you can't it's impossible to overemphasize the importance of having put this term in okay so once Maxwell did that that's the complete description of electricity and magnetism this is if you have this stuff in place you can predict all the other things you need to know so for example if you just took this piece of lecture and you said okay I got it I got I got I wrote these equations down and now pop into your time machine because I know you keep one in your garage right so hop in your time machine and go back in time 300 years you would change the world right you would have in your power to do things like oh my goodness based on this I can create I can create a Telegraph's alright I can set up a whole Telegraph system of telegrams and this is what these equations right here what enabled us this is why we as a world laid down that big ol cable in the Atlantic Ocean connecting two continents and allowing Telegraph communications that was enabled by Maxwell's equations this is what enables the telephone okay so communications over whether the over wires or without wires so this enables your cell phone technology it enables everything from the electronic circuits that are inside the little chips that you can't see all right to the wireless communications that let your phone connects with the cell phone towers it also enables light so the fact that light is propagating in this room is controlled by these equations radio okay so radio communications are enabled by having put all these pieces together satellite communications I mean our entire you know modern society basically is founded on these equations up here you can't overemphasize it could Maxwell possibly have foreseen all of the societal implications of figuring this out no way all right did he know he was going to make a global society by having done that nope so who knows all right if you decide to you know switch into into basic sciences and do something pretty amazing like that who knows what the implications would be for your work in the future as well actually I should also tell you since you're sitting right here in Purdue all right there's a once a long long long time ago there was a man here named Edward Purcell who got his Bachelor of Science degree in electrical engineering at Purdue and he went on to get a Nobel Prize alright so sitting amongst you right here maybe there's another Nobel Prize winner right now who's going to do something else amazing like that all right so adding time to amperes law what do we need to do to do that we had amperes law last week in the previous weeks but it didn't have explicit time dependent other than the current the current was assumed to be steady state all right so now let's think about what happens if we think of well let me set up a conundrum first so first I'll set up a paradox and then let me show you how max will solve the paradox so here's an amperes law which as we know it's incomplete it doesn't have explicit time dependence other than that steady state current but if things are changing with time we're going to get a contradiction so the law stated in this way gives a paradox let's say that we apply this to a capacitor all right so what does equation means what amperes law means is that if there's a current running there's a magnetic field curling around it okay so current running magnetic field curling around it current running magnetic field curls around it what happens when we get to a capacitor all right when we get to a capacitor of currents running and gets to the capacitor the current doesn't go through the capacitor right the current just charges up the plate so think of it very much like a water hose carrying water up to a buck and then it dumps the water in the bucket okay so let's think that of applying this to a capacitor so what are the equations tell us the equations say the line integral of the magnetic field looped back around itself anytime you see a circle on an integral it is something that's closed it's a closed integral so this is a closed line integral comes back to itself and so I take that line integral and what I've told you before is that this current enclosed here's the way you know whether the currents enclosed or not take the past if the line integral is going to go over pretend that that's a bubble wand okay now you're going to take that lond and dip it in bubble solution and then the soap film that comes out of that if a current Pierce is that silk film it counts alright so when we had the big long wire and we thought about taking the line integral of magnetic field along the big long wire clearly the current cuts the soap film we counted that as I enclosed and we knew that we had to have a curling magnetic field around it so now do that here in the capacitor oh we get something kind of strange all right so what this law would tell us is that if the soap film passes in between the plates of the capacitor there's no current piercing the soap film if there's no current piercing soap film we would get zero contributions and we would find that the integral the line integral of B dot DL around the boundary of our soap film right around the bubble wand itself would give us zero but now let's take the same soap film we're just going to blow on it a little bit okay so just blow on it a little bit extend it so that it passes the capacitor now catches some of the current okay so it's the same bubble