CHAPTER 2 Special Theory of Relativity
|
|
- Emery Colin Owen
- 5 years ago
- Views:
Transcription
1 CHAPTER 2 Special Theory of Relativity Fall 2018 Prof. Sergio B. Mendes 1
2 Topics Inertial Frames of Reference Conceptual and Experimental Inconsistencies The Michelson-Morley Experiment Einstein s Postulates The Lorentz Transformation Time Dilation and Length Contraction Addition of Velocities Experimental Verification Twin Paradox Space-Time Doppler Effect Relativistic Momentum Relativistic Energy Computations in Modern Physics Electromagnetism and Relativity Fall 2018 Prof. Sergio B. Mendes 2
3 How to describe an event? The precise description of an event must be characterized by its location in space rr and its time tt. zz The set of (calibrated) rulers and (synchronized) clocks form a frame of reference that can be used to characterize events. Fall 2018 Prof. Sergio B. Mendes 3
4 What is an Inertial Frame of Reference? One in which Newton s laws are valid!! aa mmgg + TT = FF nnnnnn = mm aa FF nnnnnn = 00 vv = 00 aa = 00 FF nnnnnn = 00 vv = cccccccccccccccc aa = 00 Such a frame of reference is established when a body subjected to a null net external force, FF nnnnnn = 00, is observed to move with constant velocity (either rectilinear motion at constant speed or at rest). Fall 2018 Prof. Sergio B. Mendes 4
5 A non-inertial observer reaches conclusions that don t agree with Newton s laws aa = 0 aa FF nnnnnn 00 FF nnnnnn = 00 aa = aa aa aa FF nnnnnn = 00 aa = aa Fall 2018 Prof. Sergio B. Mendes 5
6 Therefore, we must use an inertial frame of reference to describe the laws of Mechanics Fall 2018 Prof. Sergio B. Mendes 6
7 Q: Well, how many inertial frames of reference are out there? A: An infinite number. Once we have found one inertial frame of reference (K), then any frame of reference moving at constant velocity vv oo with respect to K is also an inertial frame of reference. Fall 2018 Prof. Sergio B. Mendes 7
8 Proof Consider that K is at rest. Consider that K is moving with constant velocity vv oo with respect to K. Consider that the two frames of reference coincide at tt = 0. rr = rr vv oo tt K rr vv oo tt K rr xx = xx vv oo,xx tt yy = yy vv oo,yy tt zz = zz vv oo,xx tt vv = vv vv oo aa = aa tt = tt Fall 2018 Prof. Sergio B. Mendes 8
9 aa = aa mmm aa = mmm aa FF = FF mm = mm If Newton s laws are valid in one frame of reference (K), then they are also valid in another frame of reference (K ) moving at a uniform velocity relative to the first system. So, both are inertial frames of reference. Fall 2018 Prof. Sergio B. Mendes 9
10 Principle of (Classical) Relativity The laws of Mechanics (Newton s laws) are the same in all inertial frames of reference All inertial frames of reference are equivalent. Inertial frames of reference are related by: rr = rr vv oo tt tt = tt Galilean Transformation Fall 2018 Prof. Sergio B. Mendes 10
11 2.1 Conceptual and Experimental Inconsistencies Fall 2018 Prof. Sergio B. Mendes 11
12 Conceptual Inconsistencies: Although Newton s laws of motion had the same form under the Galilean transformation, Maxwell s equations did not. SS EE rr. ddaa = QQ iiiiiiiiiiii εε 0 CC EE rr. ddss = ddφ BB dddd SS BB rr. ddaa = 0 CC BB rr. ddss = μμ oo II + μμ oo εε oo ddφ EE dddd FF = qq EE + qq vv BB Fall 2018 Prof. Sergio B. Mendes 12
13 Experimental Inconsistencies: 2 EE xx 2 = εε 2 EE oo μμ oo tt 2 2 BB xx 2 = εε 2 BB oo μμ oo tt 2 In Maxwell s theory, the speed of light in terms of the permeability and permittivity of free space was given by: cc = 1 εε oo μμ oo Fall 2018 Prof. Sergio B. Mendes 13
14 The carrier medium for light, ether: Following the tradition of the time (that every wave has a medium to carry its propagation) the luminiferous ether was considered as the carrier medium for light propagation, where cc = 1 εε oo μμ oo Ether had to have such a low density that the planets could move through it without loss of energy It also had to have an elasticity to support the high velocity of light waves Fall 2018 Prof. Sergio B. Mendes 14
