Special Relativity | Lecture 8

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Special Relativity | Lecture 8
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(June 5, 2012) Leonard Susskind covers more topics in electromagnetism and relativity, including dynamics of the electric and magnetic field and the related effect of charges on the electromagnetic field. In 1905, while only twenty-six years old, Albert Einstein published "On the Electrodynamics of Moving Bodies" and effectively extended classical laws of relativity to all laws of physics, even electrodynamics. Professor Susskind takes a close look at the special theory of relativity and also at classical field theory. Concepts addressed here include space-time and four-dimensional space-time, electromagnetic fields and their application to Maxwell's equations.

Original­ly presented in the Stanford Continuing Studies Program.
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want to talk about before going into Maxwell's equations swift talked about emotionally charged particle electromagnetic field we have not talked about the dynamics of the electric and magnetic field was a field equations or for that matter the related effect charges on the electromagnetic field where do a bit of that tonight this time not starting with reaction principle of prewar will come back reaction principle but from calm but 1st I thought was jumped to some very formal mathematical technology I have occasion West weak actually while I was in Belgium visitor euro missus already was already hero family who are the people who 1st brought together probably the very 1st physics conferences that took place ever in euro making hundreds I had occasion to look at our our 1st paper I had that I had to struggle through some of the German are just the because some historical or discussion came up about and I looked at it and Harry read the 1st basic paragraph the bible a paper English is on electrodynamics moving objects and Einstein expresses I read this wall of gold worth more than 1 moves forgotten about Express is extremely clearly won his logic and was what is motivation like the back to it just discussed it for a moment because it is deeply connected with the transformation properties of electromagnetic field and I want to go through it if you have access starring 1st paper it's 4th plus worth reading you Canary I'm sure you could understand are bird nothing else just reading of 1st paragraph was it as it's marvelous I I wish I had here I would read you the 1st paragraph book Berber Adar abound but he expresses b puzzle which by that turned the longer puzzled him but the the puzzle of the year In the variance of certain electromagnetic phenomena that fact he described in different ways in different reference frames and 1 reference you will blame assert effects are magnetic field and another reference frame you blame it on electric field and he says doctors no real difference between the phenomena are viewed frames but we describe them in a completely different way there's something there that we should understand that we should understand the transformation from 1 frame to another and 2 different descriptions give rise to exactly the same phenomenon or is talking about relativity and the 2 phenomena he's talking about a single phenomena is talking about is of course the effect of electromagnetic fields are moving and stationary charges the easier us more explicitly talks about charges movie wires and of what's currents that doesn't matter much what the charges on moving in wires or whether it is moving in free space he might have talked about charged particles moving in space just maunder charged particles the reason we didn't was probably because at that time it was not easy to see of the motion of a charged particle was much easier to a push a particle for Wyeth charged particle for wire but in any case it was the situation was talking about his 3 dimensions of space off let's suppose that work recalled we could call X X Y feared if this would be why this would be busy Entebbe supposing or to correct however Murray notes of course is why which is probably inconsistent with the right here role but never mind are X for seemed there let there be on magnetic field point thing the blackboard another word BUY on magnetic field and only component of the electromagnetic field while would create such a magnetic field well it could be the polls of Mac the talk polls on magnetic car called over here and the polls Over here With the magnetic field in the blackboard North South Hong magnetic field Comey but I know it doesn't look like it's coming out of a black for think of it as our of blackboard along the direction wiII but there will also be a charged particle and the charged particle is moving along the x-axis moving along the X axis misdirection in our frame of reference although it is the
magnetic field and the particle was moving right now and Einstein was a regulation he really was thinking about 80 wire in this direction the wire contains of course we know what the wire wire contains however use the Adams mailed and place in a wider they can move at electrons and electrons and conducting wanted to move up and down so really was only electrons America Einstein talk about electrons and protons and neutrons so what we just talked about current a wire but it might be just talking about charged particle and then the wine is made a wise or the charged particle is made From here all the x axis what happened what happens is a forces exerted on the charged particle we call it the effect warrants I'm more words call repertoire and know about it it was the low rents Cross electric charge and deep across the magnetic field of the magnetic field was out of the blackboard the particle is being forced to move all along the x-axis there will be a force which is both perpendicular to the velocity and magnetic field maybe out fossil this way so as a 4 somber charged particle your Upward Bound a better man I am if you configure our instantly which way it is with a preventive care for the moment the idea is that the Pennzoil overcharges plus a minus it right OK but there's a charge or longer along Of those start course along the y-axis which for example would cause a current the floor along the wire that a description from 1 frame of reference in which effect on charges the effect the current is produced by the motion of the charge in a magnetic field by scientists as where 2nd will also now another factor this time keep the wire fixed don't move the wire but Mova magnet move the magnate and what happens when you move a magnet and you have a wire cracked well according them according to Arthur I'd starring and everybody else I'm moving magnetic field like that will create what they call the EMF an from electromagnetic force which effectively means an electric field that creates an electric field mode taking a magnetic field which pointed into or out of a black hole of moving the magnet creates an electric field vertically which has the same effect causes charges flow or exert on a stationary particle words if your station particle removing magnet it would have the same effect I think that is what we have these the phenomena saying of vertical for us we have to look properly different descriptions in 1 description of the magnetic field exert forces are moving charge and the other b molding magnet creates an electric field and the electric field exerts a force on the charged particles that total these are the same effect as must be the same effect the only explanation is what looks like a magnetic field pure magnetic field in 1 frame of reference In another frame of reference mainly in this case the frame of reference that happens to be moving with a particle looks like a bit at least a bit also electric field sold what said is electric and magnetic fields are part of the seem complex of objects which transformed into each other on there a he didn't call and Lorentz transformations probably would prefer that they will coincide transformations and draw up a motion OK so kindly in detail mathematically Ruiz transformation when we have a magnetic field of Forget charged particle now we have a magnetic field in your frame of reference is a magnetic field pointing and that'll just a magnetic field and I will along with the charged particle can we see that I perceive an electric field to we see it from a transformation properties on the electromagnetic field OK so for that we need to explore a little bit the transformation properties of electromagnetic field turned firm we did some of is that I wanna come back to it and talk about this particular effect God let's remember what the electromagnetic field cancer is 1st of all that's a thing with 2 win the FA Nuno that