wand I've just deformed the soap bubble a little bit I can do that the math allows that okay so we deform that and now we're catching some current now we'd get a current enclosed and now we would find that integral B dot DL is not zero okay and this actually says makes much more sense because you know how could I possibly say that all this current leading up to the capacitor and this large curling magnetic field around it it's not going to just go away when I get to the capacitor okay so the curling magnetic field continues on unabated so this is what we call a paradox all right we've got a contradiction in these lines of reasoning and so something is a myth the equation has to be updated and what's missing is the time dependent term that Maxwell added to these equations so here's what we need to do to fix it we're going to add slime to amperes law and what we're going to see is that we need to take into account that while the capacitor is charging the electric field inside of the capacitor is increasing which means there's a region of space where the electric flux is increasing so that time dependent electric flux is what's going to save the day so let's calculate the electric flux inside the capacitor and then we'll calculate the time derivative of the electric flux and we'll see that it's the piece we need to solve the puzzle so inside of a capacitor the electric field is one of our epsilon not Q over a all right and this is the electric field deep inside a pair a large parallel plate capacitor you know there are also fringe fields that are typically negligible so we'll focus on this piece that's deep inside the capacitor so the electric flux now through our surface alright what we're showing here the dotted line is where the silk film is now inside the capacitor and I've got the ring okay over which I'm going to measure V dot DL is far away from this alright so the piece of the electric field then that contributes is not over the entire area of the silk film but it's only going to be where there's actually strong electric field so here interval a dot NDA is going to end up being the electric field inside the capacitor times the area of those plates so e times a and that gives me the e itself inside of a capacitor is Q over epsilon naught a times this area of the capacitor plate that's the flux okay that's the electric flux through that surface and now the A's cancel and I get Q over epsilon naught where this Q now refers to the charge on a plate it's not where our soap film is but it's the charge that's next to the soap film so now find the time derivative of that so now I DV by DT of electric flux which is just going to be D by DT of Q over epsilon naught epsilon not so constant it comes out of the derivative and DQ by DT you might remember is current okay it's just another way to state current current is charged per unit time moving by which is formerly a time derivative of the charge so DQ by DT gives me current and again this is current in the wire not on the surface so it's something that's happening nearby the surface not right on the surface but the current that's coming up to the capacitor is charging the capacitor and as the capacitor charges the electric fields inside of it grows and so of course we can then attribute right the current coming in is what gives me the changing electric flux inside so that's that's why those are determining each other so the flux inside kinda takes the place of the current so if I add then in this piece here epsilon naught V by DT integral e dot NDA that gives me the full amperes law okay which now gets to be called the ampere Maxwell law so basically in outside the capacitor so if I think now the geometry we had before where I had I'm going to be taking integral B dot L around the line right a closed line that then forms like the ring of a bubble wand dip that in bubble solution and take the soap film so now whether the soap film goes straight through the capacitor or whether the soap film is extended a little bit to go beyond the capacitor and catch the current either way it's going to give us the same integral V dot L so the soaps are going back to our geometry here right the soap zone will either catch the changing electric flux in between the parallel plates of the capacitor in which case BAM it gives the right contribution or if we deform it a little bit it'll catch that current piercing it so independent of which contribution it catches it still gives the right integral B dot DL okay so that completes the equation do you have any questions about that so far okay all right so this is now the full form that's the ampere Maxwell law that was the last piece in the equations so this this is it this completes it this this slide now this is the slide that if you snatched it and you ran back in time 300 years you too could transform society okay so this is everything there is to know about electricity and magnetism contained in those four laws we also implicitly need the force law right because the force law forces QE + QV cross B that tells us how these electric and magnetic fields affect charges it tells us how charges of affect electric and magnetic fields okay so all that together is everything we need to know and as we said this it basically empowers a global society right it gives us everything from light to radio communications the wireless communications to the Internet