15 An Absolute Inertial Frame of Reference Ether was proposed as an absolute reference system in which the speed of light was this constant and from which other measurements could be made. The Michelson-Morley experiment was an attempt to show the existence of ether. Fall 2018 Prof. Sergio B. Mendes 15
16 2.2 The Michelson-Morley Experiment Albert Michelson ( ) was the first U.S. citizen to receive the Nobel Prize for Physics (1907). He built an extremely precise device called an interferometer to measure the minute phase difference between two light waves traveling in mutually orthogonal directions. Fall 2018 Prof. Sergio B. Mendes 16
17 The Michelson Interferometer Fall 2018 Prof. Sergio B. Mendes 17
18 How does it work? 1. AC is parallel to the motion of the Earth inducing an ether wind 2. Light from source S is split by mirror A and travels to mirrors C and D in mutually perpendicular directions 3. After reflection the beams recombine at A slightly out of phase due to the ether wind as viewed by telescope E. Fall 2018 Prof. Sergio B. Mendes 18
19 A Typical Interference Pattern Fall 2018 Prof. Sergio B. Mendes 19
20 The Analysis assuming the Galilean Transformation!! Time tt 1 from A to C and back: tt 1 = ll 1 cc + vv + ll 1 cc vv Time tt 2 from A to D and back: tt 2 = = 2 cc ll 1 cc 2 vv 2 ll 2 cc 2 vv 2 + ll 2 cc 2 vv 2 So that the change in time is: tt = tt 2 tt 1 = 2 ll 2 cc = 2 ll 2 cc = 2 ll 1 cc Fall 2018 Prof. Sergio B. Mendes vv2 cc vv2 cc 2 1 vv2 cc 2 2 ll 1 cc 1 1 vv2 cc 2
21 The Analysis (continued) tt = tt 2 tt 1 = 2 ll 2 cc 1 1 vv2 cc 2 2 ll 1 cc 1 1 vv2 cc 2 Upon rotating the apparatus by 90º, the optical path lengths ll 1 and ll 2 are interchanged producing a different change in time: ttt = ttt 2 tt 1 = 2 ll 2 cc 1 1 vv2 cc 2 2 ll 1 cc 1 1 vv2 cc 2 Fall 2018 Prof. Sergio B. Mendes 21
22 Difference in times upon rotation: tt tt = 2 ll 1 + ll 2 cc 1 1 vv2 cc vv2 cc 2 ll 1 + ll 2 vv 2 cc 3 Fall 2018 Prof. Sergio B. Mendes 22
23 Increasing the pathlength: Fall 2018 Prof. Sergio B. Mendes 23
24 Crunching the numbers: vv = mm/ss cc = mm/ss ll 1 ll 2 = 11 mm tt tt = ll 1 + ll 2 vv 2 cc 3 = ss λλ = mm cc = mm/ss TT = λλ cc = mm mm/ss = ss ffffffffffffffff oooo aaaa iiiiiiiiiiiiiiiiiiiiiiii ffffffffffff = ss ss 0.4 iiiiiiiiiiiiiiiiiiii rrrrrrrrrrrrrrrrrrrr 0.01 Fall 2018 Prof. Sergio B. Mendes 24
25 The Experiments on the relative motion of the earth and ether have been completed and the result decidedly negative. The expected deviation of the interference fringes from the zero should have been 0.40 of a fringe the maximum displacement was 0.02 and the average much less than 0.01 and then not in the right place. As displacement is proportional to squares of the relative velocities it follows that if the ether does slip past the relative velocity is less than one sixth of the earth s velocity. Albert Abraham Michelson, 1887 Fall 2018 Prof. Sergio B. Mendes 25
26 2.3 Einstein s Two Postulates 1. The principle of relativity: The laws of physics are the same in all inertial frames of reference. There is no way to detect absolute motion and no preferred inertial system exists. 2. The constancy of the speed of light: Observers in all inertial frames of reference measure the same value for the speed of light when propagating in vacuum. Fall 2018 Prof. Sergio B. Mendes 26
27 Consequences: Source and Observer at rest: c Observer in motion with respect to the Source: c Source in motion with respect to the Observer: c -v v Fall 2018 Prof. Sergio B. Mendes 27
28 Consequences In Newtonian physics, we previously assumed that tt = tt. Therefore K and K would always agree if two events happen at the same time (simultaneous) or not. Einstein realized that events considered simultaneous in K may not be in K. Fall 2018 Prof. Sergio B. Mendes 28
