Quantum Entanglements, Part 3 | Lecture 2 + 3

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Quantum Entanglements, Part 3 | Lecture 2 + 3
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reversed a lower price for less time we discuss influence of electric and magnetic fields on charged particles with what they do in electric and magnetic fields warn charged particles that push them around and they exert forces effect even brought some equations what's right the basic equation the basic equation is the summer charged particle F is equal to the electric charge of the charged particle or that there that's there because of charged particle was charged with didn't have charge it would be for starters and then that gives multiplied by the electric field but just Corby electric field class something that depends on the magnetic field and the thing that depends on the magnetic field as the cross fraud of the velocity with the magnetic field that's the basic influence of electric and magnetic field on charged particles to exert forces on the force them and call them around store for the missing piece thus far is what is it that determines the electric and magnetic fields in space what is it that determines whether it electromagnetic waves or it determines if AT electric charge produces an electric field himself for 1 the equations which govern the fields themselves those equations a cold Maxwell's equations of positive of equations we want to discuss this evening they the equations which tell you how charged particles affect field is the equation which says how fields effect charged particles the other half of the coin is how charged particles affect field was it necessary that charged particles affect fields what kind of action and reaction if electric and magnetic fields could influence particles with doesn't mean it means an electromagnetic wave King come here charged particle send flying off some of the direction in which an aide at energy of a charged particle well if an electric field Canadian energy to which I particle that it had better be better charged particle consort energy out of an electric field otherwise energy would be conserved so it stands the reason that a kind of action and reaction of electric and magnetic fields can do things the particles charged particles and charged particles charged current electric charges had better be able to do things for electric and magnetic fields so that's this topic for tonight but as always in this course the basic question is whether the laws of physics and in this case the laws of electric and magnetic field of the scene in all reference frame so that's the thing we been concentrating on the study of relativity is the idea that the basic laws of physics are the same in all reference right now 1 of the things are going to write down Maxwell's equations in a moment but 1 hobby basic things rats Art we learned by back step or basic questions as to the light all all of relativity theory have endured light metals equations of the theory of light and Maxwell's equations tell you that light moves with the speed of light and Maxwell equations of the same in every reference frame it stands to reason that light will move with the speed of light in every reference frame this was precisely the puzzle that Einstein wrestled with so hard how is it possible to get a description of nature in which like was of the single oddity in every frame of reference that beefing which ensures that physical laws of physics are the same in every reference frame and past lectures we saw the work that made is that these equations of physics should be expressed in terms and quantities which have definite transformation laws when you go from frame to frame that means scalar depart change it means vectors for vectors it means cancers and all the things that we discussed last time when we discussed the transformation properties of the various things and physics but I just remind you what this equation looks like if it's expressed In a cold variant form in a form which is afraid sane every reference frame it looks like an expression involving not electric and magnetic fields will yes electric and magnetic fields but cancer a field Cancer and I'll remind you of a while what that field cancer is but it's an ant asymmetric field cancer which is built top Out of electric and magnetic fields and multiplied by the fall of vector of velocity Sosebee news component of before the new component of the forest it is written in terms of a 4 by 4 and isometric cancer who's transformation properties we and before vector of velocity in this form the rents law is written in form which is manifestly the same every reference right so that some gold is to write all the laws of physics in particular the Maxwell equations that AT and a friend a born which is the the same in every reference frame which means writing it in terms of vectors enters so for now
let's just talk for a little bit before we get into it deeply let's talk about the kind of affects the charges have on electric and magnetic fields not that electric and magnetic fields have on charges of charges have electric and magnetic fields 0 1st of all we know that we have an electric charge will quality charge q q stands for the electric charge that we have an electric charge of certain magnitude it creates around it an electric field divorcing pictures of this electric field spreads out away from the charge very much like a fluid emanating from a From a pipe might spread out exhibit spreads out 3 dimensions that's the electric field created by a point charge another example as if we have an electric current moving wire electric currents are nothing but moving electric charges electric charges moving through a wire those electric charges their motion creates a magnetic field and the magnetic field there's a field which circulates around the wire that so did examples of how charges and current create field what happens for example if you move the charge supposing I suddenly moved the charge from 1 point to another point vendor field has to rearrange itself instead of pointing out this point it was 0 . that at this point but I can't do that instantly why not because somebody out here it is not allowed to see an effect before World life could have been transmitted this point so the field could not suddenly rearrange when you move a charge it really Rangers by spreading out away by spreading out a wave of rearrangement so that I will after the charges been moved nearby the charge the field been rearranged but far away just as it was before and then as time goes on a wave spreads out away from the charge that signals the field far away from real range to accommodate the fact that the charges moved if you move the charge back and for what was spread out as electromagnetic waves to vibrate the charge back and for then what you will see going out an electric and magnetic wave that spreads out just like a light wave in fact that's 1 unintended Nintendo simply moved would back and forth like the same thing is true all currents in white if you suddenly reversed the current on a wider you would reverse the sign of the magnetic field and the magnetic field circulating around this way magnetic field circulate around in the other direction but that's signal get very far In too short a time Einstein's role was no signal ever gets faster than the speed of light so it takes some time 1st somebody far away to discovery that the current has been reversed that means that the magnetic we the change in the magnetic field the fact that its cheese direction spreads out on a wave and again that's an electromagnetic waves and if you'll ocellated