okay everything you need to know to run the power grid everything you need to know to make motors and all that great stuff everything you need to know to make integrated circuits do you have any question so far ok this is our last piece of the puzzle now ok it's got a star because this is it and I just want to remind you of the pieces that are in here ok so of course is a theoretical physicist I like math and you know so some people look at the motor luta and think that's you know very beautiful thing I look I also agree that's very beautiful thing but I look at Maxwell's equations and I have the same you know reaction some people have to the Mona Lisa or - a beautiful sunset I go oh that's just the prettiest thing I've ever seen so it is it's so pretty okay so what's pretty about it right so so what is it I've got two different kinds of integrals okay so I've got a flux type of integral okay so this is about areas so flux through areas and these are enclosed areas and I get one from the electric field one for the magnetic field so it's nice and balanced that's very symmetric that's very beautiful and I have two another kind of integral for curl okay so a line integral the curls back around itself okay this is told me about curling electric fields and curling magnetic fields and they're balanced there's one of each so that's very symmetric as well and now I have this nice symmetry here as well between changing fluxes okay so this is the magnetic flux and if it changes the magnetic flux changes it causes a curling electric field and then we know over here if the electric flux changes it causes a curling magnetic field and so forth I should also remind you of how these integrals are related to each other right this closed line integral all right makes essentially like the rim of a bubble wand and then a soap film going through it that's this area here and you can deform that soap film and the math still works okay so this area is just some area that ends on the rim defined by that same thing here so very beautiful equations and you can already see you could kind of get a feedback effect once you have these two time-dependent terms there right so let's just think about kick-starting something so let me just kind of kick-start a magnetic field somewhere and I'm going to have a region of space with a changing magnetic flux okay that changing magnetic flux causes an electric field to curl around itself okay so once you start a curling electric field well now you've got a changing electric field you've got a changing electric field there's some region where there's a changing electric flux all that changing electric flux starts off a curling magnetic field the curling magnetic field is now changing which means there's a changing magnetic flux somewhere which goes on as it just loops back around right this causes that to happen causes that to happen causes that to happen and so forth so you can already see how you can get something that that is a self propagating wave and if we have time at the end of lecture I'll show that to you the next thing I'd like to show you though is how to convert these guys into their differential form so here I'm going to take a different path from your book okay I'm going to skip to the end of the book and show you how to convert these integral equations into differential equations once we get the differential equations in place you remember I've already told you how to solve differential equations right when you get a differential equation you knock on the door of the math department and you say here's my differential equation what's the solution please mathematicians and they tell you what's the solution if it's all right there are differential equations out there that we don't know solutions to yet but if it's a solved equation you ask the math department so what we'll see is that when we convert these guys into their differential form we'll get a differential equation that turns out to be the wave equation all right so we'll see that we can actually just predict light waves electromagnetic waves out of this stuff so let me show you how to get the differential form basically it amounts to taking derivatives of these integral equations but let me show you how to do that and the very controlled calculus methods that you already know okay when you're going to take a calculus limit what you need to do with you is you define geometries and then you're going to shrink those geometries down and then in the limit calculus happens right so let's think about Gauss's law first so Gauss's law tells me that I need to think about some sort of charge okay and then if I enclose that charge of an area the electric flux poking through that area is directly proportional to the amount of charge that's inside the enclosed area so now think about a geometry that's going to be easy to take limits of right if I'm going to take a calculus limit set up the geometry to be easy so in this case I'm going to think about a region of space enclosed in a box now I'm going to make the box a nice square box okay so it's got its got a Delta X that's lined up on the x-axis a Delta Y is the distance along the y axis Delta Z is the distance of this box along the z axis and I want to think about taking Gauss's law and just divide it by the volume of that box okay so I take the electric flux divide it by Delta V I took the right-hand side 1 