WAV01: Maxwell's Equations

ready let's do it today's lecture the first part is entitled Maxwell's equations we're going to put all of the pieces together right now you are seventh level field masters you have become that I now confer that status upon you all and by virtue of your performance over the last two tests you are now seventh level field masters now we now that we can walk we're going to start to run on this lecture we put all the pieces together and in doing so we recount one of the greatest scientific synthesis of concepts in the history of mankind this it's actually a pretty neat lecture because we get to put all the pieces together just like James Clerk Maxwell did back in the 1860s when he was working on this keep in mind this is about a time of the American Civil War almost 150 years ago now and we will get to see him or step through what he was thinking as we put this the system of equations together and perform what we call a synthesis that was taking a bunch of things that were known polishing them up a little bit putting them together and uh you know discovering something new this is really neat because it you know we're using the power of the theoretician the power of the pencil he's getting down on a piece of paper putting everything together and what we'll be seeing especially at this lecture and in the next segment how in a way he's calling into existence all of the modern technology we're using today how do i phones and and smart phones transmit to cell towers you know how do i Ethernet cables work other than anything electronic work and to that we'll end this is actually going to be one of your exam questions that I will put the answer of the board as we work through this lecture today the question will be memorize and recount Maxwell's equations both an integral differential and word form and describe the name and the units given to all the variables in that system of equations that's it that is 53 points I just give it to you out of maybe 150 or 200 but still it's a substantial portion if there is one thing that Georgia Tech students know how to do its memorize the hell out of something and I'm counting on that I'm counting on that so that that uh it's not easy there are still people that will make mistakes silly mistakes they'll either forget to study or they'll forget their units but I'm yeah that's right and I'll give a little crib sheet on line with my answers all you got to do is just spit it out well a few times you can get away with that in this class so that's Maxwell's equations so let's go through first and say what is it that we know what is it that we know so far in this class well we know Coulomb's law which says that a closed surface integral of electric flux density vector is equal to the volume integral over the volume that that surface encloses of charge density volume charge density so let's let's pretend this was the test let's pretend this was a test and I'll go through ad this will be the integral form part of the paper this one will be the differential form so it's all labeled as properly Maxwell's equations and my corresponding Coulomb's law that I've written there in differential form we know this to be the divergence or the source enos of the electric flux density vector is equal to the charge density the volume charge density remember coulombs per meter cubed electric flux density vector is coulombs per meter squared and when you differentiate it with respect to position as this operator does you get coulombs per meter cubed it's like dividing through by a meter and then let's go over here and write the word form and this may be one of the most important things to know from this class besides the equations because the word form lets you explain it to other people what does that do they might want to take a crack what does that equation mean in word form how would you describe this Randy do you want to give a try I saw your hand bravely go up and then shrink down really quickly so I'm going to pick on you you can do it the radio of density is is equal to the value inside yeah yeah that's right the gradient of the electric flux density is equal to the volume charge at sea but you really just read the equation what does it mean what does it mean it was like there was a I had a physics teacher in high school who was going over Newton's law did I ever tell this story no okay he's an F equals MA and somebody said what well sir what does that mean and he took look at it after a minute of thinking missus well it means F is equal to Ma and then he continued onward thanks thanks a lot thanks a lot notice it's hard though because you know how would you explain this to somebody who wasn't so technically wasn't an electrical engineer you know somebody who didn't understand science like maybe an industrial engineer that was caught on tape oh crap so sell with some gradient that's that's right that's right the sourcing is hundra go ahead can we say that it's actually coming out of a surface around the surface it's actually what's actually inside that volume yeah yeah yeah that's kind of like that that's a good way to describe the integral form it says the same thing but that's really my my joking explanation of Gauss's law right it's what's inside that counts electric flux comes out of positive charges and goes into negative charges so you can tailor this however you like the expression the expression that is kind of succinct and and communicates very clearly with an economy of words what that equation means is this electric charges spawn electric flux like that that word spawn like the salmon like the savage who would spawn me means come out of right so positive charges spawn electric flux negative charges actually sink electric flux they do the opposite that's why they have the opposite polarity that's fundamentally what those equations mean electric field and hence electric flux comes out of positive charges and goes into negative charges starts and positive ends on on negative that's all it means that's all it means then and that equations really just give that a little bit more mathematical