Electromagnetism and Relativity

Transcription

1 Chapter 6: Idea 5 Eletromagnetism and Relatiity The fats are relatie, but the law is absolute. When you understand this statement, then you understand Relatiity! Introdution We hae taken an historial approah to the study of Physis. We are now at the turn of the last entury the 0 th entury that is (so around 1900AD) There were three major sientifi aomplishments in the 0 th entury And two of them were in Physis! 1 3 The Big Three The Struture of DNA Watson and Crik, the Double Helix Quantum Mehanis Unierse shaking Idea #6 The physis of the ery small: atoms The Theory of Relatiity Unierse shaking Idea #5 The physis of the ery fast Introdution Relatiity is an old idea Not inented by Albert Einstein But perfeted by him Been around in arious forms sine the Copernian Reolution Questions about the moing Earth and the appearane of the night sky Introdution Relatiity does not say Eerything is relatie. In fat it says some things are not relatie and explains the onsequenes of this If offers important new insights on many fundamental quantities Spae, Time, Mass, Graity Introdution The modern Theory of Relatiity is Einstein s resolution to some problems Some inonsistenies in Physis It hanged our understanding di of basi ideas Spae and Time Motion Graity Spae And it hanged how we think about them Introdution The Theory of Relatiity desribes how different obserers iew an eent Desription depends upon your point of iew Depends on the Rl Relatie Veloity Vl between the obserer and the eent By different obserers we mean obserers with different Relatie Veloity to the eent Introdution So Relatiity is a theory about motion! Different obserers different desription Different different Relatie Veloity To understand Relatiity we need to understand Motion So we need to understand Spae and Time! 7 8 9

2 Relatiity Motion: moing through Spae and Time Veloity = Distane Traeled Elapsed Time Distane Traeled a hange in Position a Spae Interal Elapsed Time a Time Interal Spae Interal Veloity = Time Interal 10 Relatie Veloity Position, Time, Veloity are all measured relatie to something We desribe our Position (Loation) as 10 miles due east of somewhere Or with Latitude, Longitude, Altitude relatie to the Earth s enter The distane traeled is the Spae Interal between start and finish 11 Relatie Veloity We measure Time relatie to some starting referene point Relatie to some date: 0 AD Relatie to some time: Class started at 11 AM We measure the elapsed Time from when the lok starts to when it stops The elapsed Time is the Time Interal between start and finish 1 Relatie Veloity Ready, Set, Go! One hour later When we say our speed is 65 mph we mean 65 mph with respet to the road 50 mph 50 mph What is the relatie speed between two ars? One going north at 50 mph relatie to the road One going south at 50 mph relatie to the road Let s see 13 Eah ar s speedometer reads 50 mph 50 mph Relatie to the road! mph 50 miles 50 miles 100 miles 15 Relatie Veloity So eah ar is moing at 50 mph relatie to the road But they are moing at 100 mph relatie to eah other One hour later they will be 100 miles apart! Their Relatie Veloity is 100 mph! Trik Question? What is your eloity right now? The answer depends upon your point of iew You are at rest, so your eloity is zero with respet to the room Your eloity is about 67,000 mph with respet to the Sun Whih is the right answer? Relatie Veloity The right answer to the question How fast are you going? is relatie to what? The answer depends upon your point of iew!

3 Absolute ersus Relatie To fully understand the Theory of Relatiity you must understand these two onepts! In Physis, we measure or alulate many quantities Position, Veloity, Aeleration, Time Mass, Momentum, Fore Kineti Energy, Potential Energy Do all obserers get the same result? Absolute ersus Relatie If a quantity is ABSOLUTE all obserers agree on its alue Its measured alue does not depend upon the relatie eloity between obserer and eent Also alled an Inariant quantity Absolute ersus Relatie If a quantity is RELATIVE all obserers do not agree on its alue Its measured alue does depend upon the relatie eloity between obserer and eent Also alled a Variant quantity Galilean-Newtonian Relatiity Relatiity is the answer to this question: What is the true eloity of the Earth relatie to absolute spae? First we must answer another question: What is absolute spae? Frames of Referene Frame of Referene a 3-dimensional objet used to desribe motion A kind of map of 3-D Spae It is a mathematial deie Remember the equant, eentri, et Motion is measured relatie to a partiular Frame of Referene Usually we are at rest relatie to our own Frame of Referene Frames of Referene Eah Frame has a Coordinate System A measuring deie with numbers that is attahed to the Frame of Referene Defines an origin where the zero point is! We an measure the loation of any eent by determining its three oordinates Coordinates where it is loated in Spae 3 4 Frames of Referene To measure an objet s motion relatie to a Frame of Referene we speify how its oordinates hange as time goes by Motion hange in Position Measure Motion hange in Coordinates 5 Frames of Referene Example: How fast an I ross a room? Referene Frame: Room We measure my motion relatie to the room Coordinate System: Floor tiles We use the tiles to make the measurements Origin: start of the first tile 6 Ready, Set, Go! Initial Position x = 0 tiles Let s start at the origin! Initial Time t = 0 seonds 7