over epsilon not charge enclosed and divide that by Delta V so far I haven't done anything magical I just divided Gauss's law by Delta V ok next some calculus is going to happen so now I want to take the limit that that volume goes to zero and as that happens we'll see that some differential equations come out so take the limit as Delta V goes to zero okay on the right hand side here as I take the limit that Delta V goes to zero I'm going to enclose less and less charge all right but in the limit that that becomes charge density so this is charge per unit volume and I'm just taking a smaller and smaller volume and that becomes Rho where Rho is charge per unit volume that's just that's what charge density means okay so this becomes charge density okay that's just by definition you have any questions about that part okay so it's just by definition there's no questions I mean we're good or I need to spend more time it okay we good going to go on okay all right and sing some thumbs up so we're on the left-hand side of the equation now and that gives me integral e dot d NDA over Delta V but I'm going to take the limit now as the volume goes small all right so now I go back to my geometry okay 50% of physics is drawing your diagrams really carefully so now I just look back at my diagram okay I've set things up in a way that's going to be easy to calculate I set things up so that I have an electric field that's only in the x-direction in this region of space okay and I've set my box up so that it's along the x y&z axis and so I have a case that's easy to calculate the electric flux so there's pieces of the electric flux that are on say this plane of that plane and there's no contribution right parallel nuts catch no fish so in the electric field is parallel to the area there's no flux contribution so I'm only going to get a flux contribution from the back end here where e^x is poking into it and from the front end here where e^x is poking out of it so let me take this piece first this gives me a contribution I've labeled it over here e to okay that's what's coming out of this side of the box and what's the area it's poking over can you kind of just see from my diagram this e to is poking over what's the area of that square I set it up to be easy so what's that what's the area of that square right there yeah exactly so it's Delta Y times Delta Z that gives me the area so that's right here that's e 2 times Delta Y Delta Z on the backside which you can't quite see in the diagram on the backside it's the same area but the flux is coming in flux coming out gives a positive contribution flux coming in gives a negative contribution so there's this negative e1 Delta Y Delta Z okay the other thing I'm going to do is write down the volume of the Box in terms of Delta X Delta Y Delta Z what's the volume of my box in terms of Delta Y Delta X and Delta Z yeah I just multiply them together right so Delta x times Delta Y times Delta Z that's the volume of this box that goes right there and you might see already all right but I a delta Y / Delta Y and a delta Z divided by Delta Z so I'm just going to cancel cancel cancel and now my limit of Delta Z says the volume shrinks down what's shrinking is Delta X Delta Y and Delta Z are shrinking all at once all I'm writing down over here is the limit as Delta X goes to 0 because I didn't have any Delta Y or Delta Z dependence left ok so limit as Delta X goes to 0 of e 2 minus e 1 over Delta X that right there if you're a math major you're very excited because if you're a math major you say that's a derivative ok I know what that is so let me just remind you of the definition of a derivative all right so the definition of a derivative DF by DX is defined as the 3 bars mean to find us limit as Delta X goes to 0 Delta F over Delta X which you could also write as the limit of f2 minus f1 over x2 minus x1 the limit as the distance between the points goes to 0 graphically if this is a graph of f of X some function versus X so there's a squiggle and this is point one and point two what this is telling you is take rise / run and then take the limit of those pieces go together which gives you the local slope so I hope all that's familiar from your calculus class right and then you can look at that calculus definition and say oh I have a limit as Delta X goes to zero of the electric field ok difference in electric field divided by Delta X that take that limit it gives me a derivative so by definition then this is ze x over DX ok so I know what the D by DX because there's a delta X in the denominator how did I know to put this as the X component that's how I set up the problem okay I set up a physical situation where my electric field is parallel to X ok so that's why there's an X sitting there right but that's the that's how this flux is going through that box and what we see now so far is that ok this side of the equation has turned into de by DX and now that equals Rho over epsilon not do you have any questions so far okay all right complain for thoughts or okay all right now I set this up in the x-direction and because I set it up in the x-direction I got a de X by DX if I had set this up in the y-direction I would have had at ve Y by dy so I've set it up in the z direction I would have had a de Z by DZ so in a general case all those guys contribute and I get de X by DX plus de Y by dy plus vez by DZ is Rho over