charge back and forth a pure silica current back and forth you send out an electromagnetic wave that's the basic physics that we want to describe body metals equation before we do I want to do a little bit a review from last time 1st I just want to remind you about wave equation equations describing fields In this case the scare described fields that could produce weight the 1st example we talk about last ours was a scalar field of field doesn't have any indices Nuuanu with just a single number 32 point in space each point in space there's a quantity he'll find it depends on X team meant to vibrate can shake it can do all sorts of things that conform wave fields the last time we discussed the kind of equation a kinder wave equations that govern such fields are as remind your more Denio it is being used stands for the derivative with respect for the for directions of space-time ex ante Newell runs over 4 values x y and z see incidentally remember that X here really stands for x y and z perhaps I should call an ex cortex ballots but continue cortex but a really stands for the components space and also he stands for the time being by means the derivative with respect 1 of the use of that that's the ordinary derivatives we can differentiate 5 a simple way the equation of very simple wave equation would just say that the derivative of the field 0 that's not very interesting wave equation and the reason is what it says is the field was absolutely constant of derivatives the X and Y in busy in the time derivative 0 0 that means the field of changeable With anything and so that's a very boring wave equation completely a less boring equation we get if we differentiated again and now using rotation that we describe last time Morse a recurrence of past where upper index and a lower index differ only by the sign of the time
component so that this would really needed a 2nd derivative with respect the time field minus the 2nd derivative with respect to x squared minus the 2nd derivative with respect but why square minus the 2nd derivative respect C squared physical 0 not us as usual because I'm lazy I said the speed of light see equal to 1 you looking on exam exercise figure out where the speed of light goes to court let's imagine there were a way moving down z-axis wave moving down z-axis moving them the z-axis and then we find only depends on it doesn't depend on X and why it may also depend on time before wave is moving near standing still watching it 1st you ought to go up and down and down so will depend on time and it will also depend on sea where you are but it won't depend on the other direction so moving them the z-axis we can completely ignore these 2 Chan which of the derivatives of flight in the direction which doesn't change occurs is not there Our wave equation for way of moving along Z along the the z-axis sorry should be easy you shouldn't show illegal 0 what kind of solutions for such an equation had we talk about last time and there are 2 basic solution this equation has 1 of them is a function of Zepa plus 2 a mail order as a function of my to freeze 2 solutions which represent waves moving right along the z-axis and waves moving this 1 represents waves moving the right along with positive z-axis this 1 represents waves moving along the electoral what functions any function any function any shape wave profile initiate wave profile and allow it to just moved uniformly along busy direction that constitutes a weighty moving down the z-axis along the positives in direction as a function losing minus wave move in the other direction then receives the same function of plus so that's the basic facts that we learned last time about waves and 1 particular special kind of array acinar cosine wave but just right down for future reference wave which really has a way he looked away advanced by you guessed that assigned co-starring so for example a waiver of this type would the right to be a solution of the wave equation which has the form cosigned of Z minus zeal dynasty because it will incur rights and furthermore because throwing constant here usually called Katie the wave number of any kind the big constant is incident way the short of wavelength of a wave of Keyes Big Wave looks like this case all the wave looks like most the basic facts that I want to remember about wave equations and in particular wave solution that their functions as you might easy plus at different wavelengths and they could be signs signs Motorola right removed alive but that Alito normally before we go ahead with Maxwell's equations just to remind you just 2 hours a week as we went through this I will remind you remove the mathematics elementary mathematics 1st of all is the notion of across product of 2 vectors of we have any 2 vectors diesel ordinary vectors not for vectors now ordinary three-dimensional vectors and three-dimensional space and we have the crossed crop b . 80 b cross a Excuse me crossed from of 2 vectors the across product is itself another vector of we're to electors each pointing in some direction will given away we can multiply them together and make another vectors the cross product is defined by its components so for example X component I'll be crossed Eddie is defined as a definition the find the bees BY planned a X minus B X Prize a wife and there are few once defined the other components you just cycle through 0 x y and z so for example b crossed AT a the white component of extra white equals beat well why ghost Z X Ruth extra white busy busy backpacks because they SRY goes they X ghost Myers be Y X why a ghost city likewise of Alaska doing was wants a won't go again b cross a Z component of is equal to X Y them at a White z electron most BX BX X Y limits for most busy miners busy a X these uh definition these definitions of across product between 2 vectors notice of the cross product itself has components and so it must be a vector now another way of multiplying 2 vectors together is the byproduct the byproduct the byproduct of 2 vectors is a scalar a dot B and only has 1 component of sustained in every reference frame are saying when you rotate the reference frame and its definition is a X B X plus a Y BUY plus a Z be redial Beckham AX BX a force using busy that's the bar product and it's a scalar that's 1 set Our concepts that that we've been through several times now the onset of concepts Corolla vector and the emergence of a vector nearly think about that Bill stop out of derivatives of the components of the vector partial derivatives with respect that x y and z while a half til imagine you're here is that the ridden with respect to X and renewed with respect Hawaii and derivative with respect disease can that those symbols on the 3 components of a vector then not the 3 components of the derivative but just pretend to 3 components of a vector