4 Fie seonds later Veloity Veloity = Distane Traeled Elapsed Time Frames of Referene My Veloity is desribed ompletely in terms of the Frame of Referene x = 7 tiles x = 7 tiles The result would be different in terms of a different Frame of Referene or with respet to a different obserer Change in Position 7 tiles = = seonds tiles seond Beause Veloity is RELATIVE! A ariant quantity t = 5 seonds Elapsed Time 8 t = 5 seonds 9 30 Frames of Referene Bak to our question Absolute Spae The Frame of Referene that is at rest relatie to all other Frames of Referene Eery obserer would always agree on any measurement made relatie to Absolute Spae Frames of Referene There are two kinds of Referene Frames Inertial Referene Frames Moe at a onstant eloity Constant speed and diretion The Earth is a good approximation Non-inertial Referene Frames Moe at a hanging eloity Also alled Aelerated Referene Frames Relatiity The Theory of Relatiity Desribes how obserers iew an eent from different Frames of Referene By different Frames of Referene we mean Frames of Referene with different Relatie Veloities to the eent Absolute ersus Relatie An ABSOLUTE quantity has the same alue in all Inertial Referene Frames Example: the speed of light A RELATIVE quantity has a different alue in different Inertial Referene Frames Example: an objet s eloity Relatiity Veloity is a RELATIVE quantity Your eloity relatie to an eent obiously depends on your eloity relatie to that eent So the alue we measure depends upon what we use as a referene Reall our two ars: 50 mph relatie to road, 100 mph relatie to eah other Relatiity Sine Veloity is relatie, then Momentum: m Kineti Energy: ½m Potential Energy: E KE are relatie too. So?

5 Example Two kids are playing ath in the bak of a pikup truk. They gently toss a ball bak and forth They an throw a ball about 5 mph while the truk dries at 100 mph. Do not try this experiment at home. It only gets worse Example One of the kids gets arried away and throws the ball out of the truk. This happens as the truk passes a pedestrian waiting to ross the street, and the ball strikes the pedestrian in the head! How fast is the ball moing when it hits? Example To the people in the truk, the ball is moing at 5 mph. Its Veloity is 5 mph relatie to the truk. To a person on the sidewalk, The ball is moing at 105 mph. Its Veloity is 105 mph relatie to the sidewalk. Ouh! Somebody all 911!! 105 mph 100 mph Example In the truk frame of referene, the ball has a small Veloity and a small Kineti Energy In the sidewalk frame of referene the ball has a huge Veloity and a huge Kineti Energy Fats ersus Laws The Fats are Relatie Different alues for Veloity, Kineti Energy But the Laws are Absolute Newton s Laws are alid in both frames Energy is Consered in both frames All obserers agree: Energy was onsered They just don t agree on how muh was onsered! Galilean-Newtonian Relatiity Absolute Quantities in Newtonian Physis Relatie Quantities in Newtonian Physis Galilean-Newtonian Relatiity Desribes whih quantities are Absolute aording to Newton s Laws Inluding the Laws themseles Newton s Laws are Inariant in all Inertial Referene Frames Length Time Mass Fore Aeleration Laws of Mehanis All obserers always agree on the alue of these quantities Stationary obserers and Moing obserers Veloity Momentum Kineti Energy Potential Energy Position Measured alues of these quantities will be different For obserers in different frames of referene

6 Galileo Gae the first sensible answer to our question about Absolute Motion He said Mehanial Experiments annot detet Absolute Motion There is no way to detet Absolute Motion by doing a Mehanial Experiment Galileo Reall our demonstration about falling balls A Mehanial experiment Do the experiment in the lab Both land at the same time ( ½ seond) Do the experiment in an airplane At an altitude of 7 miles aboe the ground Moing 500 mph relatie to the ground Galileo I get the exat same result in the plane They take the same time to land as in the lab They still land simultaneously The results must be the same otherwise I ould tell from the experiment that I was moing Priniple of Relatiity Aording to Galileo The Laws of Mehanis are not hanged by inertial motion There is no way to detet tinertial motion by doing a Mehanial Experiment You annot detet inertial motion unless you look out the window See a different referene frame! Priniple of Relatiity Inertial Motion onstant Veloity We annot feel Inertial Motion We only feel the Aelerations The hanges in Veloity Example: Only feel the bumps and turbulene To reiew: Relatie Veloity Speed of one obserer as measured by another The Theory of Relatiity it Desribes how different obserers with different relatie eloities iew an eent A theory about motion, spae and time Referene Frame A 3-dimensional objet used to desribe motion A kind of map of 3-D Spae Inertial Referene Frames Inertial means onstant eloity Constant speed in a straight line No aelerations! There is no way to detet inertial motion by doing a Mehanial Experiment 5 ABSOLUTE Absolute ersus Relatie Measured alue does not depend on relatie speed All obserers get the same result RELATIVE Measured alue does depend on the relatie speed Different obserers different relatie speeds with respet to the eent get different results 53 The Fats are Relatie Different obserers, different measured alues The Law is Absolute All obserers agree that Energy is onsered. Howeer, a new field of study was emerging to hallenge these onepts Mid 1800 s 54

7 Eletromagnetism Galileo s Priniple of Relatiity Mehanial experiments annot detet Inertial Motion When Maxwell deeloped d his theory of Eletromagnetism, it raised a possibility Can Eletrial or Magneti experiments detet Inertial Motion? James Clerk Maxwell ( ) Born in Edinburgh, Sotland Brilliant but shy Sotsman Studied Math, Astronomy, Chemistry, Eletriity/Magnetism Died at age 48 of abdominal aner James Clerk Maxwell James Clerk Maxwell Published his first paper when 15 years old Math paper on oals Graduated from Trinity College (England) in 1854 Degree in Mathematis Mostly self-taught though James Clerk Maxwell Mathematially proed the rings of Saturn had to be small partiles (not solid rings) in order to be in a stable orbit Confirmed by Voyager I spaeraft in Marh Helped formulate the Kineti Moleular Theory James Clerk Maxwell Most important work (in 1873) 4 equations linking Eletriity and Magnetism New field alled Eletromagnetism (E&M or EM) Called Maxwell s Equations today They proe that light is an E&M wae! One of the greatest mathematial ahieements of 19 th Century Physis! Eletromagnetism Maxwell s theory explains both Eletriity and Magnetism A ombined (unified) theory More on this in a later hapter Before Maxwell, E&M were onsidered separate, distint phenomena Maxwell showed they are related Gae a unified theory of E&M Eletri Charge Stati Eletriity That shok you get from a rug Holds a balloon to a wall Caused by the transfer of Eletri Charge from one objet to another Eletri harge annot be reated or destroyed! Another of those onseration laws Eletri Charge Eletri Charge is onsered! Eery known proess onseres harge Total amount of Charge neer aries A iolation has neer been obsered Charge annot be reated or destroyed Charge an only be transferred