structure and rigor but that's fundamentally what it means okay so that's one thing and that's what was that called what was that individual law called that we studied at the very first lecture after transmission line theory coulombs law this is all really just a different statement of Coulomb's law or Gauss's law applied to Coulomb's law okay good good so we know that we know that let's see what else do we know let's go over to here Maxwell's equations integral form and let's see what else do we also know here's a good one here's a good one and peers law if I take the integral around a closed path a line integral this is where we're counting magnetic field vectors the collinear part that are that lie tangential to ik and enclosing path then that must be equal to the enclosed current and we can write the enclosing current as a surface integral that this line integral goes around surface integral of my current density vector dotted into the surface normal differential element of integration of course we all know this to be the current I where this is amps per meter squared this is amps this is amps per meter and when you integrate it with respect to position it just becomes amps you have amps equals two amps everything is satisfied that's another elegant little integral form and of course we also know that has a differential form we can apply Stokes theorem and we can pull out the input or parts of the integrand and we get that the curl or the swirl enos of h is equal to the current density vector i always like these differential forms of Maxwell's equation because they're so succinct and cute there weren't actually succinct in Maxwell's original treatise if you go back and look at his original work he actually wrote this out operator out longhand in a lot of his equations so he'd have partial derivative with respect to x y&z operating on the x and y and z components of h so we actually have three sets of equations each spelled out with their differential partial differential operators longhand it wasn't until oliver heaviside came along a few decades later and kind of clean this up in a textbook on Maxwell's equations and he was working with and we were all the beneficiaries of that right so dr. Dorian oh yes bill from Savannah yes is that is about my friend bill that is true yes oh good thanks thanks Bill what can I do for it's just that since the unit's are part of it and when we would put the units on the Coulomb's law we know that of course it's charge but we also have to include the flux density early yeah so um I'm actually going to I'll need to erase one of my boards at the end but at the end I promise I'll write down also all of the variables that we used and then their corresponding names and charge units just like you're expected on the test so the only reason why I'm not doing at the moment is because I'm running out of board spaces I got free already and then there's that little one at the corner that's no good for anything so good good question and if I forget I won't but if I do come ding me again in about 20 minutes good good question okay so who wants to try this to describe this in inward form how about Bill from Savannah because nobody from Savannah ever chimes in to respond to my rhetorical questions and Bill you're the only one I know out there could you repeat the question oh yeah sure know that this is a what would you describe this equation in Word form give it a shot the this swirlin asst of the of the magnetic field mm-hmm is equal to the moving current enclosed yeah that's great that's great let me put it over here I'll put my version which is almost the exact same thing good job they'll suffer to that everybody thank you for being my volunteer bye my willful my willful participant magnetic field swirls around current will call it electric current or if you want to be a little maybe more technical so swirl eNOS doesn't cut it like it the the technical conferences and that sort of thing you say circulates circulates that's the right hand rule right magnetic field circulates around current now this one isn't quite done you I shouldn't put a period there is we're going to add one thing to that a little bit later a little qualification okay and then we go over here to Maxwell's equations and we also studied Faraday's law which says II filled the closed form a field integral around a a path that closes in on itself should be equal to - let me write it like this - partial derivative with respect to time not giving myself enough room we'll just do it like this - the partial derivative with respect to time of the flux integral so you take the surface integral over the surface enclosed by this line path and you add up all the magnetic flux density you make a flux integral you take the time derivative and the total voltage around that path the cumulative afield that is tangential to whatever path shape you're integrating around which of course we know to be a voltage this is volts per meter integrated along distance to give volts that is going to be equal to the time variation in the magnetic flux of enclosed by that path that is Faraday's law Faraday's law and of course we know we can apply Stokes theorem to that we did that earlier in the class and what we got was the curl of e field is equal to the time derivative of the befell V field and any given point in space remember these are all functions of three-dimensional space I haven't written them as such you know they should all be functions of that R vector XY and z XY and Z function of XY is a function of X Y Z function of XY and Z function of X Y Z I'm so mitting that because it just gets tedious to put parentheses our vector or parentheses XY and Z at the end of all these things so yeah now who would like to try the word form of this how would you describe this Carter yeah that's good that's the good changing magnetic flux will induce a voltage another way to put that and that would be fine I'll give you full credit for that if you wanted some symmetry with the one we came up here is that electric field circulates around changing magnetic flux and we know that