happens just there just a pro you put everything on the table Nuno happens to be expressed in terms of the vector potential what's right although I don't think use it for the moment he you know miners being a new means derivative with respect X no component of a vector potential might miss the the river with respect X Noel of the music
component these you means Dubai fight that's E. lets us feel Cancer Center Asymetrix just work to recall a couple of facts about Nuuanu go from 1 of 4 they run from behind X Y and Z kind 0 4 0 1 2 and 3 and half naught it and that's the kind of mixed space component or directly misses 2nd Sowerby diagonal elements of this a 0 the mixed space time components are electric fields will end and now are ordinary spatial components muon know are space him women in our spirits price that's that's a connection between the field can serve as much appeal cancer and electric field happens from this equation here by the time the roof of a N modestly not sorry and in the end In the Celeste their side has a variety of Cyrus man comes the prank derivative are ah sorry Pansy and derivative of pain North pain or just rewriting this for the north 5 locker users particularly the expression in terms of that turned to register a mark your best electric field and the magnetic field is related to the space space compartments and end the connection is that the magnetic field equal curl Of the vector potential when restricted the basic components this expression here is connected is simply vehicle components of the curl of a vector and so the corresponding Bureau of and In the magnetic field is of space component of the potential for example this would say for example an example of that would be that beast of X is equal to the derivative of respect the wife of the Z component of a vector potential minus the derivative with respect B BY component of the vector potential This is the shorthand for these equations 3 of them for BXP docket Narka summarized the connection between the electric magnetic field and the field cancer by
writing down a big matrix of 4 by 4 matrix and filling in the components with electric and magnetic fields so just who write about once I know we've done it before but with your anyway you know was a big matrix product that keep track of the elements we will let new war rather horizontally and you will run vertically but sustainable right they have the 2nd element he runs horizontally the 1st or runs vertically then the 1st weekend are just show organized the matrix has put some guidelines in it's a former for matrix I if they Barak enormous 0 0 romp in the upper right-hand corner here at all just mind you the 1st role here is the kind component for the next 1 was x y and z and likewise for the role component ex-wife z tying the mix based on components are electric fields so that means along here have electric fields he X U Y an easy minors he exited by cement that might why and my easy and then other components for space-based components are components of a magnetic field and those get filled in as a white component not worth component tulips sorry Zeke component why is it is component they cost it is a X Y component of X X Y component means is decomposed suffered miners beasts of Y and B sub x and down here mine as busy as he and by symmetry Watts Laura half of a matrix is simply reflection except for my assigned plus beast of why might be so backs Annuzzi electromagnetic field considered incidentally just but through the 1st 2 years I put Euro exercise filet MYX for it as you know what operating overseas the rule is whenever you go from a lower indexed when upper index if you have raised but kind component you get a minus review of raise the space component no minus right so in fact it's easy to work out what's best new Norwood upper components it's a fact exactly the same except for a slight difference to the lines in here With the only difference is that this is my 1st EX minors you y minus easily EX your wife why did that happen laughter because C elected component house we want Prime index in 1 space barracks so when you raise the indices you raising wunderkind component in 1 space component minus signed the medic terms are saying are unchanged easy might be and BX you could fill corner here it's just all right so that it doesn't matter which 1 of you to use your blanks stretch the euro versus feels Hatcher said the argument that components are based space components peers base so about me as a wherever they want us to take advantage of some colored hand and block out he yeah the space-based components trying x y z I was freebie green matrix I use the space space components and they could assist only of magnetic fields and zeros be mixed space time components are the ones along edges here OK now what really wanted that we want figure out how this object transforms we from 1 frame of reference for about another word given that an electromagnetic field concerned 1 frame of reference which the electric and magnetic fields would announce year the final moved relative to return what they move along the x-axis for simplicity of our move along the the x-axis what are the know components of electromagnetic field pence a new frame work goes out we simply have but remember the rules transformation or cancers of things with indices pensive adjust things with industry go back 1st through simplest all for the simplest of all kinds of course dealers is galas don't transformed the tow simple but for actors and that regular for XUX new Prime x y z TXY saying we are they transformed the fact that our transformer rents transformation keep trying is equal to teach might be next all square for my be squared ex- prime musical acts seat on a square 1 square white prime y z prior Nichols bowler can organize this tho a matrix expressed or from the right two-way wicker right using Einstein summation convention reintroduced a matrix called hello hello lower rents mule Noel X you'll remember we start Rice's short hair for this ever for equations here summary although no and we know that we can reel off the compound or we could do another way to think about say the equation is tho lineup X assist X 1 X 2 X 3 x 4 now sorry and 0 X 1 X 2 X 3 and trying the hall of fame I
don't want to put the in here this to really make it look like the 611 Rex 21 23 in order to that just formerly outside a bar and that's given by some maker L 4 by 4 matrix whose components of called New York Times column vector X nor X 1 medics who works for a day requires and they could sell read off the Matrix showed read matrix out of the out from this equation here all right back before you matrix L that's associate of Lorentz transformation is again a 4 by former matrix I put lines just a garden I am good at keeping rows and columns of meat born so Oaxaca opined 80 1 all the square root of 1 might be squared he was 1 all the square root of 1 mind be square he is a V all the square root of 1 might be squared also hear via over square mile square 0 0 0 0 0 0 0 0 0 warned warned sorry Toyoda said Specter said was a minus sign him because as I witness said that's correct of course the minor starring formally here if this
