Special Relativity | Lecture 9

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Special Relativity | Lecture 9
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(June 11, 2012) Leonard Susskind discusses plane electromagnetic waves in regards to Maxwell's equations. He then looks for a Lagrangian formulation of Maxwell's equations in order to support the laws of conservation. 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. In this course, 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.
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1st I want to go through electromagnetic waves quickly just play collector may playing PLAN PLA IBM plane electromagnetic waves how they fit there with Maxwell's equations work like I wanna quickly because norms by wild card merit but I wore adored Barham query for dressing or write down Maxwell's equations and put them on the blackboard here on a white board in there are both their called variant the relativistic form in honor of former and play there are 2 sets of Maxwell equations the 1st step so far identities that follow from the existence vector potential the course BRT identities and they are dealt about being able 0 end b . the time derivative trying be the Ferrara I suppose I should really writer arrows DBE by GTE misses a vector quantity BB body seeing is equal to a crawl on electric field of notice in every 1 of these equations and America's always and all sides of the equation all terms they always come differentiated were important I think that the only equations and these 2 equations of owns which identities in the absence would stick during absence of charges and current charges current and these ladies and the 2 other equations are Dell got equals 0 no charges and chorus works for this except even interchanged were not quite so there ever equations DVD she is equal top minus dealt across the magnetic field foes of Maxwell's equations they have a wobbly symmetry very easy to remember buildup equal 0 but equal 0 and 1 of them says that the time the river of the magnetic field the growth of electric field and the other 1 the crew of electric car into on the electric field is a Corolla magnetic field on the afternoon members wear the sign goes and that takes a couple minutes a workout but there are to these they equations on the right hand side can be a great chemical variant form which is called identity know F no stigma plus new rule of f stigma on mule plus their right yesterday plus these said where F Nuno 0 physical 2 0 that's just this is just a way of rewriting these 2 equations for him these
equations here which not identities which are required as they would change if you changed ground June we have introduced over Djindjic but we row are these equation of the form Dean U you know equals in the case where are look charges and currents equal 0 now this equation and this equation both his equations here or there may still have a right inside the memo included this could be a charge density right here so argued it would be a current irately equations and displaced no charges no car where they are will solve put this up here and by solve like found a solution or write down generic form of a clean waves solution a play wave simply means a way proceeding along some access and of course we have some freedom we have freedom along what axis we want to send the way Bullock can always rotate coordinates always work Iraq warned so that z-axis is along the direction of propagation propagation of the wave but the rest are Williams movie at moving back actress along the axis of the opposite sex rebuke all year From swallows ecological Harrah's wavelength or Edinburgh vehicles party violated by Katie with called wave number of plane wave of that form typical generic playing waiver draft form generally has the functional form of society R K X marinas 4 major seed where will have to figure out the moment where there was taken me and the team can be is fixing the wavering and way of life can be anything with Iran will discover their own major must be fixed in terms of determined and Carey said each component of the three-year-old electric and magnetic also 6 components are owed you know what I'm where changes I said the fear way was going down z-axis didn't die Zeit z-axis as a way of going down access who wave number K frequency omega show which components of the field we talking about what will assume all components of the field 0 laws which are 0 have exactly the former sorrow what's right our a genetic former of strike the x component of the field as a function are Z and she is the x component as a function of the AT&T set able to some number Epsilon X called the polarization vector but it's just a number of times the same signed signed a busy minus Omega 3 paid no attention to whether I'd opera lowercase disease saying there are EYE vehicles epsilon y a different number sigh of cases Omega seem using equals epsilon GE for the he'd here means a function at just mean numbers saying sigh Katie Pansy Omega to say for the magnetic field X or so and has a former army a new set of numbers will call on beta exercise and Kizzie minus McGerty of writing and were anyway there signed KY less McGerty and finally be easy and I think I'm regular not right your correct about our lives each other about broker your 1st of all you might have How I know 0 whereas the magnetic fields an electric field have both the same kind of dependents while I sent a plane wave ice said the plane wave I mean better to sign an army or components a our fields are some sort of sorry some sort of waves book like signs and this is a fairly generic think to right down but faces logical also them call sign right you correct cosine icon I could do something else I could make electric field signed and the magnetic fields call sign deals that will happen if I did that if I did that we got to the equations up their Maxwell equations or not that every Maxwell equations if and particularly those which relate electric and magnetic fields think about EDT equals fear of course be times when you write those equations In each 1 of them you will be differentiated either with respect as they deal with respect to you what you get if you differentiate signed with respect fizzy or see you they call signed so the left-hand side of it any equation emotion it will involve a co-star the right hand side is also going to be differentiated live for example the car operations derivative and so if I differentiate the magnetic field I will also be signs up get something consistent have that the left hand side of the equation is a cosigner inside the equations across our suppose I tried to make me electric field signed an magnetic field co-starring that I would get into trouble differentiating for example with respect the time the electric field give me co-starring but differentiating the magnetic field with