is the the process of induction in the physical world so these were all the things that we knew at the time that Maxwell came along and started to heavily invest in this problem trying to figure it out and these were kind of disjoint pieces of information we had Coulomb and a bunch of other people looking at electrostatic charge and how to describe that we had this magneto static relationship that ampere was kind of working with we had a Faraday's laws sticking magnets into coils and seeing lightbulbs light up and things like that or electric or work being done galvanic potential jumping up and down as he changed the flux through a coil and so he looked at this and he kind of arranged things in this form and he said you know what I kind of used the principle event you know the physical world should have some beauty to it some some a little bit of symmetry to it to you know we wouldn't necessarily expect that but we we don't wanna when I have some beauty in the way that the laws are written here and we look over at this system they think well what's missing what's missing we have this kind of Gauss's law relationship or Coulomb's law relationship for electric field or electric flux density this is it not reasonable to think that we should have a similar relationship for magnetic flux density in other words shouldn't you have a closing surface integral to count magnetic flux coming out and shouldn't that be equal to something like if you wanted some symmetry in the deal the volume integral of magnetic charge but wait a second up until this point and at no time period thereafter have we ever observed a magnetic charge so one of the key contributions that Maxwell made to kind of close the loop in all these equations that he brought together was to come over here and said you know what we haven't seen anything like the magnetic charge yet so this must all be 0 this must be 0 and in fact because we haven't seen a magnetic charge you know in a way we've got this thing called electric current up here we could if we wanted true symmetry we can add some magnetic current in there too right I've kind of Linda Kuk laws would kind of match you know you'd have electric current would cause magnetic field to circulate around it and we're tempted to think oh yeah if there was some something like magnetic charges and they were moving you'd have electric field circulating around that but alas because we have never observed a magnetic charge it would be foolish to think that there would be magnetic current at this point and so well let's just leave it off let's set that category with a equal to zero so this law is now complete this law is now complete and we just added a little zero there because we haven't observed charge yet and the last thing that Maxwell did was to change amperes law at a what what initially was kind of like a fudge factor but it actually makes a lot of intuitive sense and that was this he looked at this and said well here we've got time variations of magnetic flux inducing a voltage around a path it would probably be reasonable to think that you could induce a magnetic field around an enclosing path proportional to time variations in electric flux density so you take that same surface integral you count the electric flux going through it just like you did the magnetic flux for Faraday's law you take the time derivative and you should get a magnetic field that circulates around it you know you can almost make that that inference that that should be there just based on the beauty of the mathematics right oh let's make this law look like this law and yet it turns out that solved a very important paradox in physical in physics at the time and it was called the capacitor paradox so if you go over here let me just sketch a capacitor for this for the moment let's say you had this circle circuit element that was a capacitor you had current flowing into and out of it and of course we know that current flowing into and out of the capacitor is really positive charges building up on one plate and negative charges depending up on the other plate of course in this case it's negative electrons depositing up here and then depleting this piece of metal up here in in physics but this is how we mathematically represent the scenario and of course this was always a little bit of a disturbance to the scientific minds because at first blush the old amperes law didn't work for this system right you had a capacitor there was current flowing through here but if you did amperes law integral around the wire that carried that current you know do a right-hand rule saw me say this is going this way currents going down here magnetic field should circulate around that wire if you did that on the lead going away from the capacitor or the lead going into the capacitor you'd always get a perfectly behaved and peers law but if you are kind of like one of those smart aleck II kids that always asked the really difficult questions and immediately your hand went up what is your question oh no I was just saying I was like oh okay okay okay yeah thank you for that if you took your amperes law integral there something bad happened there was no current flowing through here no physical current it couldn't be it was it was a vacuum and yet if you were to measure the magnetic field just because you were taking the integral down here instead of up here down there it would still be the same it would still be the same and anything is by adding that last term that we just did Maxwell solve what was called at the time the the capacitor paradox the capacitor paradox because what he says is his in his construct that he put up on the board there he said well there are two types of current there's physical current which is the J vector the charges that are moving in the system and then there's this other term change in electric flux density which he called displacement current partial derivative of the electric flux density vector with respect to time and we can see that's a really nice way to do it because if charges are depositing positive charges are depositing up here negative charges