epsilon naught okay and then there's a nice shorthand for saying well when I'm taking the X derivative I care about the piece that's parallel to X okay when I take the Y derivative I care about the piece that's parallel to Y that's because and going from the back of the box to the front of the box I walked along the x-axis and I needed to know the flux that was parallel to X so that's why I got a derivative with respect to X there okay so there's a shorthand for this which is called del dot e alright and del itself is a is a vector derivative d by DX of the X components d by dy of the Y components D by DZ of the Z component and the shorthand is dela D but you could always just remember the longhand as well all right so what that means is take a derivative in the following way walk along the X direction and measure how much to the edited the electric field parallel to my step how much did that change okay now take a step in the Y direction and look only at the electric field parallel to my step how much did that change now step up in the Z direction and look only at the electric field parallel to my step how much did that change all that together is what's called the divergence actually of the electric field and that's the differential form of Gauss's law to have any questions about that okay how about the the curly signs in the in the derivatives if you guys seen curly signs before okay so let me have it I see a lot of heads nodding so some of you have if you haven't seen the curly signs let me just tell you what it means it's actually not bad at all the curly sign here for you know usually you see a D there ve by the X I've got a curly D the reason is to remind you that when you're working on this term you only take the derivative with respect to X and ignore anything with respect to Y or Z and when you get to this term you only take the derivative with respect to Y and ignore an easier X dependence and so forth okay so it just means do one at a time that's all it means questions alright and sometimes this is called a parallel derivative okay so that gave us that was a lot of math but what we did was we converted the integral form of Gauss's law into the differential form and in fact sometimes I said this is called a divergence because of the following idea if I have a charge density that a point charge for example alright and then I think of how do the electric field lines come off of that the electric field lines come out in such a way that they diverged right they come out they spread apart that's diverging and so this del dot e captures that piece of the electric field that's diverting alright the differential form of amperes law so here's amperes law alright in a goal line integral of B dot DL around a closed loop is equal to the sum of the currents that are piercing the area inside the closed loop plus any time derivatives of the electric flux through that soap film that attaches to the closed loop that's amperes law so now again half of physics is writing diagrams carefully I'm going to carefully set up a diagram here to where I can take a calculus limit and get the differential form so here I have current all AI coming out of the board towards you and I'm going to let integral V dot dl be this nice square I'm going to take a nice square path so I can easily take the calculus limit so I'll go along step one step two step three step four x axis is horizontal ix this is vertical and Z is out of the board towards you so now think of applying amperes a lot of this case okay integral B dot DL will be around that path I have an i enclosed all right there could also be electric flux so we'll keep that term as well and what I want to do is divide all the vampires law by the tiny area associated with that square that I wrote down so Delta a represents that area of the square so have limit as Delta a goes to zero of interval B dot L over Delta a okay then again I'm just going to then take the limit of that area goes small that's what calculus always does so on the right hand side here I have current okay I'm going to rewrite current as current density okay current density is current per unit area and now so if I take current density which is current per unit area and then multiply it by area I get back current so current is current density times area all right so that's sitting right here divide all that by Delta a the area okay and then the third term was D by DT integral e dot NDA and again I'm just going to divide that by Delta a then I'm going to take over the whole thing so it's a whole equation divided by Delta a now I'm going to take the limit that Delta a gets small and I'll do the same thing I did before re-express Delta a in terms of things that are in the diagram so let me do a little bit more simplification here first with J dot n alright I'm going to now say in my geometry that what I care about is the flux coming through the board right so I've defined an area alright in our area it's it's ends that the normal to that area is along the z axis so now if I identify n as V hat I need to take JN which becomes JZ and I need to take a dot n which becomes easy so it's the Z component of the electric field comes down any questions so far all right okay does no questions mean we're doing okay or I need to talk about something more just just give me a little thumbs up or thumbs down or Twitter thumbs up means keep going thumbs