Madoff magic Dole Cross of vector what does that mean that something that's made up out of the derivatives of the components of 80 but all I'm sorry the are the 3 components of called the vector Del Valle is just a shorthand for upside-down triangle below crossed they have met by you just go back this formula wherever you saw a stake in unit be by DXT body YOB by IDC so the kernel of 80 it then just given by Aso that's Beebe IDX I should called beach shouldn't I would have made more sense and nicer Taylor called beat track so said aid we write the 3 components of the derivative warily seed beat we just a be so that's BX with Karole Rosalia B X component of the curled go Cross B B X component of it right does this so a X you want well d by X B worry might this Debye DYT we can go through it cycle through it and find the other components of the of the kernel of baby but they're always made up a baby but they're always made up of derivatives of components of B and the same pattern the same exact half of the cost progress is made up what about still not be that's called the divergent fell but be but would come through the definition of the virgin are the oviduct product AEX firms BX that will be me by X B X close Debye BYE BYE plus debug easy easy now these 2 objects composed as derivatives of the components of vectors vectors constant and faced vector field which is constant and space highs neither neither of divergent nor world roughly speaking of that there has a divergent when it seems to be an and they have vector field as a their virgins when it seems to be emanating out of some place like bits of vector has a curl where it seems curled around from worries so if we had a magnetic field for example in a wire that was going into the blackboard bad vector would have if we have an electric charge giving rise to an electric field pointing out from some point that have a divergent that we've gone through these things before what is known as reminding you of them and now want to write down Maxwell's equation in terms of these objects Our goal is good to right Naxos equations that's easy you get more summaries T-shirt what tomato T-shirts with Maxwell's equations on them but don't understand in what was seen in every reference frame that's our goal to see that Maxwell's equations could be written in a form which is the same in every reference quite so they like and Baron curl up electric fear the electric field you construct the curled is equal time might the kind derivative of the magnetic field that's Maxwell equation number 1 Maxwell equation number 2 divergent Of the magnetic field is equal to 0 Watts says is that the magnetic field doesn't have any sources in the same sense of electric field about it doesn't eliminate out from a point it tends Mr. around things OK the next equation Is equations similar to theirs but it involves Dell across Of the magnetic field and escape equal plus E by so this took place in the very similar it's almost as tho you interchange electric and magnetic field but not quite as assigned difference between them and you just can't be changed the magnetic and electric field you get the wrong strike here on a wire story is just go about equal 0 quite parallel with a magnetic field here these are the basic Maxwell's equations but these other Maxwell's equations when there are no charges around in the absence of charges in current let me put back in what happens when they were charges and current another words these are the the call the Mac br the vacuum Maxwell equations they describe electric and magnetic fields in completely empty space with no electric charges no current let me just tell you now what happens if you a in some electric charges some electric currents and so forth they don't have an effect on our answer for example a writer Reggie at the actual thing goes the charge density the charge density is cold rolled and its amount of electric charge per unit volume at every point in space there may or may not be some charge In every will sale of space there may be some charge the charge per unit volume was called the charge density so really goes here is the charge density but there is no charge density just isn't there all here is another term and electric car and it's called GATT and I'll define for you later but for the moment I want study the Maxwell equations with out electric current in charge who want to see if we can understand why those equations other seamen every reference rate How could possibly be wiser a puzzle but to puzzle because as a city these equations describe light waves moving always with the same speed they always move with the same speed the speed of light it down the axis with the speed of light how can they possibly be the same in every reference frame after off I was a reference if I was a reference frame moving down following the way I would expect to see it moving slower nevertheless equations of the same every reference frame and every reference frame see the speed of light is exactly the same so let's see if we can recast these 4 equations in a form which is the same in every reference price when the tools to do it but we have to remember it is a P electric and magnetic fields combined together we did this last time combined together into a cancer Dornier by symmetric cancer so let's
remind ourselves of that and isometric cancer new new made up for a buyer for its matrix and its components of made up Out of electric and magnetic field since attend isometric has zeros on the diagonal and so badly In the 1st role in the 1st column on going to put the kind component the kind component the x y and z that's right to let me just show in for you my the x component of electric field minus 4 why component of the electric field and minus Z component of the electric field down here we have class piece of X plus east of why past east of the entire symmetric matrix always has the property that what above the diagonal here Is the negative of what below the paranoia about Lee's missing slots over here that's where the magnetic fields go out any of mind this beast of Missouri of whitey and Martinez beasts of X and then them here you just have be reflected plus these subsidy Martinez be of wiry plus piece of X. This is electromagnetic field cancer transforms under Lorentz transformations as AT symmetric cancer assure you have to do that last time and there is nothing new about it for the but say we would call this component over here b T X component 1st roll here refers to time the 2nd row were 1st rowers time the 2nd row was X. The 3rd rose why the 4th row Izzy and likewise for the cop we would call this E or we recall and this element over here would be cost T equals acts was Nuuanu both Ronald XYC into we fill out the matrix as the most of electromagnetic field and as I said it transforms the right way to be cancer if it another 10 servitude could make up Out of electromagnetic field it is also a tensor and it is made up by simply replacing electric field by magnetic field and magnetic field by electric eels we did look at these equations we see the electric field and magnetic field car almost symmetrically that might not be a great surprise as another cancer which is made up of exactly how all right equal another cancer it's usually called F with little twiddle on top of it and all it is is it's the same object except electric and magnetic fields interchange but not quite you don't quite a change electric and magnetic and magnetic electric change all that stricter magnetic or magnetic From my electric have that right now I don't quite you replaced electric by mind this magnetic and magnetic by electric bats in order to account for assigned different here when you do that you discover that there's another tents report they are not going to try and approved for you that this transforms a potential it's just straight forward exercise as young as I used to be a source of much exercises and used a blackboard I won't do it but what is right about where these say electric but minus magnetic wherever you see magnetic elected self F twiddle other electromagnetic field cancer again Zero or on the diagonal and and plaster b sub x the use of and B sub z them wherever we see magnetic we put back electric soul marinas Z Y mind he acts and then on the other side here you just have to remember that same symmetric so everything on a reflective side of the Matrix here it is just the opposite the negative award is in the upper half hour rate I bother writing it out these out to cancer