matrix was multiplied by X nor X 1 X 2 X 3 will also move 888 T X Y and Z decays will also known as T X Y and Z it simply reproduces the court before equations over here so this is a matrix which represents the low-rent transformations are velocity v along the x-axis by now what early duel with a cancer like if you know how it transformed I will take the council would be opera br Parisian of its white just because I already have rules of things of indices are written about how are you transformer thing with tool operate overseas it we not a transformer thing with 1 offer index is a transformation for the thing with 1 the upper index and fact it on your mood changes and makes him 1 of course said saying the summation index soared doesn't matter what I call at ports the right order right down 40 0 small obvious what do you write down that path you know trying the field cancer in the normal court units in other words in the moving reference frame here's L mule said not exactly that same object couple blackboard that's how the new index that's up that's associated with the transformation of the mule index the transformation of the new index is you put another L as planned and introduce another but some additional barracks where you multiplied by you multiply by death stigma another Lawrence away the meal index transforms it is exactly the way it would if it was only the new index news they might have said the that no index transforms is exactly the way new index would transform Newtown F can when Botha this is a president that very simple rule if you'll had an object with more industries you have a number the the rule would be the same for every index to transform the same way that sets the rules for transformation of cancers and now let's try about let's see if we can figure out what the prime cancers let's see if we can discover whether or not there is an electric field along the 8 x x 0 so Zurich the electric field 1st war we were we have a particle moving along with Texas year we're be field coming our broader y-axis normal Melbourne removed so we should see a Z component of an electric field the question is will we see component of electric field so let's try West had computer as a component of the electric field in the find reference frame we can call heap crime he Z car but nor was it was it easy is really F Norwalk disease Best the Odyssey is easier Michael about signing signs of 2 were complicated for me to keep track of the sea the plus or minus system structure of import right calf naught the city is Z component of or during the next space time component our is electric field behind version of so that's what I would like to compute now let's go equation and computer or we know we know something about on prime concern but the on prime cancer but a pure magnetic field all longer the wire axis of pure magnetic field along the y-axis be on prices so all we know that right against aside the effort said he now has only 1 component the more it F decks Brazil F X Z resort correspond the B Y F X Z is BY ONE inside will be asked X zee there will be a at also and what's now Magill mysteries Morris this newly next year matches the new index up he were looking for F norms either had better be on North index up here what about you know is associated with the 2nd index here and a vast abuse ERP here what about or industries stigma and how well they also have the match the indices already here so that means this want X and this warriors Missouri she quipped what happened the other some but you were supposed to do to posted and these formulas for well astonishing collapses to a single terms for the simple reason that this there were assuming there is only 1 component of F year namely the component F exuviae always looking for a 4 1 component on the left-hand side which I'm the computer particular component of our hands on left started and they all really is and egg zeal the right-hand side river only be components sticking out of a blackboard right now let's see what it is here's our matrix produced in x y z was L Z Z West as warning the current els easy is Warner think I erased just 1 over here but what about the mixed from Parliament naught acts as this 1 over here nor what X this 1 over here has a minus
V all the square root of 1 might be square so this product is not both friend vii plus Maria Stockport v over square former V. square and a Z goes along for the ride now about f Exeter's EU What was that that's just a magnetic field a we started were right this is just Lisa original beasts of what's the left sorry he left hand side is the electric field along the c-axis along with actress electric field Rosiak stirs Ian the moving frame of reference store particularly we see snobs euro we started with an electric field which was you're always and magnetic field only a warning VRB grew up y axis we subjected us all through a motion along the x-axis we discover but there is In fact it is each component of the electric field in no frame of reference but no but references we electrons or whatever it is is not moving well I assume that I assume that I moved fast enough to keep up with electrons and therefore there's no magnetic force its at rest my in my friend my frame moving with electron arrest but nevertheless there was forced on it and forced on it is electric some electric forests which is a fact proportional Bossidy as 1 might be squared overseas squared this is something very very close to forms of Austria's Barbara montage and so we discovered in the moving frame of reference the same effect as that would have the effect which in the stationery frame of reference was due to a magnetic field and the warming charge because an electric field acting on a stationary charge that was for content if you like of the power of the 1st paragraph of aren't starring paper of course not content was the question the rest of the paper was answer and this was the OK so that the Laos that 14 minutes smoke reports what quarterback Franken years Violette's come the Maxwell's equations now let's come Maxwell's equations I'm not going to derive them today from their action principle if we went back toward a standard methodology usually we know what the answer for the equations of motion and then we look for an action principle to describe week could go the other way next week I will go the other way and starts with the basic action principle and the rise of the equations of motion of electromagnetic field after was nor will the moon Torquay rapid field equation not the equations for the charged particle OK Iraq tool kinds of Maxwell harmony in Waxhaw equations or other altogether entrepreneur 8 hour but they can't combinations vectors outpatient converts before a 1 2 3 4 5 6 7 8 yes are these equations half of them foreigners consul free come free they are they are consequences of the definitions of the electric and magnetic fields in terms Of the vector potential promoter words chaos of the equations are simply identity used far from these equations together with someone I think duties some that their identities and your identity is an identity is ready a mathematical fact richer just follows basically from a definition but sometimes even know they just fall from definition sometimes era you know who but not obvious OK so Ali Sarwar told basic identities the 1st identity involved the divergence on curled they now does not have to be the vector potential 8 stands for 83 vector only the 3 components from armored a stands for 3 vector and yellow crossed Eddie Izzard Kerr all of a sudden free vector which