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# Quantum Entanglements, Part 3 | Lecture 2 + 3

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reversed a lower price for less time we discuss influence of electric and magnetic fields on charged particles with what they do in electric and magnetic fields warn charged particles that push them around and they exert forces effect even brought some equations what's right the basic equation the basic equation is the summer charged particle F is equal to the electric charge of the charged particle or that there that's there because of charged particle was charged with didn't have charge it would be for starters and then that gives multiplied by the electric field but just Corby electric field class something that depends on the magnetic field and the thing that depends on the magnetic field as the cross fraud of the velocity with the magnetic field that's the basic influence of electric and magnetic field on charged particles to exert forces on the force them and call them around store for the missing piece thus far is what is it that determines the electric and magnetic fields in space what is it that determines whether it electromagnetic waves or it determines if AT electric charge produces an electric field himself for 1 the equations which govern the fields themselves those equations a cold Maxwell's equations of positive of equations we want to discuss this evening they the equations which tell you how charged particles affect field is the equation which says how fields effect charged particles the other half of the coin is how charged particles affect field was it necessary that charged particles affect fields what kind of action and reaction if electric and magnetic fields could influence particles with doesn't mean it means an electromagnetic wave King come here charged particle send flying off some of the direction in which an aide at energy of a charged particle well if an electric field Canadian energy to which I particle that it had better be better charged particle consort energy out of an electric field otherwise energy would be conserved so it stands the reason that a kind of action and reaction of electric and magnetic fields can do things the particles charged particles and charged particles charged current electric charges had better be able to do things for electric and magnetic fields so that's this topic for tonight but as always in this course the basic question is whether the laws of physics and in this case the laws of electric and magnetic field of the scene in all reference frame so that's the thing we been concentrating on the study of relativity is the idea that the basic laws of physics are the same in all reference right now 1 of the things are going to write down Maxwell's equations in a moment but 1 hobby basic things rats Art we learned by back step or basic questions as to the light all all of relativity theory have endured light metals equations of the theory of light and Maxwell's equations tell you that light moves with the speed of light and Maxwell equations of the same in every reference frame it stands to reason that light will move with the speed of light in every reference frame this was precisely the puzzle that Einstein wrestled with so hard how is it possible to get a description of nature in which like was of the single oddity in every frame of reference that beefing which ensures that physical laws of physics are the same in every reference frame and past lectures we saw the work that made is that these equations of physics should be expressed in terms and quantities which have definite transformation laws when you go from frame to frame that means scalar depart change it means vectors for vectors it means cancers and all the things that we discussed last time when we discussed the transformation properties of the various things and physics but I just remind you what this equation looks like if it's expressed In a cold variant form in a form which is afraid sane every reference frame it looks like an expression involving not electric and magnetic fields will yes electric and magnetic fields but cancer a field Cancer and I'll remind you of a while what that field cancer is but it's an ant asymmetric field cancer which is built top Out of electric and magnetic fields and multiplied by the fall of vector of velocity Sosebee news component of before the new component of the forest it is written in terms of a 4 by 4 and isometric cancer who's transformation properties we and before vector of velocity in this form the rents law is written in form which is manifestly the same every reference right so that some gold is to write all the laws of physics in particular the Maxwell equations that AT and a friend a born which is the the same in every reference frame which means writing it in terms of vectors enters so for now