8 Eletri Charge Eletri Charge What holds a balloon to the wall? There are two kinds of Eletri Charge Named by Benjamin Franklin Like harges REPEL The balloon starts out Neutral Equal amounts of Positie and Negatie harge Positie Charge arried by Protons Negatie Charge arried by Eletrons Note: Eletrons are ery light so they are muh easier to moe than Protons Proton mass ~ 000 x eletron mass 64 Eletri Fore Opposite harges ATTRACT Eletri Fore Eletri Fore 65 Rub the balloon on your hair Transfers eletrons from Balloon to Hair Balloon is now positiely harged Hold balloon against the wall Negatie harges in wall attrat positie Balloon 66 Eletri Fields Eletri Fields Eletri Fields How do Eletri Charges feel eah other? A harge here an feel an eletri fore from another harge there How is the Eletri Fore transmitted? By the Eletri Field A property of Eletri Charges The Field is assoiated with a Charge Eery Eletri Charge reates an Eletri Field whih exerts an Eletri Fore on other Eletri Charges Eery Eletri Charge is influened by the Eletri Field reated by other Eletri Charges Eletri Fore on a stati Eletri Charge Fore F = q E Eletri Charge Eletri Field Eletri Fields The Eletri Field is a etor Magnitude how muh Stronger Field larger Fore Diretion whih way Points the way a positie harge would moe Away from Toward Eletri Fields The total Eletri Field is the etor sum of all the Eletri Fields of all the Eletri Charges present The total Eletri Field depends on Geometry: how the harges are arranged Kinds of Charge: Positie or Negatie Eletri Fields We an draw the Eletri Field using Arrows Arrows tell us the magnitude Closer together stronger Field Arrows tell us the diretion Point the way a positie harge would moe

9 Strong Field near the Charge Eletri Fields Repels other harges Region of Large Eletri Field Eletri Fields A region of Zero Eletri Field Eletri Fields The density of the field lines tells the field s strength Weak Field far from Charge Isolated Positie Charge 73 Dipole: equal and Charges 74 Two equal Charges 75 Eletri Fields Eletri Fields Magneti Fields Region of Constant Eletri Field Eletri Fields are a useful way to alulate the total Fores exerted by a olletion of Eletri Charges Eletri Fields are a property of stati Charges Stati not moing What happens when they are moing? Moing Eletri Charges onstitute an Eletri Current Eletri Currents are measured in Amps An eletrial unit you may know Eery known Magneti effet is due to Eletri Currents -- moing Eletri Charges! Sheets of equal and Charges Magneti Fields Magneti Poles Magneti Poles Eletri Currents reate Magneti Fields Bar Magnets Comprised of small indiidual id urrents Eletrons moing in Atoms Eery Atom is a small Magnet Permanent Bar Magnet Atoms lined up There are two kinds of Magneti Poles Magneti ersion of Eletri Charges North Poles N South Poles S Note: there is an important differene Magneti Poles always ome in pairs! No Indiidual N and S monopoles

10 Magneti Poles Magneti Poles Magneti Fields Two smaller N-S pairs 8 Like Poles REPEL N S NS Magneti Fore Opposite Poles ATTRACT N S Magneti Fore Magneti Fore 83 Eery moing Eletri Charge reates a Magneti Field whih exerts a Magneti Fore on other moing Charges Eery moing Eletri Charge is influened by the Magneti Field reated by other moing Charges 84 Magneti Fields Magneti Fore on a moing Eletri Charge Fore F Eletri Charge = q B Veloity Magneti Field Magneti Fields The Fore on a moing Eletri Charge depends on the relatie Veloity between the Charge and the Magnet So a moing Magnet exerts a Fore on a stationary Eletri Charge And ie-ersa The Magneti Fore is RELATIVE! Magneti Fields The Magneti Field is a etor Magnitude how muh Stronger Field larger Fore Diretion whih way Points the way a North monopole would moe Away from N, Toward S Magneti Fields Magneti Fields Magneti Fields Region of Constant Magneti Field Similar to field of Helial Wire Magneti Field Magneti Field urls around Eletri Current Eletri Current urls around Magneti Field Eletri Current Long Straight Wire Eletri Current 88 Helial Coil of Wire Magneti Field 89 Bar Magnet 90

11 Magneti and Eletri Interations Magneti and Eletri Interation Relationships among q, E, B Maxwell unified E&M Showed Eletriity and Magnetism are related They are all interrelated! Start with a stationary Eletri Charge Eletri Charges There are relationships among Eletri Charges Eletri Fields Magneti Fields Apply an Eletri Field Exerts an Eletri Fore on Charge So it aelerates it moes! F = ma Now we hae a moing Eletri Charge whih reates a Magneti Field 91 9 Magneti Fields Eletri Fields93 Relationships among q, E, B Relationships among q, E, B Relationships among q, E, B 1. Stati Eletri Charges reate Eletri Fields. 3. Moing Eletri Charges reate Magneti Fields. 5. Changing Magneti Fields reate Eletri Fields. 6. Changing Eletri Fields reate Magneti Fields.. Eletri Fields exert fores on stati Eletri Charges. 4. Magneti Fields exert fores on moing Eletri Charges Relationships among q, E, B Relationships among q, E, B Light Adapted from Figure 6.4, page Eletri Charges reate Eletri Fields. Eletri Fields exert Eletri Fores 3. Moing Charges reate Magneti Fields 4. Magneti Fields exert Magneti Fores 5. Changing Mag.Field reates an El.Field 6. Changing El.Field reates an Mag.Field 98 The final piee to our puzzle In his theory of E&M, Maxwell proed Light is a wae of hanging E and B fields He een predited the speed of light miles = 186, 000 se = 670 Million miles hour 99