down means I've got a question okay again sorry seen a lot of thumbs up alright so cancel the A's alright cancel the A's cancel the a is there and take the limit of Delta a goes to zero so now the right hand side became mu naught J Z there it is and the right hand side over here all the A's fell out so the limit of Delta a goes to zero was easy in this case again I'm just shrinking that loop down and I get Dez by DT times mu naught epsilon naught so it's the left hand side we're going to have to work on okay so this equation I'm going to copy on the next slide there it is copied straight down and the thing we need to work on now is the left hand side so work on the left hand side you see that's a bunch of math but before we get to the bunch of math what's going to come out as derivative right so that's the purpose of what we're doing we've divided by a delta a we're going to take the limit of Delta a goes to 0 we'll find the derivatives based on the definition of a derivative over here and then that'll that'll tell us that these the derivatives okay so let's work on this will massage this guy a little bit so I've limit as Delta a goes to 0 of line integral B dot DL over Delta a right so now I need to just look up my carefully Don drawn diagram right half of physics is carefully drawing your diagram so I have segments one two three and four and so now what is integral B dot DL mean it means walk around that loop right so walk around that Lucas movement I take B dot DL at every step so one of the steps is a long segment one alright so long segment one I'm going to take a step in the DX direction and I'll take the magnetic field that's parallel to me so I'm at position 1 so it's call that b1 and I'm taking the component of magnetic field that's parallels on my step which means I need the X component so it's B 1 comma X just means X component alright and then times Delta X there's the Delta X that was the first leg the next leg is we're going to walk up so now walk up in the Y direction and so I take a delta Y step up there's a Delta Y times the magnetic field at that spot which is B 2 just according to how I drew the diagram and I need the y-component of that so that's here B to Y times Delta Y all right now the third leg is up top now I'm going to walk backwards along the x-axis so I'm taking a step now in the minus Delta X direction all right so there's a minus Delta X right there for that which piece am i picking up I'm picking up b3 that's just magnetic field at position three so take a minus Delta X step I need the component that's parallel to the X direction which is B 3 X so there it is B 3 X there's a minus sign because I walked backwards now while carry now I'm going to down along the y-axis ok as I went down along the y-axis I get a minus Delta Y contribution times the magnetic field at position 4 in the Y direction so that's all that numerator right there ok Delta a I'm going to reexpress in terms of Delta Y and Delta X what's the area of my square in terms of Delta X and Delta Y you can just read it off the diagram right so what's that what's the area of my square yeah I'm hearing some whispers here of Delta X Delta Y that's just how I set it up so this area becomes Delta X Delta Y and now you can see in this first term here Delta X will cancel Delta X so now I just have a Delta Y and this first term is here the second term is going to land there so the first term here Delta X cancelled and for the limit as Delta a goes to 0 what that really means is shrink Delta X to 0 while I shrink Delta Y to 0 okay and in this term I don't have a Delta X left so what are you took the limit and what's left is this Delta Y right and then over here I have B 2 y minus B 4 y the 2 and the 4 just represent where we are in the diagram and the Delta Y is canceled so I have an over Delta X and I have the limit as Delta X goes to 0 all right so this guy right here I can spot as a definition of a derivative same thing here alright so in looking here I have again a function difference in a function divided by the space distance that I walked and the limit is at goes to zero that's this again it's rise over run okay it's rise over run of the magnetic field in the X direction okay and in fact it's about how it how the Y component is changing so this piece right here becomes D by DX of dy notice the Y was there so just carry the Y over this piece right here becomes V by dy of B X so BX is right here do you buddy why all right now because of how I set things up this guy has a minus sign in front that's actually that's actually physical and so what does this mean okay this is called a cross derivative because I'm taking a D by DX of the Y component and I'm taking a D by dy of the X component let me just tell you what it means physically okay this guy right here okay this X component says do you buddy X Z buddy X means as I walk along the X direction okay so they'll walk along the X direction I'm taking a derivative of B Y dy is pointing in the y axis right so CY is here at this spot and then a CY changes to visit this spot I just got a derivative in dy right the change would be what that's what that terms about so this is called a cross derivative all right and in fact in this case here it's about Z component all right so this can be re-expressed as del cross B it's a cross derivative right says when I walk along X I want to know how B Y is