quantities both of which have the same transformation properties and gondola rents transformations form of coherence of objects which are a cancer and symmetric cancer if we can succeed in writing Maxwell's equations here they are right here if we can succeed in rewriting them in terms of these objects in a way that looks called variant which looks the same little reference frames them we will have proved that these wave equations here these equations are the same everywhere is not hard to do so at sea of leading guess the answer notice of Maxwell's equations each have 1 derivative and they don't have 2nd derivatives of many derivatives of course is the river prospective teachers there was X but they never have higher derivatives they never have 2nd derivatives 3rd derivatives for derivatives and so forth so we might try to see if these equations here community written in terms of his field cancers rewrite have asked you know that's electric and magnetic field we can differentiate but luckily differentiator with respect to weaken differentiated of course with respect to some X but we want to make this equation in whatever tears we want to make it the same in every reference for and that means we want them match indices coal variant and Contra variant industries garlic about the only thing we can write down His derivative with respect X mural of afternoon who is equal to 0 1st of all let's count how many equations he can't come equations The Rock is 1 for each New remember we somehow knew knew was a summation index so that's out a free indexes this problem is 1 equation for each new and is therefore equations boastful equations we will see our say they are these 4 equations right here while I say there are 4 equations here is want and now there are 3 components Of the vector equation This is a vector equation 3 components EXE 3 components he won here those of components let's check let's actually check that these full equations of what we get from that our hard we just write out 1st of all new
couldn't be take b by deep team of F T no plus the fact let's a specialized let's take New will could be Z the Sephardic let's take it easy and see what we get me by the C of F P Z plus debug EX of F X Z plus a bloody Of f Y Z plus but z of F Z easy Z equals 0 basso this would say ritual or Lepista forever would respect disease that is easy and easy 0 because all of the diagonal elements 0 0 reconfigured this this isn't there Have Caesia 0 the world's F X as well as to what F X zee years we got what to we X role and Z column that be so let's just target area that BY what it is F Y easy Fermat we go to wife a wider role Zepa column rats Maria's be Cimex or vectors of minus abusive acts worth looking good this is Dedie by DXO BY minors baby wife of BX whose looks very much like a piece Of the curl of bees that's where it's what about F T Z while we go to the key role place now we get money list said so this is minus a derivative of the E sub Z With respect to tea all of that equals 0 well if you look at it carefully you'll see that's exactly the equation over here the Z component of time derivative of E sub Z it is busy component of the Corolla at exactly what we have here we checked these policy which component will get worse see component of the year Lizzie component just gives us back were Maxwell equations namely busy component of our Maxwell equation they also the case which is interesting it is to right down the time component here all the key component 4 Our well let's go let's continue to call new no let's work our thee kind component of this equation wears street now 1st of all it has a term which is Debye of FTT at Newark was seat but 0 FTT that's right here as 0 0 0 0 and Italy have grossed debug be X of F X T the 1st debug of F. White Post buddies 3 of zee woes effort was F X 3 white earns each other just the components of the electric field F X T minus Wilmer assigned is east of X Aziz of the other 1 of these all this says 1 package neatly derivative with respect to x he ex post the derivative with respect Hawaii Of wiII posted derivative with respect to Z of easy is equal to 0 that reference is just Dell bodies 0 for the moment we been ignoring charges and current density Georgian cards so you concede that my bed told the said 4 of the Maxwell equations 4 of the 8 Maxwell equations just have performed a relativistically called variant even every reference frame victory equation that's a vector equations because the left-hand side is a vector it's a vector because it is derivative we've contracted the index is only 1 index left over so this is a vector equation it's the same in every reference point that if you went through the entire labor of doing the Lorraine's transformation was on the coordinates and on the components of E and you would find out that after Lorentz transformation the equation was exactly the same what about the other half Maxwell equations well both have exactly the same form accept but With twiddle here Of course they have exactly the same form as little as saying his efforts it would be in be interchanged an extra miners are thrown exactly what Cedar you go from here to here I the changing electric and magnetic fields with actress signed here equations are the same so this is a a powerful way to to rewrite Maxwell's equations again without charges without current densities This is a powerful way to rewrite them is simple elegant form but it's not the simplicity and elegance which is important it's the recognition that these are for vector equations and therefore the scene in every reference frame so Maxwell's equations then a relativistically invariant they don't change from frame to frame will cut back charges and current awhile but or maybe next time with time but Armed now that we've found Maxwell's equations reversing every reference what do they say what are the implications of these 8 equation what kind of electromagnetic fields they allow what they described so let's again forget currents in density for the moment 0 0 nothing and let's study these 4 enough for equations equations but study them and see if we can make wave equations out ever go back and what we did with a wave equation have someplace yes areas the wave equation is an equation with 2nd derivatives 2nd derivatives
mean differentiate twice Maxwell's equations of all have only 1st derivatives but they have 2 distinct field Ian be we'll find out that having musical first-order equations because they only have 1st derivatives but we have first-order derivatives first-order equations that and 1st order equations beat 2 variables within a fine out that that's equivalent the having 2nd order equations for either the electric field or the magnetic fields separately this is not hard to do this is easy to do so we need but only the clear blackboard board over here only to manipulate us a little bit I have been Belo mixed up but I want to write equation just for the electric field I wanna try isolate just electric field and see how it behaves so here's what I do let's say take this equation over here and let's differentiator with respect to time it's got only 1 time derivative in it let's differentiated again with respect to time so that we get a 2nd time there will be these