is called it a as I said it could be any vector field has to be a feel because we are differentiated but it could be any 3 vector fear wherever doesn't have to be the vector potential is everybody and not all the definition of a curl so everybody knows the definition of cruel their virgins ingredients server 1st identity is was a major of this object Watson major curled may took froze it's another vector another vector field curl is a vector field and as such it has diverted . maybe I should put over here definition of Corolla by Virgin mistresses were used them the curl over vector can be a specified by its components for example X component of the Corolla AP is by definition BY a Z minus D Z a Y N cycle of going from 1 the next S X or Y and Y to you could work out the other 2 components our long storm are just cycle but extra white but why gurus woods ego X cycle through them and you could ride out the other 2 components of the referral definition the chrome off divergence of a vector any vector now don't drive Fahey Freddy right there is a scalar quantity and given by a derivative with respect that X Of the x component forced derivative restrict YEAR component but the zeal of the G component the last concept is a gradient of the scalar which you don't worry well the district just writing out or have the gradient of a
scalar is a vector it's a vector whose components for example whose X component is just derivative of U.S. with respect X and likewise for Missouri alright to review the identities of 1st identity it I'm a curled his always 0 9 Penny approve that you prove it was better writing hold them off Everybody wants to know nobody wants a seat right now not you just literally right Our would be components of Bill crossed Eddie and differentiate them according to the rule for AT divergens and you will find out but there's a whole bunch of cancelations typically terms will involve something like the river respect that acts as a derivative respect why by DXT by of some 80 something minus the derivatives the same derivatives in the opposite order and you'll save about 0 so just simple identity diverges of leak crew is equal to 0 and the other identity is that the of a gradient 0 given any current gradient crawl Close DOS this is a vector go as director of the road from some scientific indicated that vectors s is now free vectors scalar crawl of gradient is also always 0 in particular I emphasize this is not a statement about any particular vector field would correct the prefectural it's an identity that's true for either as long as it's different from Bull and nicely smoothly behaved so that you could do derivatives OK these 2 identities are enough the proved tool or half of Maxwell's equations without doing anything else they number words they follow as definition server follow from that position is let's begin with the magnetic field which is curled they and let's take the divergent of the magnetic field but stated emergence of a magnetic field dell . beat as the source of the magnetic field and the cents Ahmed there Barry reduced the outgoing field tendency of the deal to spread out from a point show BA is equal to Delgado croat from the definite from the simple fact or from a simple definitions of the magnetic field as a crowbar vector potential right but the beverages of any curlers automatically 0 so without any further work Nolde dined their mixed nor just Pugh definition of pure kinematics of your wife the divergence of magnetic field 0 was just to get ahead of me get get ahead of ourselves from moment was a divergence of electric field truck and electric charge density that particular point charge has a divergence a spread of America electric fields were out among other things this would if it could be trusted that Burma that they cannot be magnetic charges magnetic field has no beverages and so were free store configuration with a magnetic field coming in our point for government medic Mama the Wasatch affair that I would say something's wrong with this equation Of the something wrong with this equation there must be something wrong with this definition so from plastic area straightforward point of view this equation says there foreign or magnetic charges now we don't have the comeback at some pouring understand there are ways around this bird simplistic weight or simply This is a statement but there are no sources of magnetic field known another equation 1 of them this is services 1 of Maxwell's equations is another 1 in which also follows for free but go back a definition that be electric field the struggle over here religious right as electric field equal to Kinder a vector potential that weight would do well maybe I'll just writers Debye D T V by Didier array but Martinez no derivatives I was a Norwalk with respect the exit and that is just equal to the gradient of a noise from the free Scholl point of view this space the time component of a could be thought of as a scalar last scalar from the point of four-dimensional vector spaces but it is a scalar from the point of view of 3 dimensions these vector whose components ah derivative of a nor can just be thought of as the gradient of 18 or so have the electric field 2 terms and 1 the time derivative Of the space component of the width of a of the vector potential and the other is the gradient of the time component OK now let's let's consider curl are the electric field with curl of the electric field is 1st of all crawl on the time derivative erect venture what's a 1st term but also happens to be the kind area Over curl already what is that because you could interchange derivatives when I think derivative maker crawl I'm Our rest I could the change the space derivative with time derivative and pull derivative on the outside so cruel the time derivative already the time derivative of the trial of a more over here have minus crawl Oliver the gradient of a norm the Corolla any gradient it is equal to 0 so we get nothing from here computer curled I get
nothing from here and just kind the the Corolla what's acrylamide could is just a magnetic field song here's another Maxwell acquired don't curl the electric field what's right president of minus the current derivative of a magnetic field is equal to 0 both are true Maxwell equations but as a shared from the fact the electric and magnetic field are defined in terms of the vector potential there are no more Maxwell equations about how equations however you really many equations are here what this is a vector equation has 3 components solo this is really really independent equations and this is a 4 point so therefore for for independent equations but of course everything Rev. vector notation 1 equation which is a scalar equation them and 1 more which is a vector these are I'm not sure means on not because this is missus this was born here for Macs were the Iife Maxwell equations alright family and this half it is half wit you derived from the reaction principle Mr. whose dynamical ahead that is not defined in terms of requires not purely definition it could be different if I chose on different actually God could make the 2nd half of the Maxwell equations be different but actual equations as Maxwell go there was and highly analogous to reduce basically with electric and magnetic field and changed but not quite so it's right down the other 2 were equations 1 of them involves the emergence of the electric field so we know what the beverages of electric field this it's called the charge density where I want to spend some time in a few minutes talking about what charge density and current are but for a moment and is writing down the equations a car formerly the right side is the charge density which means the amount of charge per unit volume charge a small volume of space provided by small volume of spirits from right here here and the other equation