give signed I'd be in trouble so you have to have a here have signed here you have the have signed here now I've made a number or conventional assumptions I said that the wave is moving along the z-axis there's no loss of generality marriages rotate my frame of reference on those z-axis is along the direction away on I have also let's forget the relation between electric and magnetic field I could put co-starring or I could put signed Cazier assuming it plus a conference in here they assigned as a of can I push story of forward a little bit portion of the year will sure care I can have a wave slightly ahead behind the waiver wrote down but that's a question also of a convention and I will simply if you want to start with the way we're just slightly ahead of time table 0 fine just move you z-axis onto Rosiak service there's where the way it happens to me 0 wants to do that again a way to get back was formed here so this is pretty general but no I have not in any way used any of the equations of motion so let's use the equation motions to what they say
1st of all is about but state don't about equals 0 goal he was about a bell that as partial of DD X with respect that exit was partial EUY with respect Hawaii force posture easily with respect right now he X was not depend on X Pescara why does not depend on worry about scars statistical 0 insolently but Easy does depend on and sulfide differentiated I'm going to get some but I'm not supposed to get anything I'm supposed to give Zero on the right-hand side here the only way to ensure that this to say that epsilon busy for a wave moving along the z-axis busy component of the field must be 0 otherwise the electric field will have a divergent summer some Jones this is 0 a plane electromagnetic wave moving along an axis has not had any electric field along back direction is still magnetic field that we use Dell not be equal 0 strengthening misses also recalled 0 I saw used up the Bob equation is now how about the other equations to what they say let's pick 1 West star courts take EDT OF 1 more French yacht yes it epsilon X in Epsilon White a constant they don't very bidders numbers for them Zach vectors in the XY plane vectors and the x plane which he termed roughly speaking the ratio for a beer the amount of EXE you why you can always rotate the XY plane wave is going that way you have always rotator actually use the X Y axis until 1 of these is 0 for example you could always lined up for your X axis so that e x is not equal to 0 but EYE 0 here will two-dimensional vector and the XY plane and you can always rotate coordinates so that 1 of the 2 components is equal to 0 that's again no loss of generality just remember that we've used up whatever rotational freedom we've had to set this also recalled 0 I get cannot it yet the the magnetic field I don't know what this implies about the magnetic field will find out more so this was starting point an electric a electromagnetic waves with an electric field at points along but to be X-axis and which oscillate physical exams or upstream OK let's see what's now use DDT equals might go Crosby PDT equals minus across will be end there is only 1 component still components that we have the worry about are it's a trivial matter to see the busy component of the equation this trivial what sorts for good be X and Y components of this equation 1st of all the x component the EX by he is ex-con part of no cross what CX component of dough Crosby articles Inez D DEC Martinez DEC BY Price a rally have an equation relating the EXE tea busy and you are the easier 0 move already said these EU 0 so we're best just get plus DDX by deep areas BY by dizzy were 1st said Ross visit BY cannot be be 0 we said EU White 0 but now we find out that BY cannot be equal to 0 and mother was electric and magnetic fields cannot be car Karen not be Harrow to whichever you cannot have a situation where BY is not equal to 0 if EX will be wise North so DDX by DTA so but plugger now the EX by C differentiating that gives us epsilon of score epsilon is only 1 of them was score of Epsilon and they're all we wanted differentiate with respect the kind that gives us a fact this all major differentiating signed kind of co-star of Omega over KXY the summit but with with book Everybody following alleges differentiated this with respect the tying the time Darugar me minors only a draft and that's what we have left side Syria right you right now they differentiate be why areas and that's going equal must be equal to y plus crimes K With differentiating with respect to my right here I keep playing a role in a letter he BY the Casey Ga terms Katie cars again co-stars signs canceled what we find his an equation she said the BY baroness BUY co-starring cancels BUY is equal from mine this epsilon X all or a case BY a physical from my Epsilon slips ORX or may go Over Kerry books a tough 1 all of OK a 2nd fact we can get by applying the same equation until white component parts work Applewhite component here yet so here we have DEY by DT but as Iraq which said that he why is equal to 0 0 where we are right he inside the right hand side has X B Z my ears these VEB Texas now the side right or have the siren wrong it doesn't matter it's even DXD ZOO these in all this is let's see this is a white component of curl which means I think I have a bigger have signed right here but hardly matters since left
side is not to be 0 is the right inside a 0 but the easiest office he equal a 0 then was is going to come if I plant Ian be Texas by plugging B X only wake me
0 is beta X is equal to 0 the only way but this can be a 0 1 I differentiate B X with respect to Z I'll get something 0 on must pay the X is equal to 0 so this equation tells me state X is equal to 0 per I was a sorry this that if the electric field but sunk warned that the picture of the slave what we know so far His up x-axis Our Y axis nausea actors and the way is moving all along the the z-axis sorts assigned ways along the z-axis what this daughter and we've assuming that they electric field points along the the x-axis circuits of electric field looks like sorrow off at a particular incident of Torrington your like back and moves most of course with the speed of light which we set equal quarters are assuming difficult 1 us feel electric field E and whether these equations tell us is that the magnetic field lies along the Y axis no component along the x-axis as only a component along the y-axis so it looks along wire axis sole-source sign we tried to draw a somewhat three-dimensional way they gotta be magnetic field was lying along the y-axis oscillating as you move down the exodus and electric field along the the x-axis so the 1st they learn about where you have a 1st by another thing we learned is electric and magnetic field are perpendicular to each other and there are also both perpendicular to the direction of motion of