component so that this would really needed a 2nd derivative with respect the time field minus the 2nd derivative with respect to x squared minus the 2nd derivative with respect but why square minus the 2nd derivative respect C squared physical 0 not us as usual because I'm lazy I said the speed of light see equal to 1 you looking on exam exercise figure out where the speed of light goes to court let's imagine there were a way moving down z-axis wave moving down z-axis moving them the z-axis and then we find only depends on it doesn't depend on X and why it may also depend on time before wave is moving near standing still watching it 1st you ought to go up and down and down so will depend on time and it will also depend on sea where you are but it won't depend on the other direction so moving them the z-axis we can completely ignore these 2 Chan which of the derivatives of flight in the direction which doesn't change occurs is not there Our wave equation for way of moving along Z along the the z-axis sorry should be easy you shouldn't show illegal 0 what kind of solutions for such an equation had we talk about last time and there are 2 basic solution this equation has 1 of them is a function of Zepa plus 2 a mail order as a function of my to freeze 2 solutions which represent waves moving right along the z-axis and waves moving this 1 represents waves moving the right along with positive z-axis this 1 represents waves moving along the electoral what functions any function any function any shape wave profile initiate wave profile and allow it to just moved uniformly along busy direction that constitutes a weighty moving down the z-axis along the positives in direction as a function losing minus wave move in the other direction then receives the same function of plus so that's the basic facts that we learned last time about waves and 1 particular special kind of array acinar cosine wave but just right down for future reference wave which really has a way he looked away advanced by you guessed that assigned co-starring so for example a waiver of this type would the right to be a solution of the wave equation which has the form cosigned of Z minus zeal dynasty because it will incur rights and furthermore because throwing constant here usually called Katie the wave number of any kind the big constant is incident way the short of wavelength of a wave of Keyes Big Wave looks like this case all the wave looks like most the basic facts that I want to remember about wave equations and in particular wave solution that their functions as you might easy plus at different wavelengths and they could be signs signs Motorola right removed alive but that Alito normally before we go ahead with Maxwell's equations just to remind you just 2 hours a week as we went through this I will remind you remove the mathematics elementary mathematics 1st of all is the notion of across product of 2 vectors of we have any 2 vectors diesel ordinary vectors not for vectors now ordinary three-dimensional vectors and three-dimensional space and we have the crossed crop b . 80 b cross a Excuse me crossed from of 2 vectors the across product is itself another vector of we're to electors each pointing in some direction will given away we can multiply them together and make another vectors the cross product is defined by its components so for example X component I'll be crossed Eddie is defined as a definition the find the bees BY planned a X minus B X Prize a wife and there are few once defined the other components you just cycle through 0 x y and z so for example b crossed AT a the white component of extra white equals beat well why ghost Z X Ruth extra white busy busy backpacks because they SRY goes they X ghost Myers be Y X why a ghost city likewise of Alaska doing was wants a won't go again b cross a Z component of is equal to X Y them at a White z electron most BX BX X Y limits for most busy miners busy a X these uh definition these definitions of across product between 2 vectors notice of the cross product itself has components and so it must be a vector now another way of multiplying 2 vectors together is the byproduct the byproduct the byproduct of 2 vectors is a scalar a dot B and only has 1 component of sustained in every reference frame are saying when you rotate the reference frame and its definition is a X B X plus a Y BUY plus a Z be redial Beckham AX BX a force using busy that's the bar product and it's a scalar that's 1 set Our concepts that that we've been through several times now the onset of concepts Corolla vector and the emergence of a vector nearly think about that Bill stop out of derivatives of the components of the vector partial derivatives with respect that x y and z while a half til imagine you're here is that the ridden with respect to X and renewed with respect Hawaii and derivative with respect disease can that those symbols on the 3 components of a vector then not the 3 components of the derivative but just pretend to 3 components of a vector