2nd by decree squared 2nd derivative electric field and that's going to equal Del crossed the kind derivative of B but differentiate with respect the time of the year adjusted the 2nd time the rural electric field of I differentiate with respect to TO here that differentiates the magnetic field with respect to Part OK that helped yet I still have electric and magnetic field the same equation the goal was to get rid of a magnetic field but look right he'll have the magnetic field audits turning derivatives written in terms of the electric field so I can be clever and simply get rid of BBB here it curl of 80 minus occur Levine missile it's just strips Butler's wasn't it was Dell Cross TB and now for we substitute minus curl of 80 gives us some minor can curl of each here we are an equation which only involves electric field 2nd respect feet from the left hand side and various complicated the rivers of right here so I was going to do is I'm going to take these acts component of this equation both sides of the equations are vectors he is a vector differentiating suspected time leaves of a vector and the try my dear could Kuo was also were vector so much take read X component of both sides X component of both sides what we have to calculate X component of the kernel of the world and slightly tedious operation if you like keeping those sleep for 2 minutes until I get the answer but I will do it on the blackboard for your arcade Dalcross double cross he the x component for ethanol is D X component Koral the X component of our curled is equal truly derivative Of the Zeke component with respect the whitey minus the derivative of the Y component with respect to Z that we are taking the x component of dough Cross they don't cross that's right so that we want the x component of dealt cross double cross it that means I just substitute a E were received as a substitute no crust EDT but 2nd again that's going to be by DYE Daisy which should be dealt cross E Z component of it best by of double cross e Y component of it but now we substitute our these compartments the same tried the same trick ends here is her real result the grand result Aerts baby why derivative UYA with respect to x derivative of X with respect Hawaii that's Debye DYA of this territory and the nearer to mark plus beebread easy our B X by D Z minus fees see by a V X and they think that's it that's it through terms pocket wake up now the worst is over let's see what we have were calculating X component of vector calculating the x component Rebecca the let's see what we have here we 1st of all have 2nd derivatives of E X with respect the wife who have 1st won't IDY square of the EX as sister and across of ours we've become off offer as theirs we by BY squares to do and it is but the white EX we have derivatives of respect z of E X plus 2nd removal respect busy squared of EX vests and then we get to other terms which local mysterious or are they they
are derivative will respect that acts of BUY EYE sister-in-law idea looks a little different because I've interchange B X and Y derivative right was derivative respect X followed by derivative lose but Hawaii you're allowed to interchange loans of the 1st 10 years debate EX Our BYU wide and that is another term which is deemed X I think it is easy easy money reuse top worn or Barbara looks very elegant profile is that Turner apart and subtract the sinter from the Barber of Seville will have form so let's plus Debye D X squared EX and that's now subtract Deeb ID X of Debye DX EX it added and subtracted exactly the same thing now you should be able to see a pattern of 1st of all we have white he white closed these easy distribute cricket or will sign mistake here we want you why these easy and be XCX appear inside he was that that's just below body but nobody is equal to 0 a former Maxwell's equations and so of this dx dx plus BY BY closed easy easy is just 0 what about this stir this is my 2nd respect why square x squared In squared Of the same quantity east Beck's let's put it together now let's put together what was it he is always good doing the 2nd derivative with respect that X Of he squared is my Bill Crossbill cross he BX comport and that's exactly what we computed over here have long tedious computation just give us all left-hand side erase it and was replaced by what we computed and you'll see a pattern here's a bike x squared EXE plus Dubai BY squared close debugged easy square here if you've kept track which you probably are not from the original wave equation will see that this is just a wave equation for the component acts EX is a field which satisfies Just Say no wave equation that we wrote down 4 5 P remember we wrote that wave equation for find it had exactly the same form and so it b at the electric field itself satisfies a wave equation the gold wave equation the same new wave equation which had right moving and left moving solutions which could be signed in where we learn we learned that they're all waves of electric field from property we've learned that wave equation which gives rise propagating waves a can move to the left or right applies already direction of course but which applies to arm of the scalar field also applies directly the components of the electric field let's see whether we can build ourselves an electrical weight an electromagnetic wave of this equation has solution is of the form for example X is equal to call signs of let's just take a simple example zeal minors this will be a way of moving down the axis wave equation has a solution which is a way of moving down z-axis workers coastlines you minus 2 years I could cut a constant K but let's leave our only complicates things this would be a wave whose electric field is point along the x-axis trust coordinates dizzy here X and his wife sticking out of a block or misses a wave moving down the z-axis its electric field His along the x-axis for
simplicity let's take it not have any component along the y-axis and Roxana call sign so it's electric field point up points down point up points down and propagates needless to to say with the speed of light Y the speed of light we've worked in units in which the speed of light is 1 and this is just a way of moving down these z-axis with you velocity Z proportional to t the whole thing just makes an excursion that z-axis without changing shape Betsy electric field and I've chosen arbitrarily the point the extra erection incidentally away even electromagnetic waves winner the electric field points along the x direction is cold a polarized waves polarized along the x-axis is polarize light polarized light polarized along the x-axis where about the magnetic field magnetic field B and all the way before we do that lets do we decided arbitrarily some set BY component equal 0 just do that we could tour y component in also but decided just to keep it simple was not Z component of of Of electric field where they prove to you that that has to be 0 has to be 0 Bacau Dell thought he is equal to 0 was dealt but it's the derivative I Ray X component with respect X but the x component doesn't depend on X plus a derivative of the Y component with respect the White also doesn't depend on white plus a derivative Lizzie component with respect to see these 2 terms in the divergence of electric field vanish so must the Swan what that if a derivative of easy with respect to must be equal to 0 and other words Z component of the magnetic field of electric field can not vary as you move along here it must be constant constant could be a constant electric field that's just throw away a constant electric field is not very interesting but certainly not part of a wave motion we could get rid of it Rabbi putting it between capacitor plates and canceling the uniform electric field the point is that the wave Potter pocket varies as you go down the z-axis must not be there for the C compiler so only the x component of electric field was there the Y component the Musee component which is not 0 it wasn't me the about the magnetic field well we can use this equation over here to tell us but something about the magnetic