the year of the year the waves the words so that Is it electrical magnetic fields are transverse music perpendicular could direction of motion of the of the young of away I as 1 more piece of information and he'll you lose that they good see equals Dell Crosby murders dough Crosby equation and we got a relationship between the Beta and Epsilon are getting rid of the indices was only on only data warriors nonzero only epsilon Exs non-zero so we can right there's just a former data His eagle Tom ideas epsilon may go over case that's the equation the condoms from the DDT equation was also the dean T equations that looks exactly the same as exactly the same structure up from either signed and n ERA change With no further work I can write down the corresponding equation just did change Beta and Epsilon and she's a assigned Roth says Is that fell door right you find out other equation is Epsilon equal is I think it's also minus beta Omega over K as a cover for Iraq to starring pseudoscience conspire I think to do the same mom astray relationship what does that tell you about epsilon however that Sutter what does that tell us how I see it tells us Barbara right up saw says I want to say yacht but 0 home cables 1 your trouble words card that says that only got all the cages equal toward is the inverse of its officers on May Golva Keyes Khaled from may go Delaware Omar garlic acre Mikhail Omega is a form get equals case I was the last piece of information that you know they don't the sees warned right if is warned they don't have different this has Omega has units of the numerous times K has inverse distance distance in 5 so that tells me then back simplify and just write as a sign of K Z minus escape here just sigh of Pansy minors see everywhere so the sign of K Z minus nowhere with the where would be armed with Spear of light going through this case patterns Z minus well that's of course the speed of light would go right where it needs to go north to make they units correct so the speed of light would go over Maxwell's equations tho have speeds of life my left them out because so knowing you doing relativity you don't want to be bothered receive about being 1 OK so that some electromagnetic playing way if it's transverse electric and magnetic fields are equal to each other but all Farner below the magnitude of the elected magnetic field Eagle pull each other epsilon equals beta your father uncle each other and the whole Boone 30 moves down the access so rigidly we know that they would be in a plane electromagnetic waves were always in favor of a circular polarization little more strictly but will playing our of waves yes there in phase with each other this as a ball that yacht Le Baron that that's double might be reserve places you care for Berkshire support that that electromagnetic fields How do not produce refer to as low as they are right hand Bobby energy was over here just moved stay here other managers and conserve Troy must the energy density at point changes the energy density is related to the square of electrical magnetic fields of stand at a point away goes why you 1st to see some energy and then using nor energy and then you see some energy in the casino energy density but the public cousin the total amount of energy energy over here after Morocco after a while I was just replaced by energy over here will later was replaced by energy over here the only other thing you need to about electromagnetic waves is that you can they had them Yukon you can chose epsilon any number you want big or small Badeau will follow the Ukiah and waves or subtract them different directions and you still have a solution America falls equation what I want to go through that because current ridiculous to teach a course electromagnetism and right Barham electromagnetic plane wave which I suspect many of you have seen before sure of pretty quick about it now I wanted yarn that Hill was 4
had not wide-eyed belt the 2nd or I don't need to assume nervous and that's the statement both sides of the equation on derivative so when you differentiate E and equaled or some other derivative of You can't afford the phase difference our here and here right now I'm sorry if Iraq had left had what of really there are a lot of bad news highlighted the phone talk just order said there are a beer after that the other equation up there has exactly the same form exam interchange with Edie right volunteer that bird Terrell it's easy for the form will go for as either emphasize all over and over again and all of the ideas of energy conservation momentum conservation relationship between those conservation laws and symmetry wars follow only if you start with a principal with faction or are you come right back all sorts of differential equations for various kinds of things and they won't be energy conservation of energy could be conserved on less those equations are derived From or grant from action principle research we are his inclined the deeply believe that energy conservation and we should be looking for it a Lagrangian formulation on Maxwell's equations focus electrical throw the principles and see if we can make our best guess about what will Grosjean and then what we wanted the 1st set of equations on the left Bozo identities we don't need a program may just follow from her from definition want the 2nd set of equations which at the right hand side learned but this way physical role which is also a Golkar kind of complained although for a vector current and now works for a bout put this on the left-hand side of the equation and the right hand side of the equation is the current vector the space component of a correct so that there is a vector that be erect alliance sorry Kobe equations this just becomes Janey the time component of this equation not simply gives you this 1 and space components combined conspired to give you the Everex for Volker a it is quite from our Rogers so let start with principles for 1st principle was locality all we are guards farmers thicker by Gretchen 1st before network a right locality but if we get that means what was going on at once and the time was only a related to a neighboring instant of time and neighboring grow position and that's always guaranteed if the action is so if we actually is an integral and the case of a particle and world lying in the case of field theory an integral of four-dimensional space TEXT white CBT before X on a Lagrangian density whether grants in city depends on on all of the field however many fields are you a theory let's just call 5 for the moment there are of course going the being the vector potential but for the moment with column find it could be several of them more rights several of them fire will stand for the fields exists and also derivatives of fire with respect to space to respect the time I'm laurie so use the tensions that the derivative Our affinity with respect the