changing and when I walk along Y I want to know how B X is changing okay so the cross derivative turns out you can summarize it and sell cross B I'll show you what all that means on the next slide and so for a general case then this is Del cross B is mu naught times J plus epsilon not de by DT so let me show you on the next slide what del cross B looks like all right okay so you've seen this math before for a curl this is Del cross B all right so del cross B so for you you see that before as crossing two vectors right you saw it as a cross B before now I've got this thing called a Dell right as we said before what is Del you seen Dell we've said before very go this Dell vector is dy/dx comma D by dy comma D by DZ and now I'm going to take that vector derivative and use it in a cross product so here's what that can look like okay a cross product I can always write as the determinant set up in the following way determine it of X hat Y hat V hat now I put this funny Dell vector n okay just treat it as a vector okay the Dell vector is dy DX comma D buddy Y comma D by DZ and then B I just write down the X B Y B Z so this is the first time you've seen a derivative put into a cross product but it's the same the same structure so the way you take a cross product is you copy down the first two columns okay set up your answers you're going to have some pieces here that are X component Y component and Z component if I think about drawing arrows down into the right every time I have an arrow down into the right it comes up with a positive contribution so this thing right here is X hat do you buddy Y easy so that gives me a D buddy we buddy Y of B Z in the X hat direction that's right here this guy gives me D by DZ of the X okay C buddies UV x in the Y hat direction this guy catches Z hat D by DX and dy that goes here as d by DX f dy in the Z hat direction okay and then up into the right gives me minus signs and those guys go right there so it's kind of a lot of math you can see why with all these terms we like to just express it as del cross B okay but what it tells you is is the magnetic field it's actually called the curl del cross V is called the curl of B and it actually tells us whether or not the field is curling so I'm summarizing here for you now the integral form of Maxwell's equations and the differential forms that we just found we showed it for Gauss's law but the same mass would say that Gauss's magnetic law has the same form okay we showed it for amperes law with the same math on Faraday's law would also give you the differential form of Faraday's law okay so now I want to just kind of give you a sense of how this math compares what we had before in the integral form was we had fluxes right we had enclosed fluxes so every time you see the circle on the integral it means close whatever the subject is if it's an area close the area okay how do I know what area is closed if there's an insider and outside that it's a closed area so in closed flux as an integral turns into something called divergence on the differential side and it's very physical right since this think about the top two equations at first think about Gauss's law in the integral form right for let's think about a point particle so the point particle had an electric field splaying out around it and the integral form said okay close and close your point particle with a Gaussian you know box which in this case a sphere is a good thing to use and add up all the electric flux coming out of it and outward pointing electric flux everywhere tells me there's a charge inside that's the integral form the differential form says look at it a little bit differently the differential form says okay coming out of this point charge isn't a lot as an electric field and the electric field lines come away from each other as you move away from the point charge that's diverging right as things move away from each other that's diverging and DelDOT is called the divergence it's telling you as I take these derivatives in different directions of these guys moving apart so divergence and enclosed flux are about the exact same physics when I take the integral form of Faraday's law in amperes law I got derivatives that are curly derivative okay and this they're they're physically about the same thing so in the integral form we had local in the closed path a closed circuit and is there a net contribution to electric field is there a circulation you call it okay and the differential form is about the curl of the electric field it's about taking a cross derivative okay do I have a swirling change in there and it's actually it's called a curl so circulation and integral form turns into curl whether it's Faraday sloth or amperes law you have any questions about that so far okay all right so what we'll hit next time is we'll take these differential forms all right and we'll show how these two differential forms in particular give us propagating waves you can start an electromagnetic wave and it will just keep on going so we'll do that next time and we're out of time for today so I'll see you guys on Wednesday

Channel: Prof. Carlson

Views: 51,980

Rating: 4.8052912 out of 5

Keywords: Physics, Physics 272, Purdue, University, Matter, Interactions, Electricity, Magnetism, Maxwell, Maxwell's Equations, Ampere, Erica Carlson

Id: fkfnDopQBYQ

Length: 44min 44sec (2684 seconds)

Published: Thu Aug 10 2017