looks a lot like this 1 will even be interchange but not quite as Dell Crosby plus DDE by duty nor is it a quota but her the current density I will tell you in a moment so far were discovered during means we're give it a need all the required jury know what kind of quantities J well the left-hand side is a free that vector electric field as a free that it's time derivatives of 3 vector the car over 3 vectors also or 3 vector Sergei is a 3 vector it is a three-dimensional scalar now not format for sector language more complicated and for other that the language we will discover but the 3 components of JT end the raw together 84 vector of 4 compartments 3 components of J warned component the role or illegal using is called a current density but electric condoms role called be charge density charge density and current density so let's talk about their here and there but may explain what they are very juries Aurora's quote cool just as role renew here but at rail to pools fur but teaching quarrel care we want pool what you'll know what new rule next quarter ever crazy teaching assigned Shekhar entire volunteered to do something which I know was going to be incredibly frustrating to teach quantum mechanics in the mathematics department well possibly from a operator of physics the problem students who have battled mathematician Condon Peak statistical work ethics if not be refined archery Sunbelt decided to do this job is not the riser courses lead the faculty the touring quantum mechanics they heard the 3rd trip to carry out the stairs said Edward told them they have to understand quantum field theory if they want to understand very basic girl modern mathematics a good believed so we try to rubber-stamp can't field theory and find out that because standard the throughout reason we can understand is because we don't know anything about quite character please help a prestigious 1 mechanics while giving perverse together but the gap or cake such a charge density and current density what are very so explain exactly repair 1st of all charge density is exactly where sounds like you take a regional space I'm going to draw some pictures here as time and the other 2 0 directions a space problem here is of course there are freed directions as berries and could be a little hard for us I'm not going to drive a draw four-dimensional space-time blackboard so hostile mergers that these 2 directions Our stand-in 3 dimensions of space and 1 direction of time by Whitacre will sell in space of as all square here but of course it killed as square ridiculed the secure because it has 3 dimensions and therefore it's a little volume known for small volume element we imagine charged particles charged particles have world wiII so draw from war warnings and let's imagine that Greece and time off silver with a kind of a bundle of charged particles moving so a flaw in a fluid on boon or use you think of the world lines that pair I told him they are passing through all this will look you know what I mean that means but and instead in history and since then the particles which passed so that killed us simply the particles that Cuba bacteria that instead of time all of particles of charged particles that could be electrons for example with call electrons in there we count all the charges we think of it as a toggle charge passing troupe that Lula volume of space passes through a sense they were in the past than they were in the future but at a that we drew that will love volume element they work in the volume element so you could think of as passing through the bars of the pulse charged in that volume are you provided by the volume element is charge per unit volume and it's Rob wrong is equal but scored With George within the volume of prevailing Delta killed don't be silly and that that the volume element
goes to 0 with our rafts however test more while the volume element in the past will charge in there that ratio charge for you volume is called the charge density it is literally the amount the charge for you volume now about the contents of the current density it simply this thing pronounced now here is where we get the a problem the vertical axis here this time the horizontal axis the tour started Toro sample of sorts was right was pioneer exist very very like Pa this is the X factor to the X actress I guess this year's stems from a Y and VX to use the kids rather modest coma white z-axis but so what think or will sale here in space and time which is perpendicular to the X factor that means it it's a little Cuomo be based in part space time a Q it has did you all the time taxes and has 2 edges along the Warriors reacts but they got put over here and draw purists space diagram X Y and Z this a job year really spareness for a tool dimensional Squire b y and z square it's a little square in the so a his solo school Aaron Wiesiek plane we can ask how much shot thing was a we think of it as a Windows or we know they're afraid of her onto the big trouble draw window on a wall here he is aware of this thing here a window and that we know is a longer y z-axis threats for that yet whether I guess the window here is feeling sorry the window here to the other problem is is None taxis were drawn many cases a window the window in this case is perpendicular to the X-axis that yet of the XX is outweighed the windows in a wise players like that and now I have a question how much each orange passes through that we know 0 per unit time the amount of charge passes through a small window per unit current another words it's the charge passing through a Polaroid unit space per unit time per unit area per unit time and that the charge passing through a unit area in unit price Walpole you area per unit time let's the precise NOR turning this outside this cell was perpendicular to time this cell this Troy spatial broke perpendicular to the X started has prime component and a white busy component North already so such trying it on side that's called jury J sub x defined J sub x you take a window perpendicular to the x axis perpendicular to the X-axis euros from much charge is flowing through it from the negative XI the depositor excited how much charge flows so it Our unit Party JIX it is orange area unit trying volume area 1st and this every year isn't area perpendicular to the X-axis emotive Hawaii area but perpendicular to the X-axis that's J X What about JYE injuries JAY of the same thing they amount the charge per unit area per unit time passing through we know perpendicular either through the direction or are busy direction Jason V Y Andrus from GE the formula as a same peculiar Delta Delta except those that a Lulu window perpendicular to you the wife of a z-axis that's the definition of Jerry and the fact JXJ y Genesee and roll former complex of things which in fact or for vector will see there will see why I say fortress of after but there are 4 wrecked their parents they enter until Maxwell's equations where were Maxwell's equations the we according Ontario where young allow such era the Jays a kind of floor charge they represent the flow of charge from 1 place to another Of course these windows or through windows are sorry X component of Bill Cosby boss a apart the river of the x component of E is equal Gerber floor charge trolley X Window a sign 1 more about the charge about the density and current director which is report that Chile can be derived from Maxwell's equations Yao we can derive from Maxwell's equations I won't buy what you do it the right Maxwell's equations but it's really just charge conservation so let's talk about charge conservation yet that I virus Steger a break for 5 years US request to rectify student so they war of words are almost was Waslaw direction Oh yes but that's the way to think about your right so you could think of a special volume as a kind of we'd 0 perpendicular