X mule sometimes others you 5 other times or use even more condensed notations 5 comma meal without a standard notation a comment we need derivative and some you'll means derivatives with respect for the new component of X stated and that they should end of our we it were localities sizzle Granzien depends on fight against far-right Neal said 5 Viacom but this doesn't mean 1 particular derivatives means all of the derivatives find you was a generic symbol here means curry derivatives space to remove all direct that's the conditions basically the equations of motion on differential equations that really means locally us Mexico next condition is at Lawrence and barriers book less fatso bold programs there should be a scalar Chad such amount depending on which frame of reference evaluated should be a scalar suffer example for a single simple scalar field I was just go back over again for a single simple scalar field off you could write down what different things but some simple things would be foreground contains for example could contain a functional foreign fighters soulful spillover is a function of fires a scalar cell with 2 worker Coromandel of CBU you will find that could continue will fight a function full so I couldn't contain must contain the reverence otherwise it's not very interesting were gradually sell it to contain things like it can contain the X 5 by itself balloting nonsense that's not that's not scalar that see X component of vector you can't put a D X expect Lagrangian BAA can be a lawyer Granzien but not just a simple form what you can have is a genial fire a theme you 5 or which is the same as find you calf icon new 5 come on new what's the difference between the upstairs index on the downstairs indexes of me anything to put this index upstairs it just a change signed for prime compartment store for example the subject here which is a scalar it is really a minus 5 . squared coughs D X 5 squared must be wiped I square and so forth but what just cheaters sign of car growth of 2 space will assume nor about that this could going to a grand jury and it's a good Scalia and by convention this was really more of a convention and anything else we put a minus 1 half but Johnson so all were rising for a simple scalar and this is not the only
thing you could write this is just a simple thing right Jim could be minus 1 have been you'll far the neophyte Martinez any should be among the mightiest and you are a as a server not the most many many more things because a simple last thing before we go want electromagnetism was just recall how you go from will grant GM the equations of motion His or lower branch equation for each field only written 1 field here but there could be be several fields are we might end the Lagrange genes for several different fields together we might even put them in a more interesting no way that mixes them together in various ways for independent field in this case only 1 we begin a partial derivative all around you and with respect 5 Cuyama near the river just Mr. River fear and then we differentiate back with respect to x new that's a left-hand side of the equation the right side of requires earned the derivative of L with respect of 5 this is median analog of the Lagrange equations of motion for a particle which would read Debye teed off posture Lovell with respect for the time derivative of Wharton is equal to deal by the time by a quarter past orders is give let you work out it's very easy it just gives you With before it gives these 2nd flying body he squared minus D 2nd Friday the idea x squared I won't bother writing wine Z got is equal Martinez the EU by defied what Sobel wave equation just with both before and I wanted to remind you because you do exactly the same thing exactly the same process with Our who were correct potential is more complicated of tried do it in the simplest way that I can the Oscar Moore yet some Awyie indices floating around 1 walk principle public locality Lorenzen variants and const electrodynamics 1 more principal Dejan vary so if gage invariant but that was for the moment let's set but current equal to 0 no charges just electromagnetic field day do variants tells us the Lagrangian or lose the accident the attacks accident should not change would make a gage transformation the simplest way to ensure that is make cool grungy itself as things which are Dejan various things it's a gage invariant IS communal electric and magnetic fields your electric and magnetic fields are gage invariant they don't when your edit Eddy Curry percents the electromagnetic field doesn't change with huge transformation anything you construct Warsaw March was sealed he of course was a something you can't put a price for example you can't put a new game you why not this is a scalar that's made up about feels visit these news he adult derivatives the these other components Out of the fields if I wanted to differentiate I've got put that Commerce in Commerce indicates derivative contract be careful about this just means a naught square mines X Quebecers why square procedure square man's ignored square this is still a perfectly good Lorenzen Marion but it's not Dejan every shift aII in this way this quantity which so Lober Kemper illegal or not they vary Yukon put in a component of as you know in any way that you like as far as gage invariant goes both alone isn't variants got bigger scale are out of it so how did make a scalar of a funeral while the number of ways the wills always the same contract industries Lauren this season cancer some general kind me you know that you won't make a scalar Roderick you is 1 index and then you could track them by setting this is equal and summing that's a general rule for making armed scalar at friendship was what happens if you try that where found death that would equal yes New York mule Doc start what about taking a S New meal What is that OK so what is that it is 0 is a forward all the components of S as such the diagonal components 0 0 F 1 1 is equal to 0 at show equal 0 0 33 is equal to 0 raising this index here the 2nd index at doesn't change anything it just changes a sign of the sign of 0 of 0 growth and change would you change sign a form or a 0 and so forth so picking EFU which means that not not force F 1 more employers to 2 process 33 that just . 0 nothing they snuggled scalar are for nor ranchers just as 0 well so no figure put in for wrote that a couple of things with well are the 1st month the real thing we did Mike error by 1st and war the lowest order and calf nothing O'Leary Ref works here try things with quadratic in and they had something of an you know that you know that means something is nonzero what's Europe means a serious resume work out exactly what this score years nor can 1st of all there are mixed components space time mixed components there would be F nor what is an F west of us where mu would is not party and you'll his space and was radical that's almost a square of the electric field the makes components of F electric fields the only reason the square of electric fields because to raise that kind component you have to change the sorry this thing here is equal top we elected field square so that's what kind of turned that