are depositing down here there should be an increasing with respect to time electric flux density vector between these two plates and as that increases the time derivative will effectively be the current that balances out amperes law equation it's also the current that balances out the cure cough current law you remember how to take in that in your circuit Theory class you could draw a circle around this and call this anode and say well the current coming in and up here has to be equal to the current going out of that node I said well what if you were that smart aleck you kid this is what Professor what about what about if I draw my node like that what says Kirchhoff's current law works fine provided that you use displacement current to balance it out at that point the change in the electric field rather than that just the physical charges and at first blush that looks a little bit like a fudge factor doesn't it oh let's just kind of upend this to Maxwell's equations to amperes law just to get it to work out it's interesting that when you look at Maxwell's equations and kind of consider it in that form you get a really nice affirmative result that comes out of this that we've actually seen before in this class so let's go over and and upend the differential form of them of that term to amperes law curl of H is equal to J via Stokes theorem well it turns out this is also you can also apply Stokes theorem and show that this is the partial derivative with respect to time of that electric flux density vector and now there's something you can do if you start combining Maxwell's equations let's take a look over here I am going to take the divergence of both sides of this equation so that I have the divergence of the curl of H and that should be equal to the divergence of J plus the divergence of the time derivative of that vector of course the time derivative of the vector is always a vector right you're just taking the partial derivative with respect to time of the X component you're sticking it on the X the Y component sticking it on the Y the Z component sticking it on the Z now a couple quick little sneaky manipulations the this is a partial differential operator with respect to space it's linear it does not do anything with respect to time this is a little partial derivative with respect to time that's also a linear operator and it doesn't do anything with respect to space so I can commute them and I can write this I also go over here and there's a basic mathematical identity that we can apply to this when I take the curl of a vector that operates on a vector field and gives me a vector field that represents the swirly nough sup the system when I take the divergence of swirling us what does that give me zero that's right and that's that mathematical property that has nothing to do with the physics any vector that you take the curl and then the divergence of you get zero a scalar zero as the result so what this really means is that my divergence of current is equal to minus the time derivative and instead of do put writing the divergence of electric flux density vector I'm going to substitute this Maxwell's equation in their charge now this is often reported as the fifth Maxwell's equation but it's actually a derivative from the two independent ones that we've written up here this has a special name and it's called the continuity equation but really all it says is that charge is conserved in the world which is another very gratifying result and we wouldn't have gotten there without this term added to amperes law why does it say that well look source enos of electric current if the source units of electric current is zero that means there's no source of electric current current is just can just flow around and be zero but if it's emanating from an area then clearly you're smuggling extra charges at the source of that current to supply the current with the charges it needs to move and if that's true then clearly you are depleting the volume charge in that same space that's all this means this term was zero then what would happen is the divergence of J is always equal to zero which means you always have electrostatics or a magneto statics there's no ability to start and stop currents and there's no charges charge if you were to do so you'd have to have charges materialize out of nowhere so that's how gratifying result that kind of gave some some confirmation that Maxwell's intuition for synthesizing these various different behaviors of electromagnetism that were being observed by researchers all over the world he was able to verify that that synthesis was correct in part by the derivation of the continuity equation so let's let's now complete everything because we've got it all here the Gauss's law in differential form divergence of B is equal to zero because we said there are no such thing as magnetic field magnetic charges and in fact we can write the word form of that as well there are no magnetic charges in the universe a lot of times you'll hear physicists call this magnetic monopoles monopole is just a charge by charge so Maxwell's equations weren't formed our new magnetic charges in the universe and I have to go append a little bit to this one as well the magnetic field circulates around electric current and changing electric flux and that's it that's where we go to now let's do the mundane exercise of writing out all the variable names and units that we have used in this system he field is electric field and it has units of volts per meter each field we electrical engineers call this the magnetic field the physicists call this a four-letter word the B field this is the magnetic flux density to us electrical engineers this is units of Weber's per meter squared a Weber is a volt second actually D vector we call this the official name is the electric flux density units of coulombs per meter squared J is current density units of apps per meter squared and Rho so V is volume charge density coulombs per meter cubed let's see if there anything else that I should write there do I miss any terms any variables I think we got them all