to the c axis to think about that wary and learn how you look at the port How does a lot less than a 3rd of the year the Europe yeah I I think yeah I I think arrest sensible questions but I would rephrase I would ask whether Rockport in jail on call area at the height of their for vectors will come that has defined a former component hurry up contrary Oberon next to our words I will see what this is enough will see that the complex Raul J. XJY but committee thought of as the components of a for vector called JUU were next counter wiretaps true so the study on duty relativistic formulation of the Maxwell equations poker with ticket through right Obama happy goes could answer questions but I want to Kure Britain lectures from Kabul my some of what you want PUTS of user you know have argued that show orders about black holes at the jewels all used the 1st walked by yeah it for what it is right now outside walls 1st of all I is is slowing down stop of what we asked me is it that In stop and you see around like old basic areas at all on a call of a flat yes that would it would he also got the forms on a black hole wild-card forms Kern sedimentary layers With comically slowly slowly slowly gravitate to the horizon quantum mechanically just laughter but well it
also slows down Her also slows down which means that the radiation coming in off by what the file of this informing 1st and Iowa sending out messages but suppose I was sending out messages with some frequency some as Iowa approach the horizon and I would send out messages in a way that sounds to me normal but you far away it would be rejected red shifted means that you you really are the frequency would give lower and lower and lower 3rd of the frequency of the sounds of was making Juarez find frequency they're basically would be all last way you but I would send out before your before the end of the message markets and more waves I would increase the threat of any given frequency our new would basically be all last a bit of a message that our concern that are sold for any finite falling in with a finite frequency with a finite amount of energy available and therefore can't so for our it will slow down and get rid of a and render and renter my flashlight I send out with optical radiation will look to you as our fully Iowa's armed infrared formal further structural required microwaves coming out now look like it's microwaves coming out me on sending out perfectly over good flashlight like you're seeing infrared microwave radio waves longer more Waverly radio waves of photons coming carry less and less energy and saw affected black is as does the approach he Yosef but for now wait for more complicated because you meant a rise in temperature and sorrow is ready lend radiation gets mixed up with the Hawking radiation the comes around becomes indistinguishable from the storm always level where were studied radios are crude lost their lost long before that I R the force not to request Framingham the requester issue we are black or it is our 1 proffer transit time crossed a black or transit time course a black or brown rice black has a certain radius called which children this time associated with the transit time across that they are just 1st we do you think are provided by sea yet are while I see that gives you a pilot and every such time Our it turns out 1 full-time now that's not a lot times 1 vote per cent is Our vary very weak of radiation but on top of that the radiation gets a longer and longer wavelength so that means the black or was it radiates from talking a Hawking radiation sends out longer and longer wavelengths and 1 per Verizon crossing party and that really just gets to be negligible from the point of view of anything we could reasonably measure so sorry I have to get quite while she says a this the like of a bar you're black or evaporate shrinks it speeds up duck yah yah yah OK right with defined J. so we know what's on the right side of this equation an interesting exercise fatah work our ideas most obvious from you work at our from this equation we could compute all with North new punch line go back I expect we will you warnings the conservation of charge concentration of charge it is so what says it is probable charging universe doesn't change with time but has really the concentration of charges of detailed law than just saying charge numerous doesn't change it doesn't change and it doesn't change in a special way special away or if especially if a foreign if I take on regional there's always a consolation saw the region's space from interest in and look at the total charges Ellis code Q Q was a standard terminology for electric charge Colicchio let's suppose after a small amount of time charge within this region change may be decreased a little bit better means the charge someplace else was the includes Brecht's controversial draw but the concentration of charge would be very helpful to us if it would no rules about where the charge increased for example will be perfectly consistent with the conservation of charge of every kind the charge change box over here reappeared instantly at office but then from point of view of all laboratory experiment it would be tantamount to saying that server conservation charge would need some do an experiment involved the alkali Durie the do experiment back office Opper comic lost or want conservation wouldn't have much by air from the rule was simply that the quantity total quantity of change the real rule is that the charge within a region can only change if it passes through the walls of the region another world Summit is spinning and monitoring the boundaries of the Xbox will if the charge on the inside changes it will be due to the fact that they discover charge passing through the walls of a box that allows a lot the ruling so lawyer says is the changes of charge in the interior box must be accompanied by something or other having could do with a comment on the boundaries of a box of awards changing Chavez take this to be a small box a small box the charges are that small box is basically the volume of the small box thank the volume to be 1 of those of the writer over and over again and let's say the volume to be won but think of avoiding was more open some units were small volume just the unit volume and then what is the kind derivative Of the charge in that knocks Whoa charge in that box is just a charge density if I take a volume of the box to be 1 of the volume of boxes warned charged for volumes in wrote it is just a charge in that box that kind of a chart charges of robot time derivative raw White conservation all week called the local conservation of charge the idea that charge can't change with volume on West it's discovered to be passing through walls that called local conservation local conservation areas that rolled down somehow be connected with which means equal to something or other above the boundaries of a box OK itself suppose role but said the case would inquiries supposing the charge increases that lets me the sum that floor charge coming here the show flow charge coming and must be disarmed of the charges coming in this we know because window plus charge coming in from this will know that window plus ones in the front and the back broke so let's consider the charge coming in through this window he was that that's the value of its figures to be X-axis rapidly the x axis vertical 1 will be a Y axis for G G the