field 1 of the 1st of all what about the curl of the electric field the curl electric field what components Thursday curl of electric field how well the only components can I would have to come from something like D E X by Desi but that's the kind of thing that appears In the curl of the electric field where would also have minors a Z ID X this would be the white component of the curled but zeal is easy component is there we've already established that that's there so for example of this would be the white component of the year of the curl of electric field if there is to differentiate EX with respect to z you'll get an answer you will get an answer song there is a White component of the curl of and that's all there is no other component of the corrosive lead except for the white comport with them look for some other components for Z component that would be D X by y Miners DAY BUY X well he X doesn't depend on line and BYU 0 altogether there's no other components of it all only component of the curl of E is along the y-axis electric field is along the x-axis Zia axes and the magnetic field is along the y-axis while magnetic field along the y-axis because it's time derivatives along the y-axis the time derivatives along the Y axis of fielded must be along the y-axis so we see then that be magnetic field drove a magnetic field in green light perpendicular Korean draws it is a magnetic field hearsay electric field electric and magnetic field perpendicular to each other and there are also perpendicular to of away that's what follows from these equations also the equations has wavelike solutions with electric and magnetic field oscillating both going up and down except the magnetic fields going left and right and all things occurs preceding z-axis that's the nature of electromagnetic waves electric and magnetic field perpendicular to the direction of motion all movie uniformly down access with the speed of light what's more you can be sure but if you want to take this electromagnetic waves and look at it From another frame of reference your have exactly the same properties wide because the Maxwell equations are the same in every reference for soul the properties of electromagnetic waves will be the same in every reference frame and always have this property of moving the speed of light and having electric crossed electric and magnetic field perpendicular to each other and to direction emotion that's a character light when and all other electromagnetic radiation right now outlets what's on put back the things we took out mainly let's ask what happens were have charges and currents in Maxwell's equations how we describe it I would describe it end what is it do most important isn't Lorenzen variant of origins of the same in every reference frames that is Our major good the show that the laws of electromagnetism are the same in every reference so I have to do is remind you jet the 2 Maxwell equations on the right-hand side strictly speaking before Maxwell equations on the right-hand side get modified get modified by putting a charge density and I will explain what that means variety and Ciara here plus 8 current density
the current is a current electric charge so let me explain what those things mean the charge density which is called role symbol role I am a wide charge density has always had a symbol raw associated with it nor although I know why current as a symbol J. associated with goes way back love I think it goes back to date but not that goes back for me anyway Rome the charge density is defined in the following way says you have lots and lots and lots of electrons and they could be other kinds of charged particles protons even some neutral particles of doesn't matter but that space is filled with a lot of particles so many of them charges of different kinds that that you're tempted to think of it as a continuous distribution of charge the just like go waters a bunch of molecules but when you look at more water coarse-grained way you think about waters being a nice continuous distribution of stuff that you take a little volume of space the value a space what's call volume make a small volume of calculus think differentially a small volume was based on DVD and take the charge in that small volume was that if the volume of the space is very small shrunk the smaller a smaller dimensions of course the charter will also shrink as more and more than engines but the race what's call the amount the charge in they'll call at D Q. 0 a small charge in a small volume and the ratio of the amount of charge per unit volume is called the density so it's be charged for you volume accepted each point of space evaluated separately so it could vary from place to place you can regions were heavy charge density readers of white charge density and In calculating this you and up the plots charges you subtract of the miners charges all the obvious things had the proton subtract the electrons and get a net charge density in each region based on that charge density is itself a function of position but it also depend on time how they depend kind of example charges could move from 1 place to another this hope I will charge could longer year it which case if we stood at a particular point the charge density would change it would change from something for nothing so the charge density also depends on time now like kid Penn a old ways you is a role in nature and called the conservation of charge topple amount of charge and the world must not change so how much it was the total amount of charge in the whole world you'll end up all of the charges In each will sell of space all up and becomes an integral the total charge core capital Q. 0 Fogel charge in the world is integral overall the volume of space on the charge density Will air pledges made you add up all the cells in each cell there's a charge which is equal to D which is wrote times DVD and up upload wrote times Devi's that's when girl is it just means Adam all up and that gives you the total charge in this case in the whole world we cannot for example just talk about the charge in a region of space we could pick a particular region of space with well-defined boundaries this room or some other particularly regional space and talk about the charge in that and that the region in which case this integral wooden gold also basis it would just go over the selected region in this case for example and that would be the charge inside the room that charm is not allowed to just change some Place and reappear someplace else it's not was calling the laws of physics for reasons of the Comoros clearer set and the suddenly disappear over here and reappear over here Well what's will back step 1st of all the rule on Nature Medicine empirical long nature the empirical law of nature is the total amount the charge in the universe never changes that's called conservation of electric charge but the rule is stronger than that for example you can't have a situation where a charge and the earth disappears and instantly with nothing happening in between the same charge appears on Mars that would satisfy the rules of the conservation charge to map the charge the change was lost on Earth reappeared with mom Mars and the conservation of charged was not view that something has happened in between but that's forbid that's not the way charges change and that's the way charges moved they can disappear over here and reappear over here on Alaska's a full of charging between another words if charge disappears over here and reappears of India somebody in between had better be seeing a current flow between How we express this idea that charges don't just appear and disappear and a fact they don't they don't move from 1 place to another except by virtue of current of electric charge flowing from 1 place to another up so ferment we have to invent the idea of let me explain where current is too simple idea our purchase upstairs With challenge a similar area and space as a will area and space Murray Drive will area a small area and infinitesimal Lee small area we just as normally called Sigma indicating that small differential and Sigma standing for surface area so the amount the surface areas Colby stigma Boland people write these stigma mean something a little more being needs of that now how the hell can then Area B a vector well what you do is you take that area and you assigned to it vector perpendicular to the area for the area alone will decide in a few minutes which direction or do it the left the the would do it right but for the moment just imagine signing the Every little element of area vector whose Lane whose magnitude is area in whose directions perpendicular to the