everything that we used now of course there are some constitutive relationships you know from your physics class that B is equal to permeability times H vector and D is equal to permittivity times your a field vector but of course these are material parameters that relate the the two vectors to one another that's they're sort of part of the boundary conditions you know these are the what I put here as Maxwell's equations are the the governing differential equations of electrodynamics systems and these are kind of talk about the boundary conditions and the different materials that you can have that relate the the fields together so yeah Justin questions kind of abstracting is but yeah sure from the four equations is almost two compared to something Engineering is like we're looking at in the first one you can get a blood cell studying how it moves on the second one we're studying like a small stream within the vein and then the third one was studying house blood flowing in veins affects Celaeno tissues and then the last one we're saying that you can't really affect anything hmm ah that's an interesting analogy I have to think about that one for a little bit interesting interesting I think I think you could probably draw a lot of analogies here especially with things that flow turbulence air flows water flows at country land Jeff question we're coming to crucian about like the works like so wouldn't you said that support there's let the space charges spawn ellipse book mm-hmm like when you start talking about things in the universe like I mean that dislike are you saying like electric charge or electric flux follows charges or like follows economical like a puppy among future as well hey I think it would be more accurate to say this uh the electric field is we've defined it in this class it's sort of a physical physical construct or mathematical construct the actual physical thing that you measure is force on charge which is always proportional to electric field electric field times the charge in coulombs is the force on that charge that particular charge so that electric field that mathematical construct that we use to track the charge forces in a system is you know I use this word spawn because it's kind of graphical it kind of shows you the vet if you can kind of visualize the vector field in your head pointing away and diminishing as you get away from the electric charges of the system so if I got a charge like a point charge like Coulomb's law says I should have electric field radiating away from that now are there field actually traveling around that's debatable because that's a it's a construct that we use to track forces and I would need up at another lecture too to delve into the finer details of that statement that's right gravity is a completely different force just just no yeah there are we haven't observed any in thousands and thousands of years of physics and observing the universe so we're pretty certain that there aren't any in fact if you remember back when we talked about currents and charges all magnetic forces can be explained as electric forces observed with a relativistic change in your reference yes and so as a result now I should say this you will know when you have arrived in life and you have become a seventh level wave master when you work problems in which you intentionally introduce magnetic charges and currents if you take in a sophistic your instructor will add terms to Maxwell's equations like this and that is for mathematical convenience it's really cool like again this allows you to solve some problems mathematically that you can't solve without without magnetic currents and magnetic charges you're like why do you have to make stuff up to solve a problem I don't know why but it just happens to work this way that if you introduce magnetic currents and magnetic charges you can do things like apply something called the equivalence theorem I used to do this all the time in radar theory so you know you send an electromagnetic wave and radar and it strikes an object and you want to figure out how does that object reflect the waves well when that nice clean plane wave strikes an object all hell breaks loose and you start spraying waves in every single direction with different polarizations different amplitudes different phases and it's important to model that especially if you're trying to design a stealthy fighter or you're trying to back solve the the radar cross-section of an enemy fighter plane or something like that or a missile and it turns out that to solve that problem it's very difficult because to solve Maxwell's of problem equations in a system that is not homogeneous free space is extremely difficult and the by virtue the fact that you have something other than free space like a missile or a plane or anything hanging in the middle of air you've until you've actually made the problem extremely difficult to solve unless there's a theorem that says you can replace that missile or airplane or parachuter or whatever with equivalent magnetic and electric currents that will behave exactly the same but you may then solve that system in free space that's called the equivalence theorem and so all you got to do is just if you want to measure my radar cross-section and you want to simulate this numerically what the old radar guys would do they'd code up their field solver and instead of codifying a really complicated conductive saltwater mass shaped like professor Durgin into their system what they would do is they would replace me with surface magnetic and and electric current you can only apply the true equivalence theorem by using both electric and magnetic fields occurrence and it was a mathematical convenience to do so you replace me with a bunch of those currents and then allow those currents to radiate and voila you've got the radar cross-section of Professor Durgin falling in free space and you could identify who that is on your f19 or whatever fighter shoot your electromagnetic teacher down definitely the enemy but of course there that is purely mathematical convenience we don't actually use anything