charge coming through this window over here is proportional J X Q now flat X is defined in such a way that it corresponds overflow from lower values of larger values of x particular the charge coming into a year per unit time through this window is actually minus J X and my ears at that point for neighborhood of a point of some trauma flowing from don't N them what miners JEX over here warm over here what's the charred flowing in from the the Jaeger posterior X promote the same point JOX as somewhat displaced point displaced extracts of fighting this contribution and that contributions corresponds flow of certain derivative on J X the derivative the difference between VAX Over here and over here Her is based going the derivative X with respect that looks like my derivative of J X with respect to x that's contribution from the toll faces which are perpendicular to X 2 faces which are perpendicular to X the charge per unit time flowing per unit time is proportional from my the removed JXO of respected acts What about the charge flowing in the box throat white faces who faces perpendicular poor y axes strengthening my ears derivative of JAY with respect Hawaii our worlds John flowing in he had chance in here because of the opposite sex a direction of different my sister while purple-white finally minus derivatives Our of Z were for starters it misses the net charge per unit time flowing in the box by flowing through these 3 kinds a Windows this must be equal to I change of the charge in the box so we have an equation Arroyo . is equal from my guests from 1 of the best this is the beverage of Jagr show but Jerry for wrote Dr. Cross go about Jerry is equal to 0 that equation is Our local fully neighborhood by neighborhood expression of the conservation of charge it had strong the charge with a small region will only change duo the flow of charge out of that region floor charge through walls the region Price idea off to equation is known as the continuity equation continuity I want figure Rosa continuity equation because that is the only sounds like something is necessarily continuous which not I worked for local conservation conservation both strong form of conservation which does not allow a charge visit pure 1 place and reappear another place on West in between the flow charge card OK so this is the continuity equation and it's another 1 of some other equation has to be and added to be be used but in fact it really doesn't have to be aired it's a consequence of these equations Hawaii I suggest to you the following exercise Healy have an expression for a role here we have expression for right J. you can't take that kind deliver the left-hand side give your or not you can take the beverages on the right-hand side that would give you the emergence of jet so if you could prove that as an identity that Raul . Hawesville by is equal to 0 another words that Maxwell's equations required be charged to be considered it's such an important thing so attempted to see if I can do it on a white board is really central freely central charge conservation is not an option that Burma it if it follows From Maxwell equation I will let you know it's fun that's a little bit of an exercise it uses a couple of identities your with star Eds fully central grew Hall were to hold face we have Maxwell's equations end the real questions that I want to address is can we express all of these equations and a relativistic notations which makes it clear that collection of equations is invariant under Lorentz transformations another Watcom we express them as tend to require Asian equations with indices upstairs downstairs with the rules of a Our index ology yesterday we're related show the less weight of reform that's is a far far no we have come to yet we have not we're not know exactly known as right away well this is target this is the part of this in Maxwell's on or our let's see on yes this and this 1 on the fiery Saturday they also know that Bill Cosby equals J. but he didn't have a word about the case where was independence this term here was invented by your body Maxwell without this can you would not discovered thank you discover that charge conservation for this terms of essential the prove that wrote about was built by J is equal to 0 so that Macs will is he know about charge conservation storms surrounded apparently they know about charge conservation they know about the continuity equation and that's what realize that he has to put this turned and in order for charged because With that would work on Lithuania this Tamil here so let's that my CERCLA 1st worked OK let's see what we can do about making these equations relativistically invariant from words just relativistic nor patient well this trauma here is pretty clear that pretty obvious but 3 right it performed as easy as they rolled off it is deemed by D. C Our what's Colorado that Israel where markets was former Class D by X of J X Class B by team authority by YEAR JYA class by Yi college is now so far I've not been thinking about for vector rotation and so forth new thinking three-dimensional really so from the three-dimensional point of view there was no difference between upstairs index from downstairs barracks fully look at this equation it becomes natural think of it In a tension which is so manifestly Europe's poorest unless 1st all differentiating with respect win X but some downstairs America Ford but if we right the index upstairs missed this port cover scalar obstruct save more completely lets final raw Kabila kind component of jade George take JXJ injuries put them together with Raul Roa J X J Y J Z I'm putting the index you're upstairs but doesn't matter because three-dimensional and this was would promote about but then together and call the
whole complex of 4 objects J. we're upstairs and next thing this seek ways and has performed on a 4 the additional relativistically invariant equations DJ New by DX sawmill although new Einstein summation convention equal 0 so it's natural for the takeover rolled together with Jerry as a component of for vector and 2nd hard to prove that they really do transformed car as a component for better or worse I try to do at the moment just make use of vote of notations Kirk take it as a very strong suggestion that the components of rowing GATT form a single complex knock off they do it with a fine pretty much the same way you do we in various directions space and time and ask about the charge pressing of so it's a rather met with respect is based on the fact that symmetry this just the statement that the complex right Jerry formed a 4th got his fears that this a move they fall nor is far from this part would always partially lawyers partial derivative why was it means it means a stay at a point in space a little box and U.S. how much charge is a sign that region changes now we're not only the bark from place to place was keeping the box fixed that means with differentiating with respect timekeeping x wines effects particularly box were not moving it was saying What's the rate of change of charge in that box that's a partial derivative because it asking how the charge of the charge density changes if you very kind not varying ex-wife wrote for young wrote that William robot partial river young partially right when are following some trajectory we're just looking fixed box so upset robot will doctors meant make literally a partial removal at the time of 2 good point OK the the rest remember also partial derivatives so now we have our we have the elements necessary tool we know that easier and v former eh entice electric cancer with 2 indices Jay-Z and Rome form 8 4 record from our words of cancer with 1 index and he a steadily over here are a set of 4 equations and the question is can we expressed in a way which is a In variant of the Lorentz transformation what what will come of the back Scott let's begin with the left side of the road for strives for beginning with the 2 equations In this box the equations away free equations that were room just a consequence of the definitions of B an EU in terms of vector factual war song kind of derivative of each 1 of these equations Harris on river at being on Acorn holders of Paris as a result was equal to 0 I'm really such equations or their there off