12 Light Light Light is an eletromagneti wae Osillating Eletri & Magneti Fields Traels through spae at the Speed of Light Many experiments showed the wae nature of Light This raised a new question: What is waing? Light Most Waes need a medium Sound waes need Air Oean waes need Water A Wae is a disturbane in the medium Light an trael through a auum There is nothing to disturb in empty spae So there is nothing waing So what is being disturbed???? Light So Physiists inented the Ether Light is a disturbane in the Ether The Ether is ery strange stuff Fills all of spae Massless, yet ery stiff (Light is Fast!) Wae eloity is proportional to medium stiffness Does not affet the motion of objets At rest relatie to Absolute Spae Problems Sine the Ether was at rest Relatie to Newton s Absolute Spae Deteting the Ether offered a hane to define an Absolute Frame of Referene Using E&M offered a hane to eade Galileo s Priniple of Relatiity More Problems Aording to Galileo-Newtonian relatiity All Fores are INVARIANT This sontradits adsmaxwell! Magneti Fore is RELATIVE: Depends on Frame of Referene Depends on Relatie Veloity! F = q B Veloity More Problems This ould be used to iolate Galileo s Priniple of Relatiity Measure the Fore on a moing Charge In two different Frames of Referene The Lab (at rest relatie to Earth) A Car (moing relatie to Earth) Compare results Gies information on the eloity of the Earth! Speial Relatiity The situation around 1900 was this: No experimental eidene for the ether None at all! Relatie nature of the Magneti Fore on a moing Eletri Charge iolates Galilean-Newtonian Relatiity! The Problem No experimental eidene for the ether! The experimental apparatus ould detet an effet 40 times smaller than the theory predited d Yet it deteted nothing, zero, nada, zilh, zip A null result We still detet nothing een today!

13 Albert Mihelson ( ) Albert Mihelson Albert Mihelson ( ) Graduated from U.S. Naal 1 Stayed on for 4 years as a siene instrutor Measured the speed of light so well that his alue was the standard for the next 30 years! Born in Strzelno, Poland Won the Nobel Prize in 1907 U S in 1855 at Came to U.S. age 3 Was a Professor of Physis at CWRU (1883) Met Chemistry Prof Edward Morley there First Chair of the new Physis Dept at the Uniersity of Chiago Mihelson-Morley Experiment Mihelson-Morley Experiment 111 The Problem Relatie nature of the Magneti Fore Conduted for the 1st time in Germany While sering as a Naal Attahé prior to WWI Maxwell s suessful theory of E&M Repeated multiple times in Cleeland Predited the alue for the Speed p of Light g Explained all Eletromagneti phenomena + downstream Oer a 5 year span All with negatie results = Ether drift = Light speed No ether eer deteted! ontradited Galilean-Newtonian relatiity Whih laimed all Fores are ABSOLUTE! upstream - 11 The Solution 113 Albert Einstein Albert Einstein ( ) There were many sientists 114 Reently oted top Working on these problems Some were ery lose to a breakthrough Physiist Sientist Person Many were ready for a new theory He was a generally nie guy! Unlike others we hae seen And aepted the new one quikly Een though it was reolutionary! Of the 0th entury But only one soled the problems Born Ulm, Germany

14 Albert Einstein Showed no partiular intelletual promise A somewhat typial student Did ery well in what interested him Math and Siene Did badly in most others And dropped out of high shool Albert Einstein After leaing high shool Bummed around in Italy for a while To aoid ompulsory German military serie Renouned his German itizenship in 1896 Entered ollege in 1895 in Switzerland Had to ram for the entrane exam Barely passed the non-siene parts Albert Einstein Not a partiularly good ollege student Disliked regimented style Skipped lasses often to read Physis Caused an explosion in a lab He was a horrible experimentalist! Good thing he beame a theorist! Graduated only with the help of a friend who shared his lass notes Graduated in 1900 Albert Einstein Tried to get an aademi position No letters of reommendation Told he ll neer amount to anything Not a Swiss itizen, beause he was a Jew Eentually gained itizenship in 1901 No onnetions No aademi job! Albert Einstein Aepted a job in 1901 Junior offiial at the Swiss Patent Offie for 8 years Worked on Physis in his spare time Kept a notebook in his desk drawer No aademi onnetions Just him and his wonderful brain Performing thought experiments Thought Experiments The Physis ersion of what if Imagine a physial situation Apply the laws of physis In a logial and onsistent manner Analyze what happens Speial Relatiity was the result Of suh thought experiments Albert Einstein Sometime during his drop out year When he was only 16 Einstein first asked himself this: What would happen if I was moing at the speed of light. Later in ollege he hanged it slightly: What would a light ray look like to an obserer moing at the speed of light? Light Reall what Maxwell said about Light A wae of hanging Eletri and Magneti Fields moing at the Speed of Light. Light This is alled a traeling wae A wae that traels from one plae to another Consider a simple analogy A wae moing down a taut rope Snap one end of the rope A single wae traels along the rope