couldn't be take b by deep team of F T no plus the fact let's a specialized let's take New will could be Z the Sephardic let's take it easy and see what we get me by the C of F P Z plus debug EX of F X Z plus a bloody Of f Y Z plus but z of F Z easy Z equals 0 basso this would say ritual or Lepista forever would respect disease that is easy and easy 0 because all of the diagonal elements 0 0 reconfigured this this isn't there Have Caesia 0 the world's F X as well as to what F X zee years we got what to we X role and Z column that be so let's just target area that BY what it is F Y easy Fermat we go to wife a wider role Zepa column rats Maria's be Cimex or vectors of minus abusive acts worth looking good this is Dedie by DXO BY minors baby wife of BX whose looks very much like a piece Of the curl of bees that's where it's what about F T Z while we go to the key role place now we get money list said so this is minus a derivative of the E sub Z With respect to tea all of that equals 0 well if you look at it carefully you'll see that's exactly the equation over here the Z component of time derivative of E sub Z it is busy component of the Corolla at exactly what we have here we checked these policy which component will get worse see component of the year Lizzie component just gives us back were Maxwell equations namely busy component of our Maxwell equation they also the case which is interesting it is to right down the time component here all the key component 4 Our well let's go let's continue to call new no let's work our thee kind component of this equation wears street now 1st of all it has a term which is Debye of FTT at Newark was seat but 0 FTT that's right here as 0 0 0 0 and Italy have grossed debug be X of F X T the 1st debug of F. White Post buddies 3 of zee woes effort was F X 3 white earns each other just the components of the electric field F X T minus Wilmer assigned is east of X Aziz of the other 1 of these all this says 1 package neatly derivative with respect to x he ex post the derivative with respect Hawaii Of wiII posted derivative with respect to Z of easy is equal to 0 that reference is just Dell bodies 0 for the moment we been ignoring charges and current density Georgian cards so you concede that my bed told the said 4 of the Maxwell equations 4 of the 8 Maxwell equations just have performed a relativistically called variant even every reference frame victory equation that's a vector equations because the left-hand side is a vector it's a vector because it is derivative we've contracted the index is only 1 index left over so this is a vector equation it's the same in every reference point that if you went through the entire labor of doing the Lorraine's transformation was on the coordinates and on the components of E and you would find out that after Lorentz transformation the equation was exactly the same what about the other half Maxwell equations well both have exactly the same form accept but With twiddle here Of course they have exactly the same form as little as saying his efforts it would be in be interchanged an extra miners are thrown exactly what Cedar you go from here to here I the changing electric and magnetic fields with actress signed here equations are the same so this is a a powerful way to to rewrite Maxwell's equations again without charges without current densities This is a powerful way to rewrite them is simple elegant form but it's not the simplicity and elegance which is important it's the recognition that these are for vector equations and therefore the scene in every reference frame so Maxwell's equations then a relativistically invariant they don't change from frame to frame will cut back charges and current awhile but or maybe next time with time but Armed now that we've found Maxwell's equations reversing every reference what do they say what are the implications of these 8 equation what kind of electromagnetic fields they allow what they described so let's again forget currents in density for the moment 0 0 nothing and let's study these 4 enough for equations equations but study them and see if we can make wave equations out ever go back and what we did with a wave equation have someplace yes areas the wave equation is an equation with 2nd derivatives 2nd derivatives
mean differentiate twice Maxwell's equations of all have only 1st derivatives but they have 2 distinct field Ian be we'll find out that having musical first-order equations because they only have 1st derivatives but we have first-order derivatives first-order equations that and 1st order equations beat 2 variables within a fine out that that's equivalent the having 2nd order equations for either the electric field or the magnetic fields separately this is not hard to do this is easy to do so we need but only the clear blackboard board over here only to manipulate us a little bit I have been Belo mixed up but I want to write equation just for the electric field I wanna try isolate just electric field and see how it behaves so here's what I do let's say take this equation over here and let's differentiator with respect to time it's got only 1 time derivative in it let's differentiated again with respect to time so that we get a 2nd time there will be these