appears in here f nor what is f North and that's what is current component now actually this at this term appears quite it appears twice be cause you can have me you'll be the space component and you'll be the time component here where the 1st index being car and the 2nd the next being space but as the opposite possibility 1st and that's being space the 2nd in next crime and that give twice the same thing I serve restores contains Martinez twice the electric field squared for Laos but also contains space space components for example have warned truly X Y FXY turns FXY raising space in this does nothing was there for 2 terms electric and magnetic fields as B 3 Z XYC 1 2 3 salt is Class B busy square foot but of course all possibilities and don't warned through which twice and there's because as f or 2 0 F 1 short also asked to warned I have to warn us twice these square Ortiz Eds twice squared miners beat us twice be squared minus eastward again as a conventional is a convention I mean it would affect nothing if I didn't do it would not affect the equations of motion but some lawyers along low-lying people started with the definite conventions about 4 grungy and the convention a structure that dog in begins with class might be squared off in other words is on my sign but I would make this plus saw Our equation correct 1st war Julio new hero a four-day 1st off conventionally want puts an a minus sign electric fields cons were positive side he square miners be squared times a factor of 2 began a factor of 2 or only occurs because each contribution appears twice at 42 2 times F-1 and F 2 1 plans to warn both the same thing secular as part of the convention is a factor of one-quarter minus warned quarter and that makes this ahead F. beastly square man's be square again there's no content in the other then to make it consistent with ancient conventions the miners quarter here they're 0 0 0 all the diagonal elements of F 0 0 perform 1 Due to the only off-track Norman's the off diagonal mixed time components Over electric the space-based components made much what's each school where might be square it's Lorentz invariant as local as and it is in many ways not only the simplest thing you cry this also the correct for Electra dynamics are gives rise to exactly these equations for the moment by Paramount include jade OK so let's see if we can know concede that Cecilia less of a nuisance but not too bad offered take there reasonably 1st of all what are the fields but are right there what's right there on the order Lagrange equation again or around those he equation for each field each field and write their 1st derivative of O'Brien's in with respect of Carmine then differentiate often with respect to x set that equaled the river allure Grosjean with respect 5 field you write a separate equation like this now what are the fields of talking about they offer the vector potential a norm X
Y and the distinct feels sorry for strict EU X Y and Z your 1 2 and 3 but by those are the independent field to get these In this easier do not correspond to derivative warned derivatives let's introduce a notation for derivatives of the fields a were 283 while can have a meal component of the field differentiated with respect to the enormous scored 8 new common literally means the derivative other aid with respect to x what about the field concerned this form the field cancer where what's sir what's right down here is a field tensor f communal equals a new bike VX nerve Minor ST a new X meal which is a safe areas a mule Myers a normal Commerce new pay other stations saves riding a bunch of partial derivatives fracture Morris axes With a simplified the means derivative OK so now we can write down your Grosjean and try to work out the order Lagrange equations motion and working up a lawyer Lagrange equations we would take the floor components of a key the beat independent separate feels off-field equations see each 1 of them self hilly mind this his L L is equal from minus 1 quarter but say a new wireless they know me 0 press SCO Kansas City said old or we have to put their early index upstairs of the begin this is upstairs With with fraud end I am emphasize over Norway the only difference between upstairs and downstairs assigned change when you move a time index that's all that's still abounds in minus 1 quarter at let's see if we can work out the field equations and let's pick up particular component of this that that's Laverne's store Ranjan Neukom ride it out in detail a few like in terms of derivatives of Eddie the things which correspond defied are the eighties so we should now work out the although emerge equations the being equations each component of a diesel pickup comp or Margaret quicker were were put here your rural reward for worry Yucca Flats to work out the equation of motion for a component a sub x rays Razorbacks or what the same a war but difficult work out the equation March and the 1st thing we have to do is we have to compute partial derivative of L with respect to 0 AD X mule analog winners see analog of this subject over here except where 5 given by a X and then we have to differentiate it with respect to With respect X Neil but stood that later we're a so wheat see calculators in the river will grant with respect the specific derivatives of specific components of aid but example the river all around With respect 2 0 X score more Whitey that's a fairly generic their forces the removal gradually with respect that a X Carmel why everybody they want you look at you will a ex-con war is driven lyrics restrict Why you differentiate manager OK William of a ex-wife here basically there's 1 hurdle in the summer Lemieux with Norway a ex-wife minus a UYX plans a ex-wife minus a wife X sold at work so it's right there will we get to it we get there what do what we get is minus 1 half I'll tell you why warmer 1 here from moment delivered they X with respect for wine minus derivative of the AY with respect x squared there are lots of other terms of bronze but they do not contain a X Y the other terms but they do not contain a ex-wife now why I have I have clearance of a quarter 0 2 terms that terrorist is wanted in which new was excellent who was why he and his mother taught in which she knew was why and who was X so you basically get 2 terms are for each combination new window I'd now let's what's right this outlets right this out some detailed minus a half the Bracken around with missiles a E X calm why squad heard squared best crossed Turner mind there's a X why K Y X with commas in the right place twice that fear bear cleanup clear the seller sequels minus 1 hey F derivatives of a our starts later Fernandez location a