15 Light A stationary obserer Sees a moing wae An obserer moing with the wae In the same diretion with the same speed Sees the wae just standing there This is alled a standing wae Light So, to our moing obserer moing at the Speed of Light The Light wae would be A standing wae of hanging Eletri and Magneti Fields But Standing Waes are NOT allowed in Maxwell s E&M theory! Maxwell and the Speed of Light In Theory (belieed by most to be orret), The only speed allowed for Light miles is the Speed of Light: = 186, 000 There is no allowane for the Speed of Light Relatie to an inertial obserer It was an Absolute quantity! And Einstein notied another problem While riding on the train to work se More Problems: The Speed of Light Measure the Speed of Light On a moing Train Subtrat Maxwell s alue Determine Inertial motion Violates Galilean relatiity Detet motion without looking outside + The Solution The Speial Theory of Relatiity Published in 1905 by Einstein A big part of his mirale year Published 5 papers and got his Ph.D. in 1905 Reeied the Nobel Prize in 191 Only two suh years in history of Physis Newton s year in 1666 Einstein s year in 1905 Albert Einstein All fie were landmark papers One on the Photoeletri Effet: quantum theory of light Nobel Prize material Two on Brownian Motion proed the existene of moleules Two on Speial Relatiity Our urrent topi The Two Postulates Einstein resoled all these problems By making two postulates A Postulate is an assumption He knew that if the postulates were right then all the problems are soled The Two Postulates His was a theoretial explanation On the Eletrodynamis of Moing Bodies Both postulates hae sine been onfirmed experimentally and oneptually Einstein simply reognized the way it should be and must be Knew they were orret een without an experiment! The First Postulate 1. The Laws of Physis are INVARIANT in all inertial referene frames. The Laws of Physis are ABSOLUTE. This postulate is not diffiult to aept!

16 The First Postulate This is an upgrade of Galileo s statement Laws of Mehanis Laws of Physis Now there is no way to determine Inertial Motion without looking outside Not just no mehanial experiment But no experiment at all! The First Postulate Sine the Laws of Physis are ABSOLUTE We an not determine Absolute Motion by any experiment All Inertial Referene Frames are equally alid So all Fores are RELATIVE! The Seond Postulate. The Speed of Light is INVARIANT in all inertial referene frames. The Speed of Light is ABSOLUTE. This one gets a little strange if you really think about it, whih we will The Seond Postulate The Seond Postulate The Seond Postulate Eliminates all the Speed of Light loopholes whih iolated the Priniple of Relatiity But this is new and different! Light Wae Light Wae The Speed of Light is ABSOLUTE The Speed of Light is independent of the motion of the obserer Light Soure Speed of Obserer: Zero (relatie to Soure) Speed of Light: (measured by Obserer) Obserer Light Soure Speed of Obserer: (relatie to Soure) Speed of Light: (measured by Obserer) Obserer The Seond Postulate The Seond Postulate The Seond Postulate Light Wae Light Wae Light Wae Light Soure Speed of Obserer: (relatie to Soure) Obserer Light Soure Speed of Obserer: (relatie to Soure) Obserer Light Soure Speed of Obserer: (relatie to Soure) Obserer Speed of Light: (measured by Obserer) Speed of Light: (measured by Obserer) Speed of Light: (measured by Obserer)

17 The Seond Postulate The Seond Postulate The Seond Postulate Light Wae No matter what the relatie speed between the Light Soure and the Obserer the obserer always measures the same alue miles for the Speed of Light: = 186, 000 se Explains the failure of all attempts to detet the Ether Used ariations in the Speed of Light Due to the motion of the Earth Light Soure Speed of Obserer: any (relatie to Soure) Speed of Light: (measured by Obserer) Obserer Moing in any diretion at any onstant speed The Speed of Light is ABSOLUTE!!! This has been experimentally proen true many, many times If the Speed of Light is ABSOLUTE there is no way to detet the Ether It is a useless onept in Physis Its existene annot be determined! Speial Relatiity With pure genius, Einstein had fixed it! He onluded the Ether onept was garbage Completely unneessary! But there are other onsequenes of his reolutionary new postulates The rest of the theory and this hapter are about those onsequenes! 148 To reiew: Trouble with Light No experimental eidene for Ether Speed-of-Light standing wae not allowed Measure Speed of Light while moing Detet Inertial Motion Violates Priniple of Galilean-Newtonian Relatiity Relatie nature of Magneti Fore Violates Priniple of Relatiity Contradits Newton 149 The Solution: Einstein s Postulates Laws of Physis are INVARIANT There is no way to detet inertial motion by doing any experiment Speed of Light is INVARIANT All inertial obserers measure the Speed of Light to be = miles, se This alue is independent of the relatie speed between the Obserer and the Light Soure 150 Speial Relatiity Einstein s Two Postulates had many surprising onsequenes Redefined the meaning of some basi onepts We ll need new onepts of Spae and Time The Spae interal between two eents The Time interal between two eents Speial Relatiity The Speial in Speial Relatiity Refers to the fat that it refers only to Inertial Referene Frames Speial Relatiity applies to motion at onstant eloity only! No Aelerations! Four Consequenes Four onsequenes of the Postulates 1. Spae and Time Interals. The Addition of Veloities 3. Inertia 4. Energy Plus a bonus story Now, things get triky

18 1. Spae and Time Spae and Time are abstrat onepts We need a simple definition. Einstein s definitions: Spae is what a meter stik measures. Time is what a lok measures. These pratial definitions are based on physial measurements and the Light that arries the information A Light Clok Let s start with the Time Interal between two eents as seen in two different Inertial Referene Frames Our two eents: Tik and tok! We ll use a Light Clok measure the Time Interal between two eents using the Speed of Light Length = L A Light Clok Mirror Light Soure A Light Clok Light soure emits one flash of Light The Light reflets off the mirror That s a Tik Then Light returns to the soure That s a Tok Emit Tik Tok A Light Clok The Clok Frame First we ll look at the Time Interal Between Tik and Tok in the Clok Frame: The Inertial Referene Frame that is at rest relatie to the lok. We and the Clok are in the same Frame No relatie eloity between us! The Clok Frame The Clok Frame Spae and Time Tik Tok In the Clok Frame The Clok is at the same plae when the Tik and Tok happen This Time Interal is alled the Proper Time The Time Interal measured in the referene frame where the Clok is in the same plae Basi Physis reiew for Inertial Motion Motion at onstant eloity Distane Time = Speed Drie 100 miles at 50 miles per hour The trip takes hours