like that in real life we don't observe anything like that in real life they're like these really goofy quantum mechanical physicists that say well you know beneath the envelope of uncertainty there are magnetic charges and we are going to be they make a physics that behaves as such I don't understand that stuff I don't understand if that even helps you with anything let's see you first just a really good question about the magnetic field be careful its density units is is that the same as a Tesla because in another class we call that a Tesla yes yes we're successful I believe that is a Weber per meter squared as a Tesla there there you can get yourself into a huge amount of trouble with magnetic units because you've got Weber's Tesla's or spins what are some other ones Gauss's I in this class I like the Weber or the Volt second because it shows clearly that that's a flux density Weber is a unit of flux the flux density should have the units per meter squared and so as a result I kind of like that better if you want eat something else that's fine I won't take off for it if as long as it's correct you can call it a Tesla knock yourself out so yeah oh yeah well so I think I think I understand where it's getting cookies because this is a magnetic charge like kind of like an electron has a electric charge so this is like a something that has charge for magnetism that's right that's right there you can make a Coulomb's law for magnetism wise oh I have a magnetic charge and there's B field radiating out of it what turns out we don't ever observe the B field radiating out of anything the divergence is always equal to zero the source enos is always equal to zero and so there can't be anything is like a magnetic charge it can only circulate around current that's right oh yeah you said that uh air with the surface integral of the current density will give you answer eight that's right all right rub the units with the look look spin see okay okay so so I've got that's a good question this equation basically right one of the you how did the units track in this equation it's actually where he was right to denote that the integral of magnetic field which has amps per meter around a path which sort of brings in an extra meter into the calculation so you have amps per meter times meters that gives you amps this is current that has units of amps and of course if we wanted to see that you say though this is apps per meter squared I'm integrating it over a surface area so this is going to bring in meters squared into the equation ABB's per meter squared times meter square is also amps this is a flux electric flux density integrated over a surface so you have coulombs per meter squared integrated over an area which is meters squared so when you're done this integral you have an electric flux which is just coulombs and when you take the derivative it's coulombs per second which is sams amps very good very good okay any other questions so far now think of how giddy Maxwell was when you put all these together he was really excited it's like putting together one of history's long great puzzles you know we did observe magnetic behavior and static electricity way back you know literally millennia before this happened and he was able to put this down and kind of unify all of electromagnetism everything that had been seen to that point could be explained by this in the macroscopic world as far as electrodynamics are concerned and it's like putting together a beautiful puzzle how many people have ever put together like a thousand-piece puzzle before yeah it was pretty cool did you feel like I really gratified when you put it together how many people were so gratified that they actually liked shellacked it and put it on a little cardboard or maybe a plywood and maybe framed it and hung it in their room anybody who did that raise your hand good good no no you and your you and I are very kindred spirits all of you in this section you know good synthetical mind you like to see the pieces come together this was a beautiful beautiful puzzle but this is not the end of the story imagine if you had put together that beautiful thousand-piece puzzle that made a nice picture I don't know of a barn and a horse and flowers in the field or whatever I don't know you put that together and you think I love this poem is so beautiful I want to show lack it I'm going to glue it all together I'm going to eventually hang this in my room so you ahead and do that you shall lack it you put it together you put a frame around it and now you got to put a backing so you very carefully flip it over that's usually what happens it breaks in this case however max will discover that he had inadvertently put together a far more beautiful puzzle on the backside and that's what the next lecture is going to be about and I'll give you a hint of what it is it's the chain reaction that happens when you complete Maxwell's equations because now a change in the magnetic field will change the electric field a change in the electric field will change the magnetic field which will change the magnetic flux which will change the electric field which will change the magnetic field which a chain reaction and now the wave is born in fact it would be two decades later until guglielmo marconi would prove that there were invisible waves out there and he basically proved Maxwell right that there are invisible waves out there that carry energy and later information more importantly which is the basis of all wireless technology so you get in this synthesis not only do you get all of electricity and magnetism you get optics you get radio waves get x-rays you get everything you basically get the way that we transport energy and information properly understood the to not being that different from one another energy and information in our universe and and so that's where we'll start with the next topic which is the wave equation you

Channel: Greg Durgin

Views: 109,510

Rating: 4.9025211 out of 5

Keywords: Electromagnetism, Maxwell's Equations

Id: Yn-MEMaiA0Y

Length: 50min 28sec (3028 seconds)

Published: Thu May 31 2012