direction of area that's called the surface element a vector service on vectors he said I could vector symbol above it but we are just be sir now we can ask how much charged flows through that area in a unit of Time How much charge flows the
area per unit time or better yet we can ask how much charge for kind per unit area flows through the charge falling through there will certainly proportional to the area of the areas a small amount of charge will be able flow through it for a given situation if the area's made a large a larger amount of charge me flowing through it the charge per unit area pro unit per unit coloring charge per unit area sorts write the words charge per unit area per unit time surface is called the current now the current as a vector the current as a victory you could see why that should be sold current is
kind of like a fool think of it as a flow of all the electrons and protons were flowing through space at any point the flow has a direction namely the direction of year charges of flowing in and also has a magnitude and the magnitude is the charge per unit area per unit hour per unit time flowing through that area that's the definition of the charm of current and as I said it is that if I want to know the amount of charge per unit time flowing from the surface and what I do is I take dot product on the surface element with the current of of them being vectors if examples the current is flowing parallel to the surface that means the college's perpendicular to be stigma In this product 0 Solis current flows the current flows along the surface no current flows from 1 side to the other end of the current is flowing perpendicular to the surface that means parallel to be stigma and then you get the full maximum branch of the flow through the up to to the surface element that's why about prop byproduct measures component of 1 vector in the direction of the other so this measures the component of the flaw in that direction perpendicular to the surface and it is not the charge 3 area unit time but just a charge per unit time this charge per unit time charge per unit time the surface element be stigma so I have the idea of the flow of charge and beyond sometimes called the flocks of charge flow of flocks of charge and current or electric car the old saying they notice would not necessarily talking about current pool wire a car and a wire is a special case and if we opened up a made do the big we could put lose surface elements of the area in there and see the current through a wider is made up out exactly these kind of currents what is broke charged to a wire but that's a special case we can have current flowing through space space current current flowing along sheets or just current flowing along three-dimensional space and that's the definition of day OK now what's concentrate on the following question here have a region but it has some charges and the charge America is integral over the volume of the charge density if their occurrence flooring maybe currents flowing into war out of the region let's shows let's chose to make up the boundary of the region as a whole bunch of surface elements at the boundary is made up by 0 means I mean the bank because this is a boon to the rubber of the balloon we would make up at a level surface elements here and now that by definition point there little be signal is out of show is to make them out of a book suppose angers met current flowing Out of the blue what does that say about the charge inside the and simply not a managing the balloon changing its sizes thing it's just a fixed region space if this current flowing Net current flowing out of every surface than the current than the inside must be changed the only way that the current can change and the only way the charge could change on the inside the only way the charge to change on the inside is passing from the inside of a outside so the charges is changing the charge and the inside is changing you could be certain that there must be a flow of current through the boundary and that tells you there must be a J the Vatican write down a simple equation we can write down at that time the rate of change which ends of the charge inside this region Q. by that's equal to the end there will be lover you roll by today the time rate of change of electric charge inside this region must equal the car in the passing out through the band actually would the opposite side of current is passing out the boundary with a positive sign then the charge was decreasing inside so it must have a minus sign this formula I want total charge that passes out of this region per unit time the public charge passing out of the region per unit time is just integral of the current darted into the surface we just ate up all current passing through each 1 of these service elements that seemed to grow here and this is a theorem now or few like perhaps it's a law nature rather than a that we kind rate of change of the current inside is equal to flux of charge passing out to the surface of a bloke that's given by the integral of a current now we need a truth mathematical theorems rather than a law nature mathematical theorem which won't cruel it's called doused his sorry is not called guesses what is called gasses look up Dallas's theorem sometimes called the divergent while tell you whether there's it's a relationship between these occurs is called a surface integral it's an integral Ave vector quantities over a surface of rebounding surface it can be related to an
integral although the interior and he is the of called gaps 0 because says that the integral are of any vector integrated over the surface His equal this isn't it the growth over the surface court surface is equal 20 Indigo or volume by Mike last year I would ask them what goes in there not now OK the divergence of that that divergent vector start to relationship between the surface integral of vector and the interval although the volume of the year of the divergent Our lobbyists from Margaretta prevented it's actually doesn't take remedial lines to prove but let's assume that still met them we can write the right hand side of this equation as an integral although the volume Hindu over the volume another and the role of the volume of the divergent of recurring it's a spreading out of the current spreading out of the current he notes like it's almost like this file give you a picture of divergence of vectors if you thought the vectors as a flow vector of some water perhaps our arms full vector of some water and supposing some water was being pumped Ian at each point of space some funeral pipes that you can't see bringing water in from outside and dumping it in the pipes