19 The Clok Frame The Clok Frame The Lab Frame For our Light Clok in the Clok Frame The Distane traeled is twie the Length Bak and forth The Speed is the Speed of Light Light is doing the traeling Time Interal t In the Clok Frame 0 = Total distane traeled L Speed of Light Now we ll look at the Time Interal between the same two eents in the Lab Frame: The Inertial Referene Frame in whih the lok is moing. We are sitting in the Lab The Clok has eloity relatie us! The Lab Frame The Lab Frame The Lab Frame When the Light hits the mirror the Clok has moed to a different plae Tik The Light follows owsad different e path than it does in the Clok Frame In the Lab Frame the Light has a longer path to trael Distane = D Tok The Lab Frame Spae and Time The Lab Frame In the Lab Frame The Clok is not at the same plae when tik and tok happen During the Time between the two eents the Clok has moed Now reall the nd Postulate The Speed of Light is ABSOLUTE The Light s speed between Tik and Tok is the same in both Frames That s what ABSOLUTE means! Same alue in ALL Inertial Referene Frames Time Interal t = Total distane traeled D The Speed of Light In the Lab Frame

20 Spae and Time In the Lab Frame the Light traels a longer distane: at the same speed: D > L Time Interal is Longer in the Lab Frame Longer distane at the same speed! The lok is running slower in the Lab! This is a diret result of the nd Postulate! The Speed of Light is ABSOLUTE! Time Dilation This is alled Time Dilation Dilate means to beome larger The Time Interal in the Lab Frame between the same two eents The same Tik and Tok is larger than that in the Clok Frame! Moing Cloks Run Slow! This has nothing to do with our Clok It is after all a rather strange lok This is a property of TIME Not a property of Cloks The Clok just measures TIME Time Dilation This applies to biologial loks too! Suh as reproduing bateria Humans hae many built-in loks o Daily, monthly, yearly yles The aging proess Time Interals are RELATIVE! Time Dilation The faster you moe through Spae, the slower you moe through Time! Spae and Time Now let s look at the Spae Interal between two eents as seen in two different Inertial Referene Frames Same two eents: Tik and Tok! We ll still use the Light Clok Measure the Spae Interal between two eents Using the Speed of Light Spae and Time A Light Clok Length Contration Now we ll lay our Light Clok on its side So it is moing in the diretion that is parallel to its Length Before it was moing perpendiularly The analysis is similar to the Time Interal We ll skip the details At rest Moing Rest Length = L 0 Length = L This is alled Length Contration Contrat means to redue in size The Spae Interal between the same two eents The same Tik and Tok is shorter for the moing Clok

21 Length Contration If the objet is a meter stik its Rest Length is 1 meter Moing at 60% of the Speed of Light its eloity is = (3/ 5) We would measure its Length to be 4 /5 meter Only 80% of its Rest Length! Moing Objets Are Shorter! They are shorter along the diretion of the motion so their height is not affeted at all! This is a property of SPACE Not a property of objets The objet just oupies the SPACE Spae Interals are RELATIVE! Length Contration The faster you moe through Spae, the less Spae you oupy! (along the diretion of motion only) Spae and Time Both of these effets are a diret result of the nd Postulate Remember what speed means Speed of Light = = Distane Traeled by Light Elapsed Time Spae Interal Time Interal Spae and Time Both interals hange so that their ratio The Speed of Light Remains INVARIANT Spae and Time are interrelated Both are part of one entity alled Spae-Time We lie in a 4-dimensional Unierse! 3-D Spae + 1-D Time = 4-D Spae-Time Spae and Time The onepts of Spae and Time Spae interal between two eents Time interal between two eents are no longer separate! Spae-Time interal between two eents Spae-Time Einstein showed the Spae-Time Interal between two eents Is INVARIANT! All obserers agree on the Spae-Time Interal Between two eents This too is a diret result of the nd Postulate The Speed of Light is ABSOLUTE 187 Spae-Time Interal Spae-Time s= T L The Speed of Light Time Interal Spae Interal 188. Addition of Veloities How does the speed of an objet in one Inertial Frame of Referene transform into another Inertial Frame? By the Addition of Veloities Remember the kids playing ath in the truk? Let s assume this time that you (a pedestrian) are playing ath with someone in the bak of the truk. 189

22 Addition of Veloities Aording to Isaa Newton If u is the speed of the ball relatie to the truk If is the speed of the truk relatie to you Then the speed of the ball relatie to you is u Addition of Veloities u u + is the speed of the truk relatie to you I m open! u is the speed of the ball relatie to the truk u+is the speed of the ball relatie to you 191 Addition of Veloities Now suppose we replae our thrower with a Light Soure Aording to Newton If is the speed of the truk relatie to you If is the Speed of the light relatie to the truk Then the speed of light relatie to you is + 19 Addition of Veloities + is the speed of the truk relatie to you is the speed of light relatie to the truk +is the speed of light relatie to you 193 I m open? Addition of Veloities This iolates the nd Postulate!! You get a different result when you measure the Speed of Light Suppose the truk kis ery fast Very, ery fast Its speed is ½ : half the speed of Light Then you measure 3 / as Light Speed Bigger than 194 Addition of Veloities Spae and Time are RELATIVE We need a rule for adding eloities that inludes the different rate of Time and the different size of Spae for the two different Referene Frames Einstein proides us with suh a rule Of ourse It s his theory, it s his solution! 195 Addition of Veloities Aording to Einstein If u is the speed of the ball relatie to the truk If is the speed of the truk relatie to you Then the speed of the ball relatie to you is u+ u u u u/ Addition of Veloities u u/ is the speed of the truk relatie to you u is the speed of the ball relatie to the truk is the speed of the ball relatie to you Yikes! 197 Addition of Veloities Why don t we notie this different rule? There is an extra term in Einstein s rule: It is ery small at eeryday speeds The is a huge number: m /s u 1+ = 1+ a tiny number u 198