2nd by decree squared 2nd derivative electric field and that's going to equal Del crossed the kind derivative of B but differentiate with respect the time of the year adjusted the 2nd time the rural electric field of I differentiate with respect to TO here that differentiates the magnetic field with respect to Part OK that helped yet I still have electric and magnetic field the same equation the goal was to get rid of a magnetic field but look right he'll have the magnetic field audits turning derivatives written in terms of the electric field so I can be clever and simply get rid of BBB here it curl of 80 minus occur Levine missile it's just strips Butler's wasn't it was Dell Cross TB and now for we substitute minus curl of 80 gives us some minor can curl of each here we are an equation which only involves electric field 2nd respect feet from the left hand side and various complicated the rivers of right here so I was going to do is I'm going to take these acts component of this equation both sides of the equations are vectors he is a vector differentiating suspected time leaves of a vector and the try my dear could Kuo was also were vector so much take read X component of both sides X component of both sides what we have to calculate X component of the kernel of the world and slightly tedious operation if you like keeping those sleep for 2 minutes until I get the answer but I will do it on the blackboard for your arcade Dalcross double cross he the x component for ethanol is D X component Koral the X component of our curled is equal truly derivative Of the Zeke component with respect the whitey minus the derivative of the Y component with respect to Z that we are taking the x component of dough Cross they don't cross that's right so that we want the x component of dealt cross double cross it that means I just substitute a E were received as a substitute no crust EDT but 2nd again that's going to be by DYE Daisy which should be dealt cross E Z component of it best by of double cross e Y component of it but now we substitute our these compartments the same tried the same trick ends here is her real result the grand result Aerts baby why derivative UYA with respect to x derivative of X with respect Hawaii that's Debye DYA of this territory and the nearer to mark plus beebread easy our B X by D Z minus fees see by a V X and they think that's it that's it through terms pocket wake up now the worst is over let's see what we have were calculating X component of vector calculating the x component Rebecca the let's see what we have here we 1st of all have 2nd derivatives of E X with respect the wife who have 1st won't IDY square of the EX as sister and across of ours we've become off offer as theirs we by BY squares to do and it is but the white EX we have derivatives of respect z of E X plus 2nd removal respect busy squared of EX vests and then we get to other terms which local mysterious or are they they
are derivative will respect that acts of BUY EYE sister-in-law idea looks a little different because I've interchange B X and Y derivative right was derivative respect X followed by derivative lose but Hawaii you're allowed to interchange loans of the 1st 10 years debate EX Our BYU wide and that is another term which is deemed X I think it is easy easy money reuse top worn or Barbara looks very elegant profile is that Turner apart and subtract the sinter from the Barber of Seville will have form so let's plus Debye D X squared EX and that's now subtract Deeb ID X of Debye DX EX it added and subtracted exactly the same thing now you should be able to see a pattern of 1st of all we have white he white closed these easy distribute cricket or will sign mistake here we want you why these easy and be XCX appear inside he was that that's just below body but nobody is equal to 0 a former Maxwell's equations and so of this dx dx plus BY BY closed easy easy is just 0 what about this stir this is my 2nd respect why square x squared In squared Of the same quantity east Beck's let's put it together now let's put together what was it he is always good doing the 2nd derivative with respect that X Of he squared is my Bill Crossbill cross he BX comport and that's exactly what we computed over here have long tedious computation just give us all left-hand side erase it and was replaced by what we computed and you'll see a pattern here's a bike x squared EXE plus Dubai BY squared close debugged easy square here if you've kept track which you probably are not from the original wave equation will see that this is just a wave equation for the component acts EX is a field which satisfies Just Say no wave equation that we wrote down 4 5 P remember we wrote that wave equation for find it had exactly the same form and so it b at the electric field itself satisfies a wave equation the gold wave equation the same new wave equation which