sex crime or why square is coming from square areas binomial feral that the miners twice D a X Y a white X and all Western years plus a wife X squared this is the only terminal grungy and which contains a X calm why it also contains other things but only terms which contains a so far this is just around the terminal lounging or interested now wanted differentiated with respect to this particular component it appears here with circle of places where it appears appears here squared and appears he multiplied not by itself a X Y and a X a different objects is derivative of AX with respect Why is derivative of a wire with respect the X different objects were searching out the dependence are X Y so what's the derivative of this will grant GM was a spectator AXY very simple reduced Madrid young age the derivative of a Coverdell with respect a ex-wife from here Martinez a X Y what happened to a factor of a half as justice a square solely square with differentiate square of something you get to so this is a X Y and then we also have this 1 over here which of miners were differentiating with respect they ex-wife that gives us a Y X soros get plus a Y X but the the long time to get there but the answer is very very simple the derivative dealt with respect a X Y it is just Mike Harris F X Y the AU X with respect the white man's their wireless suspect X His ex-wife so it was a long but let me write down the general formula follow Mr. Blake was also equaled the minus F X Y space components Lauber down for free so we're right there now the general formula with the River Po grungy with respect that Eddie mule Carla New are early that is equal from my so f you know and we went through the same exercise and is that every component we discover the differentiating Lagrangian with respect to the derivative of a with respect and would just give us as you know what's next that the next step is right over here With differentiate poor Burns in with respect farm you win then differentiate again and job Debye X are L my idea aII new common no that's just minus the derivative of S with respect the X Noel turning up here differentiating with respect X Noel just gives us a left here sigh of Maxwell's equations was nowhere else is their woes the derivative Granzien with respect to the field themselves that would mean the removal of rising with respect that undifferentiated Eddie but undifferentiated a doesn't appear on about Jim that's it that's all areas of oil or Iran's equation is Mrs. equal to 0 right that is that How or xxx this it assault story of books much more complicated than that is just take half of the squares of all the elements for granted from let you work out your over grunge equations and it's just 2 of the Maxwell equations that told the Maxwell equations that happened that have the Jaisalmer right side we haven't included among the other Maxwell so equations have to do anything their identities of from our definition of a relationship to work any questions of 4 Murrell it was not hard but you have to sit down and do what you have to sit down and don't work out the the details are it because you with a dozen the ads show it doesn't help the what do I have no doubt that doesn't help watch me do it even if you manage the forward did you will forget it fired minutes but you won't forget of the steps yourself young viewers are on the top of not says was head John square over here there was a Oh I'm sorry yacht in writing this over he acknowledges pulled out 1 terms I just pulled out when turned 1st that about wraps are passed out of 5 young guard for supper but I just pulled out 1 terms for examination work now you use a lot fun you will would have you have got the same sort of thing you will find the held by a dealer nor I missed end where the U.S and Nora which would have set a component of the electric field when I used to space components I get magnetic field was once based on timing and like field read the change but as you keep track of the suns carefully you'll find votes that this is general yes yet
got a rogue also rose here early in the season you find out cancel out but Xiao got their 1st we would be there is not a game of like a home for the moment I Commack for the moment tardy are assigned him right now is that there is so there are no gage invariant things that I can make up just hours these days that don't involve as you know I'm giving you 1 example and fact is the example that really does give rise Arthur the real Maxwell equations today the Lagrangian the subject here the question now is how you get the right hand side of the equation where did that come from laughter and something the Lagrangian you must have something around you have something had better involve the current before component of a column doesn't Bob Corker born the current it's quite clearly North going to give you what you want sell let suppose that this pretty established electric car that electric current means a charge density and the official U.S. time of flow of charge space in components so let's go back up and recall the principles of charge conservation before we do anything else we have 4 components Jamil which equals the prime component which is charged and end the space components and more displays components are the space components of a charge for you area per unit time passing through a little window know Orient along an axis a charge are through what a move when the Oriental the am axis or perpendicular to it the charge per unit time per unit area and that's called jet OK they if you take a little and you say the charge in and course to charge America's proportional times the volume of the XTY busy that's a charging and the rate of change of the charge it is just a tiny derivative rock firms do you want review say only source Cheney Of the charge in that region is charges which has through ruled the walls of the religious then that gives you the continuity equation country work that out last time is robot a change of changes charge and will sell class dell . J E equals 0 Gove RJ means partial J X with respect to x was partial JAY with respect Hawaii of partial Jacuzzi respect his straw how to read this this is a change is charge will use the test will solve this it is the difference between the charges or charges passing out through B X Windows this the charge passing out to Hawaii and that Windows and soulful and this is just a balance of charge inside the box pastors through were members of the box Our can also write this and relativistic notation Dean J. new accords Dini Virginia recalls OK Rhapsody fear equation call the continuity equation but represents charge conservation represents the idea but no charge disappears on accounted for is moving out of the region the fact your aware that charge