23 Addition of Veloities u Newton Einstein 60 mph 30 mph 90 mph 90 mph 186 mps 18.6 mps 04.6 mps mps Addition of Veloities The effets of Speial Relatiity are notieable only at ery high speeds Een at 186 miles per seond ( 1 /1000 ) Newton and Einstein are ery lose The human speed reord: 1 /7000 The Apollo astronauts returning from the moon 6.89 miles/se = 4,800 mph There seems to be an upper limit on speed A osmi speed limit 00 Addition of Veloities So what about our moing Light soure? Aording to Einstein: 1 3 u+ + = u ( 1 )( 1+ ) 3 1+ The nd Postulate holds! = = 01 Addition of Veloities 3. Newton s Laws and Inertia Newton s Laws Een if the truk is moing at Light Speed Aording to Einstein: u+ + = = = u ( )( 1+ ) 1+ The nd Postulate still holds! 0 The Speed of Light is a natural speed limit No objet s speed an eer exeed or een equal the Speed of Light It is a onsequene of the nd Postulate But there is no speed limit in Newton s Laws 03 Reall Newton s nd Law F F = ma a = m Relates ause and effet Cause: Fore Effet: Aeleration (hanges in motion) 04 Newton s Laws Apply a onstant Fore produe a onstant Aeleration Aording to Newton If you push long and hard enough your speed will exeed the Speed of Light For an aeleration of one g (9.80 m/s ) it takes about a year to get to Light Speed Speed Newton s Laws Can t happen! Newton Newton s Laws This is not allowed in Speial Relatiity But een Einstein an t hange how hard you an push or how long you an push So how an he limit the speed of the objet? Where does the osmi speed limit ome from? So what does happen?? 05 Time06 07

24 Newton s Laws Aording to Einstein Aeleration is RELATIVE it s based on a hange in eloity after all As the speed inreases the aeleration must derease So the Objet s Inertia must inrease Inertia is an opposition to a hange in motion Newton s Laws Mass is RELATIVE! As the objet s speed inreases so does its Mass So for the same amount of Fore there is less Aeleration F a = m Speed Newton s Laws Einstein Newton Inertia is the objet s resistane to Changes in its motion and is determined by its Mass 08 Moing objets hae more Mass! 09 Time 10 Newton s Laws 4. Energy Energy The faster the objet is moing the more Mass it has The inrease in Mass produes smaller and smaller Aelerations The speed is always less than Speed of Light Mass is RELATIVE Under Newton s Laws the Kineti Energy is Mass is ABSOLUTE 1 m Einstein showed that Mass is RELATIVE More eloity more Mass Een more Kineti Energy Einstein also showed that E = m Mass is a form of Energy All Mass is equialent to Energy This is a onersion formula Mass Energy Energy One big differene between Newton and Einstein Aording to Newton an objet at rest has no Energy Aording to Einstein een at rest, an objet has Energy A LOT of Energy Rest Energy Energy E = m 0 0 Rest Mass Speed of Light squared Energy The is a huge number: m /s So a small amount of Mass an be onerted to a huge amount of Energy One kilogram of Mass (about. pounds) ompletely onerted to Energy would run the entire U.S. for 9 hours

25 Energy The Relatiisti Fator: γ The Relatiisti Fator: γ Unfortunately (The Big IF!) there is only one way known to ompletely onert Mass to Energy Combining Matter and Antimatter Just like on Star Trek Not yet feasible tehnologially or eonomially Antimatter is expensie to make, hard to handle 17 All of these new relatiisti effets are extremely small for eeryday eloities There is a way to alulate how small The Relatiisti i i Fator: γ (gamma) γ = The bigger the Relatiisti Fator the more important Relatiity is For speeds small ompared to Light Speed we hae γ 1 For speeds large ompared to Light Speed we hae larger and larger γ 19 The Relatiisti Fator: γ At a speed of 1 /10, we hae γ = about 67 million miles per hour The preditions of Newton and Einstein are only different by ½ perent About 1 part in 00 So if Newton says the answer is 00 Einstein says it is 01 0 The Relatiisti Fator: γ At the human speed reord: 1 /7000 about 5000 miles per hour = γ = The preditions of Newton and Einstein are different by only 1 part in a Billion! Too small to notie without experimentation Can be measured with atomi loks though 1 γ Small γ ( 1) for small speeds Large γ for speeds near! / The Tale of the Traeling Twin Let s do our own thought experiment Suppose we hae two twin astronauts Eah is 30 years old at the start of the trip One traels to a some star and returns The other stays on the Earth When they are reunited whih twin is older? 3 The Tale of the Traeling Twin Let Twin A be the Spae Traeler So Referene Frame A is her point of iew Then Twin B is the Mission Controller Referene Frame B is the Earth s point of iew Let s see what Speial Relatiity has to say 4 The Tale of the Traeling Twin Let s go to the star Vega Distane: 5 light years (in Frame B) Let s trael at a speed of = % 9% of the Speed of Light At this speed we hae Relatiisti Fator: γ = We expet Relatiity to be important here 5