had right moving and left moving solutions which could be signed in where we learn we learned that they're all waves of electric field from property we've learned that wave equation which gives rise propagating waves a can move to the left or right applies already direction of course but which applies to arm of the scalar field also applies directly the components of the electric field let's see whether we can build ourselves an electrical weight an electromagnetic wave of this equation has solution is of the form for example X is equal to call signs of let's just take a simple example zeal minors this will be a way of moving down the axis wave equation has a solution which is a way of moving down z-axis workers coastlines you minus 2 years I could cut a constant K but let's leave our only complicates things this would be a wave whose electric field is point along the x-axis trust coordinates dizzy here X and his wife sticking out of a block or misses a wave moving down the z-axis its electric field His along the x-axis for
direction of area that's called the surface element a vector service on vectors he said I could vector symbol above it but we are just be sir now we can ask how much charged flows through that area in a unit of Time How much charge flows the
area per unit time or better yet we can ask how much charge for kind per unit area flows through the charge falling through there will certainly proportional to the area of the areas a small amount of charge will be able flow through it for a given situation if the area's made a large a larger amount of charge me flowing through it the charge per unit area pro unit per unit coloring charge per unit area sorts write the words charge per unit area per unit time surface is called the current now the current as a vector the current as a victory you could see why that should be sold current is
kind of like a fool think of it as a flow of all the electrons and protons were flowing through space at any point the flow has a direction namely the direction of year charges of flowing in and also has a magnitude and the magnitude is the charge per unit area per unit hour per unit time flowing through that area that's the definition of the charm of current and as I said it is that if I want to know the amount of charge per unit time flowing from the surface and what I do is I take dot product on the surface element with the current of of them being vectors if examples the current is flowing parallel to the surface that means the college's perpendicular to be stigma In this product 0 Solis current flows the current flows along the surface no current flows from 1 side to the other end of the current is flowing perpendicular to the surface that means parallel to be stigma and then you get the full maximum branch of the flow through the up to to the surface element that's why about prop byproduct measures component of 1 vector in the direction of the other so this measures the component of the flaw in that direction perpendicular to the surface and it is not the charge 3 area unit time but just a charge per unit time this charge per unit time charge per unit time the surface element be stigma so I have the idea of the flow of charge and beyond sometimes called the flocks of charge flow of flocks of charge and current or electric car the old saying they notice would not necessarily talking about current pool wire a car and a wire is a special case and if we opened up a made do the big we could put lose surface elements of the area in there and see the current through a wider is made up out exactly these kind of currents what is broke charged to a wire but that's a special case we can have current flowing through space space current current flowing along sheets or just current flowing along three-dimensional space and that's the definition of day OK now what's concentrate on the following question here have a region but it has some charges and the charge America is integral over the volume of the charge density if their occurrence flooring maybe currents flowing into war out of the region let's shows let's chose to make up the boundary of the region as a whole bunch of surface elements at the boundary is made up by 0 means I mean the bank because this is a boon to the rubber of the balloon we would make up at a level surface elements here and now that by definition point there little be signal is out of show is to make them out of a book suppose angers met current flowing Out of the blue what does that say about the charge inside the and simply not a managing the balloon changing its sizes thing it's just a fixed region space if this current flowing Net current flowing out of every surface than the current than the inside must be changed the only way that the current can change and the only way the charge could change on the inside the only way the charge to change on the inside is passing from the inside of a outside so the charges is changing the charge and the inside is changing you could be