wait your next box and you discover the kind passing through this window here just Eds charged to elect box so charged it disappears in this box goes into marks match charge conservation alright now Dili the continuity equation with another Turner Nordic Dejan variance playing scalar how we can write it down involving the vector potential will check and see if the gages doesn't look gage invariant the begin where false the actions associated with colors as usual as he and the world of D for acts of some kind of Lagrangian underwrite them very very simplest thing I can unimagined Jury prime is a new was of course means JUU your of position space time position your position that is a scalar bits before vector product of 84 vector promoter for that's a scalar malls of current also vector potential but doesn't look gage invariant browser Our core a check of the gage invariant well check gage invariant incidentally Our by conventions Murray aside from that might assign ultimately used to Benjamin Franklin cough was a it was a convention OK on White roses why is will gradually right here Of course JMU August turned around June is January aren't let's further let's do a gage transformation withstood gage transformation what happens what happens because the minus integral saying it will be 4 X J and now work PM he knew when it had something they knew from your head d s body dioxin your dream you as this is a change Barbara's resources for change change In action under a gage transformation pondered gage transformations GTE a change doesn't look like 0 it it was not 0 than actions like gage invariant but it is 0 With
checkered of 0 Australia White 0 who I think there will be 4 x was amenities means and the XCY busy detail whatsoever Baron the XVY easy but now we Harris D. s VAX North plus J. the essay by X Warren right bust-up about back and this is integrated or what all space I'm going to make 1 or assumption on going to assume that if you go far enough away there is no current flowing back all the currents in the problems are contained within a bigger laboratory goes to 0 far away it if not the current flowing into infinity you have to take it especially recount of any ordinary experiment than a laboratory currents move and so forth charges a rare but you can imagine the laboratory being isolated and sealed so that no current call out of the box In that case let's see what gives let's take our a particular firm let's take the Inter Grove J 1 D S by DAX warned as 1 particular terms and lasted the integrated X warnings XPX fierce RDX XTY busy Vt Everybody Norman Lucinda will buy part of our allure the integral here forget the DXT wide easy just forget that would doing integral over X J. DST X is not going to let my heart rule is you flip the derivative the other factors changes sign with integration by parts many times this remind you if you have an integral over X terms a derivative of respect X the newly and the girl just by flipping the derivative the other factor and changing the sign of was minus the Robey Jerry wanted by VAX warned Best the X and then the rest of them BY museums for black if I did the same thing with with the terminal with a 2nd terms the 2nd terms freeze J 2 0 S YBX till for happened this through Jomo contributions and the other contributions would be DJ tool by DX till I was worried that the 3rd term yes by X 3 times J 3 would DJ 3 by the X-rays and finally there's the time component also time integration here if you do a whole thing you find out that the change in the action integral need JBO by DX middle fans S R we started out with Jay-Z turns DST acts we integrated by parts and that put the derivatives on JET instead of on that's all we did the trek that they meet the revelers I'm jury a sorely have here and that's equal to 0 because of the continuity equation so radical 2 0 so the change in the action if the current satisfies the continuity equation all the current of the continuity equation limit In this peculiar looking right here is gage invariant this does is so Souz looting wagers that have still the GATT left I wrote Dr. Fausto Barger equals 0 OK let's go back young young that's a 30 hour bus ride that's a 1st here if war for a price for the summer more for musical 0 that guarantees current OK that guarantees the gage invariant of this salt we can and this tour around however as is to the equations a watcher what we have here anything that involves a derivative of a we don't have anything we just a small something but we just it is something that a itself was a large equations over brokers who writes for over equations have all left here v New Moon are of partial L with respect to a new York is left him started and the right hand side is partially L with respect a the term terms composed by a core component will write down your Lagrange equations each component of the field 80 molest is arguing or that gave us that gave us minus partial of F Newman who with wrist said parcel of and I think lizards respect X Newell so we got from the previous terms on or around June was a source of wealth we go back he is the new terminal around what happens if we differentiator with respect we just taking this germs nor Ron gin and differentiating a with respect to a component of AD you just get the corresponding component the journey sorrow we're this is equal collecting my J. know the lesson here the proper Russell lessons 1st of all you could write Maxwell's equations around you have some poor 2nd Lanzhou has a very simple form minus a quarter at Union Square minus J
3rd and to gage invariant if NOAA OAS the current does conserved what happens if the current does not closer and the gage in middle our that gage invariant and what you will find our that's that's a disaster because it means that the equations of motion on my future if the equations of motion amount gage invariant something really strange because the left-hand side of the equation of motion was certainly gage invariant that only involves electromagnetic field was so that means something very strange is going on that just can't be the right equation is operatic work because they were not door the conservation electric charge Yukon on imagine a world in which charges disappear if what would happen if charges disappeared well maybe not so maybe nothing so terrible but Maxwell's equations could be right Maxwell's equations could right you couldn't arrived it's a little exercise Yukon derive the concentration of charge from Maxwell's equation what's wrong a vigorous contriver direct travel do it from a lessee of Regan derive the continuity equation From Maxwell's equations facts arrears Maxwell's equations Raul because Dell . ie parents J. he quotes DDT gross big OK so let's 1st figure . 