26 A Light Year Frame of Referene B Frame of Referene B A Light Year The distane Light traels in one year It is a unit of Distane, not Time! As measured by loks on Earth This trip takes a time of Approximately 6 trillion miles Or 6 million million miles So it takes the Light from Vega 5 years to reah us here on Earth 6 t B L B L = B 7 t B L B 1 = =50 years, month Twin B is 50 years, ½ month older So he is 80 years, ½ month old! 8 Frame of Referene A Frame of Referene A The Tale of the Traeling Twin As measured by loks on the Ship This trip takes a time of L LA t A = = years, 3months Twin A sees Earth and Vega moing so the distane between them is ontrated In her Frame of Referene They are loser in her frame: L A < L B t A L A L = A 9 Twin A is years, 3 months older So she is 3 years, 3 months old! 30 So the trip takes less time in her frame Same relatie speed, shorter distane 31 The Tale of the Traeling Twin By traeling at a ery fast speed ery near the Speed of Light Traeler A slowed her rate of time Moing fast through Spae Moing Slowly through Time The Tale of the Traeling Twin This effet is sometimes alled The Twin Paradox Paradox: something that seems to ontradit or oppose ommon sense Why? The Tale of the Traeling Twin Twin B sees Twin A moe off and return So Twin A s loks run slowly Twin A ends up younger Twin A sees Twin B moe off and return So Twin B s loks run slowly Twin B ends up younger Is eah point of iew is equally alid?

27 No! The Tale of the Traeling Twin Speial Relatiity says All Inertial Referene Frames are equally alid Twin A uses two Inertial Frames One out and one oming bak She turns around So there were Fores and Aelerations No longer inertial referene frames!! Minkowski Diagrams Also alled Spae-Time Diagrams A graph of Time ersus Distane Time is plotted on the ertial axis Spae is plotted in the horizontal axis Let s look at our two Twins Minkowski Diagrams Time Axis Twin B Light Twin A Can t get there from Here Start Here Spae Axis 37 Minkowski Diagrams Minkowski Diagrams Minkowski Diagrams Twin B stays at the same plae Only moes through Time Twin A hanges loation in Spae And also moes through Time We see that see used two Inertial Frames! Time Axis Twin B Twin A Both leae at the same time Trael in opposite diretions They both stop at the same Time Where they turn around and return home She will be younger! Both will be at the same age upon return Let s look at some different trips 38 Spae Axis Start Here Minkowski Diagrams Minkowski Diagrams Summary of Speial Relatiity Time Axis Twin B Twin A leaes first Twin B stays home for a while, leaes later Moing Cloks run slower Time Dilation Twin A Traels faster and athes up to Twin A Meet at the same plae at the same Time Moing Objets are Shorter Length Contration Spae Axis Start Here 41 Twin B will be younger Can proe this after muh muh math! 4 Spae, Time, and Mass are RELATIVE Spae-Time is ABSOLUTE 43

28 Beyond the Speial Theory of Relatiity The Speial Theory of Relatiity only oers the speial ase of inertial referene frames non-aelerated e ed referene e e frames It gies the orret equations for transformations between two referene frames moing at onstant eloities with respet to eah other Beyond the Speial Theory of Relatiity Einstein wanted equations whih were muh more general than these speial ones so he deised a general theory to oer all referene frames, inertial and non-inertial. This theory is alled the General Theory of Relatiity and is MUCH more diffiult mathematially. Took 10 years of work to deelop this new theory (published in 1916). The General Theory of Relatiity Called GenRel for short He started with an obseration that both Newton s nd Law and his Uniersal Graitation Law both ontained the same quantity the objet s mass Inertial mass in the former F = m a Graitational mass in the latter m1m F = G r The General Theory of Relatiity Sine these were two separate laws, the masses did not neessarily need to be the same Preision experiments were arried out whih proed the two masses were in fat the same Einstein thought that this was not oinidental and that the aeleration in the nd law was related to the graitational aeleration in the graitational law. The General Theory of Relatiity Proposed his Equialene Postulate It is impossible to distinguish a graitational fore from an equialent aeleration indued fore If you were standing in a spae ship moing with an aeleration of g, then you would feel the same fore as if you were standing on the Earth s surfae You ouldn t tell the differene! The General Theory of Relatiity A fore is being simulated by an aeleration Any effet whih ould be desribed by an aelerated referene frame ould also be desribed as a graitational effet The General Theory of Relatiity As seen from outside As seen from inside Graity is being simulated by the spaeship s aeleration Path of a ball thrown horizontally on the Earth s surfae! For different horizontal speeds 50 The General Theory of Relatiity In an aelerated referene frame, een a beam of light would be bent This led Einstein to onlude that a graitational field would alter (bend) the path of a beam of light! 51 The General Theory of Relatiity Einstein interpreted the bending of the light as representing a urature of spae itself Sine the motion of the ship ours in ured spae, graity was just an effet indued by moing through this ured spae. Large onentrations of mass (stars, planets) ure the spae around them Any motion through this spae auses an aeleration due to the urature whih we feel as graity Just as you feel a fore pushing you outward as you round a ure in your ar 5

29 The General Theory of Relatiity The General Theory is muh more than this; I e just srathed the surfae The General Theory has been tested many times and has always been found orret! 53 The General Theory of Relatiity A star s light is bent around the Sun as seen during an elipse (in 1919). True position Sun Earth Apparent position 54 The General Theory of Relatiity The preession of the planet Merury s orbit an only be ompletely explained by the General Theory. The graitational pull of the other eight planets an only aount for 93% of the preession. The relatiisti effet of the Sun uring the spae near Merury exatly aounts for the remaining 7%! 55 The General Theory of Relatiity General Relatiity says that the properties of spae are dependent on graitational fores and the presene of matter It also says the properties of spae-time are determined by light rays Eletromagneti waes The General Theory of Relatiity Einstein belieed these effets were related He spent the rest of his life trying to unify them into one oerall theory The effort ontinues today to disoer this single unified-field theory More on this in Chapter 9 1/31/1999 Albert Einstein