certain that there must be a flow of current through the boundary and that tells you there must be a J the Vatican write down a simple equation we can write down at that time the rate of change which ends of the charge inside this region Q. by that's equal to the end there will be lover you roll by today the time rate of change of electric charge inside this region must equal the car in the passing out through the band actually would the opposite side of current is passing out the boundary with a positive sign then the charge was decreasing inside so it must have a minus sign this formula I want total charge that passes out of this region per unit time the public charge passing out of the region per unit time is just integral of the current darted into the surface we just ate up all current passing through each 1 of these service elements that seemed to grow here and this is a theorem now or few like perhaps it's a law nature rather than a that we kind rate of change of the current inside is equal to flux of charge passing out to the surface of a bloke that's given by the integral of a current now we need a truth mathematical theorems rather than a law nature mathematical theorem which won't cruel it's called doused his sorry is not called guesses what is called gasses look up Dallas's theorem sometimes called the divergent while tell you whether there's it's a relationship between these occurs is called a surface integral it's an integral Ave vector quantities over a surface of rebounding surface it can be related to an
integral although the interior and he is the of called gaps 0 because says that the integral are of any vector integrated over the surface His equal this isn't it the growth over the surface court surface is equal 20 Indigo or volume by Mike last year I would ask them what goes in there not now OK the divergence of that that divergent vector start to relationship between the surface integral of vector and the interval although the volume of the year of the divergent Our lobbyists from Margaretta prevented it's actually doesn't take remedial lines to prove but let's assume that still met them we can write the right hand side of this equation as an integral although the volume Hindu over the volume another and the role of the volume of the divergent of recurring it's a spreading out of the current spreading out of the current he notes like it's almost like this file give you a picture of divergence of vectors if you thought the vectors as a flow vector of some water perhaps our arms full vector of some water and supposing some water was being pumped Ian at each point of space some funeral pipes that you can't see bringing water in from outside and dumping it in the pipes aided by different places in space so the water is better verging How would away from the tips of the pipes and going out and filling up space the divergent of the water on the bridges of a flow of the water is basically just beat water coming out of these individual pipes what this says is that the amount of water that's coming out of these little pipes ending at this point and flowing out ways were adult has out of the surface vessel out of the surface and that's this expression he ever beverages be integral Over the boundary arm of week should 1st floor it is also illegal to the integral over the interior of volume of the divergence of the vector so answer general fears about vectors it doesn't have anything particular to do with electricity it doesn't have anything particular to do anything special other the mathematical properties of vectors personally have removed theorem that integral over any volume and think any volume whatever we had to go over any volume of the time rate of change of the density of charge is integral over the volume of the divergent Of the current if Julie in the goes like this equal but their equal not only for some particular volume but their true it's true that every volume that you can think of and there's only 1 way that that can't be true effect any region that you could pick me into go over the volume of 1 thing is equal to the into global volume of another thing the only way that that can be true is further things themselves to be equal that again as a mathematical theorem but it's more or less obvious that affect any region the interval of 1 quantity is equal to the role of the quantity 2 quantities have to be equal we can't do it in the conclusions about the connection between charge density and current it says that the derivative with respect to time of the charge density this plus bringing everything over the left hand side of the divergence of the current what's right about derivative with respect to acts of J sub X derivative with respect the wife of J White Post derivative with respect Z J sub z equals 0 not this is to be true in every frame of reference and I didn't specify anything particular about a particular frame of reference when I told you this little story be chill every frame of reference it better