aided by different places in space so the water is better verging How would away from the tips of the pipes and going out and filling up space the divergent of the water on the bridges of a flow of the water is basically just beat water coming out of these individual pipes what this says is that the amount of water that's coming out of these little pipes ending at this point and flowing out ways were adult has out of the surface vessel out of the surface and that's this expression he ever beverages be integral Over the boundary arm of week should 1st floor it is also illegal to the integral over the interior of volume of the divergence of the vector so answer general fears about vectors it doesn't have anything particular to do with electricity it doesn't have anything particular to do anything special other the mathematical properties of vectors personally have removed theorem that integral over any volume and think any volume whatever we had to go over any volume of the time rate of change of the density of charge is integral over the volume of the divergent Of the current if Julie in the goes like this equal but their equal not only for some particular volume but their true it's true that every volume that you can think of and there's only 1 way that that can't be true effect any region that you could pick me into go over the volume of 1 thing is equal to the into global volume of another thing the only way that that can be true is further things themselves to be equal that again as a mathematical theorem but it's more or less obvious that affect any region the interval of 1 quantity is equal to the role of the quantity 2 quantities have to be equal we can't do it in the conclusions about the connection between charge density and current it says that the derivative with respect to time of the charge density this plus bringing everything over the left hand side of the divergence of the current what's right about derivative with respect to acts of J sub X derivative with respect the wife of J White Post derivative with respect Z J sub z equals 0 not this is to be true in every frame of reference and I didn't specify anything particular about a particular frame of reference when I told you this little story be chill every frame of reference it better have the form of some invariant equations and it's not hard to imagine how you make an invariant equation that you 1st of all say but the current but 1st of all you say that the current JXJ together with the row for quantities Raul J. X excellent X upstairs now JAY Genesee former a 4 vector they formed a format there J. Neil when you goes from time off and the 3 components of space that is what is suggested by having these 4 components rope JXJ wife injuries and they end this equation which is called the continuity equation call the continuity equation that takes the a simple form that Dean New Jersey New is equal to 0 that say it says derivative respected T of JET plus derivative with respect to X J X Loblaw blob is equal to 0 which is exactly what the so the way they make this continuity equations BAA equation that's true in every frame of reference is to think of the 4 components of current and density as a four-day that's not too surprising current current represents the velocity of the motion
of the charges as we've seen over and over again the velocity vector Is it for that their velocity vector of an object the relativistic velocity vector of an object is a former actor and so was not so surprising that the floor fluid should also be described by a 4th and for that there is a former rector of current density well OK now we have any object before a vector of current and velocity that we can try put into Maxwell's equations somehow let's go back to Maxwell's equations which seems to have disappeared after blackboards sagas works here that have been around the end of it so it's right Maxwell's equations again this time putting near of the charges in currents amigo Dell thought he is able to roam 0 . be physical 0 across busy Hooper minus by IDT draw vector equations and bill Crosby physical baby city plus J. wedge aid is exactly the same J. that I defined here the current per unit area for time what I assert her which were reject these equations are exactly the same as the following equations being f Nuno whatever you know it is ever muumuu is poor by 4 and symmetric matrix of cancer which is built by a bout of electric and magnetic field components the fact is equal to Jane New Norris the left-hand side of contracted the new index so the new index is up really a real index here it just as 1 new index right him the with 6 OK let's check reduced simple called variant where this is clearly a core variant equation differentiating it can't give you a vector misses a back and sell this equation would be invariant saying every reference for assuming a J transforms as a vector so let's just check that we get the same equation a month and the quality of the ones the the ones it's relevant here oddly ones with the right hand side the ones with current and charge against there are 4 equations CFO for a different values of therefore equations here 1 2 3 4 so let's check to see if any of them others 1st of all let's take the kind component time component would be deemed U. S. new is equal to GATT always JT that's kind component of the lovely for vector on current sore right knee inside we have role for what about the left hand side doable left inside as we have the Golan rewrite the field cancer but we don't need the whole thing really need to remember years let's say that the army in 3 hour that we have the components of the electric field 0 X EYE and easy formed here we also have minus EX minus EYE my easy and then we have a magnetic field sitting in here and various ways OK so let's we have we have only the time components of af that means the components along was 1st role here to write this as D X F X Frost BYD FYT applause Desi and GET missing 1 yes missing 1 missing D T F T but FTT is 0 already bad elements of the maker of 0 no FXT FYT and those of a component of the electric field source says this is just D X he ex post BYE BYE was easy easy another words it Is No . 80 a left-hand side Bell but right here Rob Rico go through the other components words exactly the same thing instead of taking the time component here we could take space component for example that the steeple the J X and we would discover the x component of this equation so the upshot of rivers that the connection between charges and feel the way in which charges charges and current electric charges electric currents the way they generate fields is Lorentz invariant it's the same in all reference right on the other hand we've already seen the way that the fields influenced the charges as voices on them also the same in all reference price so we see that basically all of the equations of electromagnetism can be written in a form in which we can immediately checked the cancer equations and therefore the same every reference frame that was the real goal of this show that the laws of electromagnetism saying every reference frame No. 1 and No. 2 could demonstrate that the laws of electromagnetism lead polite waves moving with the speed of light from most of the things as follows light travels with the speed of light every frame of of reference that sort of closes up and this is up V Cast started by Einstein on rewriting all laws of physics in a way which is St. in every race In every frame of reference power a Dedham I'm her
a room the preceding programmers copyrighted by Stanford University please visit us at stanford .


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