4 wrote Dr. wrote . is equal to Del e . right to differentiate prime differentiated both sides the if go due next Jake Young the logic here we have gelled he docked here we have a doctor let's see what happens if we take the emerging Soviets Watson beverages of always 0 0 bridges Kurrowah Stories 0 so Delgado but now care source out so we have to go back JET equals about you guys but that's exactly what we have here so this gives us I hope opposing Eurostar because it should remind jail happened here that I make a mistake waters that far but at 1st roh it caused our . a E . His equal to lizard go crossed a matter J where are they write that's soccer make any difference suburb golf I think it's right . or Sabah saga of their birth birth figures tried out were good say so wrote Dr. is Dow about E . that's for sure if role was equal to go about it 80 and that's equal Mexico from nothing yet upset Maria eat dog Is this so let's try adult about you got there is no server Kroger 0 and that's gonna Delvalle by Jerry I got Cyrus a reserve on March orders I have robot sequel to Fidel dot J 0 0 0 0 0 2 the hosts feelings I have a feeling was a sorry nearly a year alomar I suspect is assigned area here I don't know I can't do it on our real time right here this the sum sign area because the right equation has to be wrote . plus budget a 0 some are still with us areas but you State you apart from sign error Lee conservation or the continuity equation could be derived directly from the Maxwell equations so another words of Maxwell's equations of are just to because conserved look she is a reminder is doing yeah Isuzu 1 more day sort out got affected anything else we set our room and they right they can't have Maxwell's equations and have charged not conserve but remote west and the final Western it is that the charge conservation is intimately connected somehow with a gage invariant the warrant further charge conservation lesser charge conservation mean the continuity equation this 1 he without the continuity equation Maxwell's equations could be right of all those things occur Guerra and that but are at Her no all what higher-order watch your what's that they brought back then we try our vessels or that food firefighters Woodlot here I am and what do they get for metals where their How yet you could start with the Maxwell equations instead they imply conservation of drawn knowing that you could say I want to know the charges conserved I know that the term J . 8 which is gage invariant and then discovery that this is the right hand to reproduce Maxwell's equations but no 1st it is bad for
brought that is how we wish for us not she heard shot down your yard can the robber from Mattel's equations but dumb really just says charges from disappeared from just calmly disappear and they only disappear our region the the so Israel connected together cut Our you can stop logic this way and say the basic principle is gage invariant that requires Jacob be conserved and then use this to derive Maxwell's equations and in fact then from Maxwell's equations you could see the GAO's conserved or contradiction or contradiction solver axles equations and say what blow grungy with Jimmy Maxwell's equations and were moving you discover Maxwell's equations is J. satisfies a continuity equation tells you on this thing which I have to put in the back of grungy and in order to get out the right hand side happens to be gage invariant sold it depends on what you think is fundamental to think gage invariant fundamental that Vijay has to be conserved continuity equation if you think the conservation of charge charges should disappear on was fundamental then the continuity equation is just an expression of our conservation just charges don't disappear and that tells GAO's concerned or are you could think of the Maxwell equations more fundamental and again using the Maxwell equations Uberaba continuity is bad if they were contradictory is good but they have and what the was minimal logic I would say the minimal logic is the combination of gage invariant Sorensen variants grant formulation lead you through this combination and the new of course but that's a matter of taste for you could start of places on the factors things that many of the questions now that shows a day for now also gave the historical anachronism have no what is armed with a narrow canyon at a age that has 0 Gates usually good 0 0 walk 0 what say 0 more because that breath a lot of credit go ahead yes it did keen from the state to not armed it came from a mistake which I cannot explain to you now I can explain to you perhaps pending what lecture again next week I will try to explain it but it came from the state of gage actually have do with measurements of length Arafat did learns nothing to do do with measurements of Lane has to do with measurements of phase where phases of the Shorty wave functions scrapped with mixing require McCann-Erickson were Herman while made a mistake he thought had to do was something that very quickly corrected himself the term stock on but you can call it that question said was correct that are wrapped DG amend a defined 0 0 of something 0 of what well so was 0 of this still feel diaspora pump pump Rory about who will understand you do very people away perhaps next time our it does have a deeper meaning the does are much deeper meaning and will come forward next time only our any other questions about what we did to Michael Young said that will the that is provisions while less I it is a bit of a young kid that where can I around your CD over that is how you're rescue a commuter changed yes yes yes We are you can invent different vector potential With the property that these equations over here become equations motion and acquires JB calling 0 become CEO of the Bianchi identities that's not a convention that was wrapped around had to do with the absence of magnetic monopoles which nobody believes anymore more so yeah some centers convention yet it is this interchange electric and magnetic bomb that 1st are firmly may beat the real physics stars have an interchange symmetry between electric and magnetic electric charges becoming magnetic charges now it's not really a cemetery because there was a secretary who discovered magnetic charges already there would have been a similar poll electrons they would await the same thing as electrons if there was a real symmetry between and no such magnetic things exist our souls it is not the symmetry in the usual sense nor will there right now I'm I'm too tired for any other questions OK we've actually come a long way we've come through brand formulation the mechanics of ground you formulation of field theory Lagrangian formulation elected dynamics gage invariant Lorenzen variance thought staff but a lot of stuff learned learned as I said I think the only way really digester but go home with the equations and work them out yourself starting from step for step 2 over 2 will block for more please visit us at
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