have the form of some invariant equations and it's not hard to imagine how you make an invariant equation that you 1st of all say but the current but 1st of all you say that the current JXJ together with the row for quantities Raul J. X excellent X upstairs now JAY Genesee former a 4 vector they formed a format there J. Neil when you goes from time off and the 3 components of space that is what is suggested by having these 4 components rope JXJ wife injuries and they end this equation which is called the continuity equation call the continuity equation that takes the a simple form that Dean New Jersey New is equal to 0 that say it says derivative respected T of JET plus derivative with respect to X J X Loblaw blob is equal to 0 which is exactly what the so the way they make this continuity equations BAA equation that's true in every frame of reference is to think of the 4 components of current and density as a four-day that's not too surprising current current represents the velocity of the motion
of the charges as we've seen over and over again the velocity vector Is it for that their velocity vector of an object the relativistic velocity vector of an object is a former actor and so was not so surprising that the floor fluid should also be described by a 4th and for that there is a former rector of current density well OK now we have any object before a vector of current and velocity that we can try put into Maxwell's equations somehow let's go back to Maxwell's equations which seems to have disappeared after blackboards sagas works here that have been around the end of it so it's right Maxwell's equations again this time putting near of the charges in currents amigo Dell thought he is able to roam 0 . be physical 0 across busy Hooper minus by IDT draw vector equations and bill Crosby physical baby city plus J. wedge aid is exactly the same J. that I defined here the current per unit area for time what I assert her which were reject these equations are exactly the same as the following equations being f Nuno whatever you know it is ever muumuu is poor by 4 and symmetric matrix of cancer which is built by a bout of electric and magnetic field components the fact is equal to Jane New Norris the left-hand side of contracted the new index so the new index is up really a real index here it just as 1 new index right him the with 6 OK let's check reduced simple called variant where this is clearly a core variant equation differentiating it can't give you a vector misses a back and sell this equation would be invariant saying every reference for assuming a J transforms as a vector so let's just check that we get the same equation a month and the quality of the ones the the ones it's relevant here oddly ones with the right hand side the ones with current and charge against there are 4 equations CFO for a different values of therefore equations here 1 2 3 4 so let's check to see if any of them others 1st of all let's take the kind component time component would be deemed U. S. new is equal to GATT always JT that's kind component of the lovely for vector on current sore right knee inside we have role for what about the left hand side doable left inside as we have the Golan rewrite the field cancer but we don't need the whole thing really need to remember years let's say that the army in 3 hour that we have the components of the electric field 0 X EYE and easy formed here we also have minus EX minus EYE my easy and then we have a magnetic field sitting in here and various ways OK so let's we have we have only the time components of af that means the components along was 1st role here to write this as D X F X Frost BYD FYT applause Desi and GET missing 1 yes missing 1 missing D T F T but FTT is 0 already bad elements of the maker of 0 no FXT FYT and those of a component of the electric field source says this is just D X he ex post BYE BYE was easy easy another words it Is No . 80 a left-hand side Bell but right here Rob Rico go through the other components words exactly the same thing instead of taking the time component here we could take space component for example that the steeple the J X and we would discover the x component of this equation so the upshot of rivers that the connection between charges and feel the way in which charges charges and current electric charges electric currents the way they generate fields is Lorentz invariant it's the same in all reference right on the other hand we've already seen the way that the fields influenced the charges as voices on them also the same in all reference price so we see that basically all of the equations of electromagnetism can be written in a form in which we can immediately checked the cancer equations and therefore the same every reference frame that was the real goal of this show that the laws of electromagnetism saying every reference frame No. 1 and No. 2 could demonstrate that the laws of electromagnetism lead polite waves moving with the speed of light from most of the things as follows light travels with the speed of light every frame of of reference that sort of closes up and this is up V Cast started by Einstein on rewriting all laws of physics in a way which is St. in every race In every frame of reference power a Dedham I'm her