for such equations his 1 and there are 3 more he because they so Iraq for such equations and that I will just show you how to write them out in cancer repression and then we can check quickened check them 1 by 1 and see that the same set of equations here I don't think there's summer trickster billion but the answer was more interesting mature or the the fact of course saw me right now for d said now said look be anyone before TXY rosary 0 0 1 2 or 3 F room towel plus DDO prairie get us with a car certain were plus now d Powell F. now what are the stars of curbing Louis Segura Ross certain no equal 0 you take 18 3 of these components stigma newly towel for any combination our 1 2 0 0 3 2 3 1 1 0 this is equal to 0 that's equal Norway these equations lots check West just check the I do this is just to go through them and checked in particular lapse take the case of young let's take the case 1st World all 3 indices Sigma in towel Our space component OK let's see what this as a signal a new era and Powell space of his DEA well well I won't write new Carter reports are works were on young or not us up or but say for 6 Sendak ache woes next who equals y Cairo equals let's try that combinations you 0 yes then this is the removal of respect the X novices ever new house so that FY easy FYG is what it's a component of the magnetic field which component but BX on this 1 over here loom physical Kauai services plus the removal of respect Hawaii how the stigma of work towers Signora Z and X that be worried With the check signed with a Google was assigned an Alaska 1 power was equal to Z area stigma and who are wire services busy radical Brazil I simply don't Barbie is equal to 0 so if we take all 3 of these could be different components and pure space alike no it's how we just get equals 0 we get exactly the same thing insolently if we took the X sorry notably X and seemed be wife would change because they're OK on let's try another 1 mistake 1 of the components to be a prime component which was which what you have of a preference for which 1 for how poor time powers time yeah web stakes turned new wood To be YNX next again so what with the says OK these city merged Arevalo respect why are no Tarriaro whatever Lutao 1 way In time and 1 leg in space that's a electric field electric field armor mixed space-time components that are so this that try to keep track of Charles assigns moved to tricky keep
track of the signs with just a our new rules with signs narrow path and you'll Tal is east of X this the revenue respect or why east of X whether plot summary assigned those who pool or to keep track real time on a blackboard what about this war is also my ears derivative with respect that Air text called effort see your that he's why a load there was a mighty assigned between so what is this a component of him at a piece of the curled if a component of the colonel iii which 1 haven't I kept track from just want OK service workers the river respected shaky of yes Sigmund that's the magnetic field in this direction F X Y is a component of a magnetic field equals 0 and again this summer signing and kept track of but that's just the bit Corolla V minus the car and the river of these equal to 0 you're not sure these equations you could just see that there are some repetitions In general it doesn't matter our which way you said the stigma Cowarts YTU TYX equation is just fine these equations all of them for all possible Sigmund when Tower either just give you 0 times you just give 0 in particular of any tool of the components of the same gives you every single time not the same you get 10 you this equation of this equation solve the 1st equation here can be represented in a completely relativistically invariant cancer notation visit cancer with 3 components 3 industries Signal paralegal earned you can check this another way you could remember that f is derivatives of a new war with respect X towel minus the of a Powell with respect to x but this 1 they have another 1 yet another 1 over here and you could simply wore them more hours and you will discover termed by terms they canceled just cancel among themselves you give 0 so these equations here the ones which call for free the ones which just just determined by the fact that F determined as defined in terms of a vector potential they are quoted for this equation for really very integral check back see our transformed and this equation of this complex equations has a made name is called the beyond identity what's a special case of identity called the Young-Kee identity Georgia was Ericha it the identity as a symbol identity in general artillery it has a certain geometric mean Geometric mean use interesting work for us tonight was again 1 of the things that expresses is the fact that the magnetic field has no beverages serves no charge densities sorry magnetic charge densities but there are no magnetic charge densities nonmagnetic currency was thought that by analogy with this equation you could put a magnetic charge density here the analog room electric charge density and magnetic current over here or finding is that the Bianchi identities is a statement that there are no magnetic charge densities or magnetic cards next time we will come to these equations these all the equations which of arrived From an action principle these are the largest identities you go for free these are the ones Rob arrived From the action principle or we'll do it next time or pay about table fish for tonight we questions yacht so was that part state that is not how they get re-elected Richard that were best Copper River rushed and it for there is normal you do it on just discover that the 1 to free that that any the repeated you give 0 security XYY comport due 0 and discovered the 1 threecom all a piece of the equation I was 2 3 1 I was so was only a was only a handful of independent ones and that you could check on his user looks like for 64 equations here for postwar plans for we are not really or what you know are the only for mainly black guys are clearly is 64 was goal right whether they deserve a good number on goal a good number of images repetitions of premium of our ones right that's true young but now John the what if John Paul well we saw at 1st but file do his or but it was that what about which was 1st older fire say it thought we think we think there is no different from those say are yes there was such a thing as we never challenged city we would put them on our scoreless electric charge electric current magnetic charge laid their however I can remember borrowing your I'm not sure of the signing of this was Marines America's them off there are completely power invariant was not I respected changing needs to eat BBT doesn't quite work why not because of the minus sign here a plus sign Iraq over differently on New Mecca a plane was axes are in the EU OK now is changing even B would be reflecting and the B & B and that's not what to do with doors rotate the buyers main degree uses egos Toby would be goes for minors EDT that's the way our egos Bibi goes minors and either electric charge goes to magnetic charter electric charge was rings charge so here many many ways of expressing the Lowry warm barked if you will back and you say that be really came from a vector potential then at least 7 naive level it would save the magnetic charge and the cholera as jewel there was no Cecil union delicate everything EU mediator change of roads in those of us there is a lot more than that would get the chiefly by regulators know if not all rule combat Mike them were I will try the government while also scored and were discussed how they fit in those pictures of Gaza to interesting fact with discovered a magnetic my barter and he woke America 2 or BERLIN March quarter focus for more please visit us
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