a room the preceding programmers copyrighted by Stanford University please visit us at stanford .
Münztechnik
Summer
Magnetisches Dipolmoment
Lichtgeschwindigkeit
Reaktionsprinzip
Computeranimation
Teilchen
Bildfrequenz
Fahrgeschwindigkeit
Herbst
Klangeffekt
Array
Relativistische Mechanik
Kaltumformen
Behälter
Elektronisches Bauelement
Licht
Elektrizität
Abend
Übungsmunition
Magnetische Kraft
Druckkraft
Schiffsklassifikation
Gleichstrom
Großtransformator
Elektromagnetische Welle
Lineal
Starter <Kraftfahrzeug>
Gewicht
Gasbohrloch
Elektrisches Signal
Bergmann
Lichtgeschwindigkeit
Leisten
Elektrischer Strom
Begrenzerschaltung
Bildqualität
Satz <Drucktechnik>
Feldeffekttransistor
Woche
Rotationszustand
Discovery <Raumtransporter>
Bildfrequenz
Gasturbine
Passfeder
Vorlesung/Konferenz
Brechzahl
Klangeffekt
Stunde
Array
Wellenlänge
Kaltumformen
Elektronisches Bauelement
Licht
Metallschicht
Hydraulikleitung
Rauschzahl
Druckkraft
Übungsmunition
Magnetische Kraft
Elementarteilchen
Weiß
Eisendraht
Dieselaggregat
Gleichstrom
Profil <Bauelement>
Mikrowelle
Scheinbare Helligkeit
Elektromagnetische Welle
Rucksack
Ringgeflecht
Initiator <Steuerungstechnik>
Frequenzsprungverfahren
Fliegen
Magnetisches Dipolmoment
Lichtgeschwindigkeit
SEED
A6M Zero-Sen
MAC
Elektronen-Energieverlustspektroskopie
Bildfrequenz
Nachmittag
Ausgleichsgetriebe
Array
Kaltumformen
Elektronisches Bauelement
Licht
Diesellokomotive Baureihe 219
Serientor-Sampling-Leitung
Magnetische Kraft
Gleiskette
Werkzeug
Eisendraht
Vakuumphysik
Matrize <Drucktechnik>
Leistungsanpassung
Magic <Funkaufklärung>
Ruderboot
Direkte Messung
Kraftfahrzeugexport
Elektrofahrzeug
Schnittmuster
Elektrischer Strom
Optische Kohärenz
Triebwerksgondel
Feldeffekttransistor
Spiralbohrer
Gasturbine
Walzmaschine
Brechzahl
Bahnelement
Klangeffekt
Tagesanbruch
Stunde
Kraft-Wärme-Kopplung
Windrose
Elektrizität
Gleitsichtglas
Schiffsklassifikation
Schlagwerk
Großtransformator
Onkotischer Druck
Source <Elektronik>
Magnetspule
Feinkohle
Fahrzeugsitz
Magnetisches Dipolmoment
Target
Bergmann
Störgröße
Feldeffekttransistor
Bildfrequenz
Gasturbine
Elektrische Stromdichte
Passfeder
Fertigpackung
Brechzahl
Bahnelement
Ausgleichsgetriebe
Array
Gasdichte
Kaltumformen
Pulsationsveränderlicher
Waffentechnik
Elektronisches Bauelement
Diesellokomotive Baureihe 219
Bett
Kardieren
Band <Textilien>
Übungsmunition
Magnetische Kraft
Großtransformator
Jahr
Magnetspule
Holzfaserplatte
Gewicht
Werkzeugverschleiß
Schnittmuster
Minute
Feldeffekttransistor
Strippingreaktion
Zylinderblock
Gasturbine
Fuß <Maßeinheit>
Tagesanbruch
Array
Kaltumformen
Kraft-Wärme-Kopplung
Waffentechnik
Windrose
Elektronisches Bauelement
Lenkflugkörper
Eisenbahnbetrieb
Magnetische Kraft
Gleiskette
Gleichstrom
Profil <Bauelement>
Jahr
Mikrowelle
Elektromagnetische Welle
Ersatzteil
Direkte Messung
Bergmann
Lichtgeschwindigkeit
Chirp
Leitungstheorie
Bildfrequenz
Zirkularpolarisation
Gasturbine
Elektrische Stromdichte
Fahrgeschwindigkeit
Ausgleichsgetriebe
Zugangsnetz
Elektronisches Bauelement
Licht
Donnerstag
Elektrizität
Tag
Divergenz <Meteorologie>
Magnetische Kraft
Steckdose
Weiß
Plattieren
Gleichstrom
Jahr
Mikrowelle
Elektromagnetische Welle
Ersatzteil
Gewicht
Erder
Magnetisches Dipolmoment
Direkte Messung
Bergmann
Ancher, Mortlock & Woolley
Satz <Drucktechnik>
Minute
Förderleistung
Computeranimation
Teilchen
Grau
Marsmond
Zelle <Mikroelektronik>
GIRL <Weltraumteleskop>
Bahnelement
Array
Eisenkern
Gasdichte
Großkampfschiff
Basis <Elektrotechnik>
Energieeinsparung
Weltall
Elektronik
Übungsmunition
Weiß
Nassdampfturbine
Motor
Gleichstrom
Jahr
Scheinbare Helligkeit
Absolute Datierung
Lineal
Feinstblech
Messung
Elektrisches Signal
Direkte Messung
Kraftfahrzeugexport
Konfektionsgröße
Elektrofahrzeug
Energieniveau
Förderleistung
Wasserbeckenreaktor
Computeranimation
Grubenstempel
Flavour <Elementarteilchen>
Gasballon
Energielücke
Bahnelement
Stunde
Array
Konzentrator <Nachrichtentechnik>
Elektronisches Bauelement
Tag
Gas
Elektronik
Übungsmunition
Satzspiegel
Eisendraht
Gleichstrom
Scheinbare Helligkeit
Jahr
Walken <Textilveredelung>
Intervall
Gagat
Lichtgeschwindigkeit
Leisten
Leistungssteuerung
Leitungstheorie
Woche
Bildfrequenz
Monat
Elektrische Stromdichte
Fahrgeschwindigkeit
Wasserfahrzeug
Feile
Array
Eisenkern
Kaltumformen
Seil
Rollsteig
Elektronisches Bauelement
Diesellokomotive Baureihe 219
Hydraulikleitung
Divergenz <Meteorologie>
Formationsflug
Magnetische Kraft
Weiß
Jahr
Stückliste
Walken <Textilveredelung>
Ruderboot
Direkte Messung
Elektrischer Strom
Bildqualität
Förderleistung
Blei-209
Gasturbine
Frost
Brechzahl
Energielücke
Bahnelement
Kommandobrücke
Klangeffekt
Steckverbinder
Stunde
Gasdichte
Waffentechnik
Elektrizität
Nassdampfturbine
Source <Elektronik>
Mikrowelle
Magnetspule
Computeranimation