29 Because speed of light is absolute then simultaneity is relative K K K K Fall 2018 Prof. Sergio B. Mendes 29
30 Two events that are simultaneous in one reference frame (K) are not necessarily simultaneous in another reference frame (K ) moving with respect to the first frame. This suggests that each coordinate system must have its own set of observers with their own set of synchronized clocks. Fall 2018 Prof. Sergio B. Mendes 30
31 2.4 Lorentz Transformations The special set of linear transformations that preserve the constancy of the speed of light between inertial observers. Fall 2018 Prof. Sergio B. Mendes 31
32 Two inertial frames of reference KK and KK 1. The axes along (xx, yy, zz) are parallel to the corresponding axes along (xxx, yyy, zzz). 2. The relative motion between the two inertial frames of reference is along the x-axis (and x -axis). 3. Consider that the origins OO and OO of the two systems coincide at tt = ttt = 0 As seen by KK As seen by KK OO OOO vv oo vv oo OO OOO Fall 2018 Prof. Sergio B. Mendes 32
33 As seen by KK KK: (xx, tt) vv oo vv oo vv oo KKK: (xxx, ttt) vv oo vv oo Fall 2018 Prof. Sergio B. Mendes 33
34 As seen by KK : KK: (xx, tt) vv oo vv oo vv oo KKK: (xxx, ttt) vv oo vv oo Fall 2018 Prof. Sergio B. Mendes 34
35 As seen by KK: A flashbulb goes off at the origins when tt = ttt = 0. vv oo Fall 2018 Prof. Sergio B. Mendes 35
36 As seen by KKK: A flashbulb goes off at the origins when tt = ttt = 00. vv oo Fall 2018 Prof. Sergio B. Mendes 36
37 According to Postulate 2, the speed of light will be c in both systems!! The wavefronts observed in both systems must be spherical with respect to their own coordinates and time. KK OO KKK OOO xx 2 + yy 2 + zz 2 = cc 2 tt 2 xxx 2 + yyy 2 + zzz 2 = cc 2 ttt 2 Fall 2018 Prof. Sergio B. Mendes 37
38 xx 2 + yy 2 + zz 2 cc 2 tt 2 = 0 0 = xxx 2 + yyy 2 + zzz 2 cc 2 ttt 2 yy = yyy zz = zzz as in Galilean transformation xx 2 cc 2 tt 2 = xxx 2 cc 2 ttt 2 Fall 2018 Prof. Sergio B. Mendes 38
39 xx = γγ xx vv oo tt linear modification in Galilean transformation due to symmetry xx = γγ xxx + vv oo ttt = γγ γγ xx vv oo tt + vv oo ttt solve for: tt = 1 γγ vv oo 1 γγ 2 xx + γγ tt Fall 2018 Prof. Sergio B. Mendes 39
40 xx 2 cc 2 tt 2 = xxx 2 cc 2 ttt 2 xx = γγ xx vv oo tt tt = 1 γγ vv oo 1 γγ 2 xx + γγ tt Fall 2018 Prof. Sergio B. Mendes 40
41 xx 2 : 1 = γ 2 cc2 γγ 2 2 vv 1 γγ2 2 1 oo 1 γγ 2 = cc2 γγ 2 vv oo 2 1 γγ 2 1 = vv oo cc 2 2 tt 2 : cc 2 = γγ vv oo cc 2 γγ 2 γγ 2 = 1 vv 2 oo cc 2 2 xx tt: 0 = γγ 2 vv oo cc 2 1 vv oo 1 γγ 2 γγ 2 vv oo = cc 2 1 vv oo 1 γγ 2 vv oo 2 cc 2 = 1 γγ 2 1 γγ = 1 1 ββ 2 ββ vv oo cc Fall 2018 Prof. Sergio B. Mendes 41
42 Lorentz Transformations: xx = γγ xx γγ vv oo tt tt = γγ vv oo cc 2 xx + γγ tt γγ = 1 1 ββ 2 ββ vv oo cc Fall 2018 Prof. Sergio B. Mendes 42
43 Inverse Lorentz Transformations: xx = γγ xx + γγ vv oo tt tt = γγ vv oo cc 2 xx + γγ ttt γγ = 1 1 ββ 2 ββ vv oo cc Fall 2018 Prof. Sergio B. Mendes 43
44 Relativistic Factor γγ ββ = Fall 2018 Prof. Sergio B. Mendes 44
45 2.5 Time Dilation and Length Contraction Fall 2018 Prof. Sergio B. Mendes 45
46 Time Dilation: vv oo Fall 2018 Prof. Sergio B. Mendes 46
47 Time Dilation: tt = γγ vv oo cc 2 xx + γγ tt ttt 1 xx 1 = xx 2 ttt 1 = γγ vv oo cc 2 xx 1 + γγ tt 1 ttt 2 = γγ vv oo cc 2 xx 2 + γγ tt 2 Fall 2018 Prof. Sergio B. Mendes 47
48 Time Dilation & Proper Time: ttt 1 ttt 2 ttt 2 tt 1 = γγ tt 2 tt 1 tt 2 tt 1 = proper time, time duration measured at same location Fall 2018 Prof. Sergio B. Mendes 48
49 Length Contraction: vv oo Fall 2018 Prof. Sergio B. Mendes 49
50 Length Contraction: xx = γγ xx + γγ vv oo tt xx 1 = γγ xxx 1 + γγ vv oo tt 1 xx 2 = γγ xxx 2 + γγ vv oo tt 2 tt 2 = ttt 1 xxx 2 xxx 1 = 1 γγ xx 2 xx 1 xx 2 xx 1 = proper length, length measured at rest Fall 2018 Prof. Sergio B. Mendes 50
51 Fall 2018 Prof. Sergio B. Mendes 51
52 Fall 2018 Prof. Sergio B. Mendes 52
53 2.6 Addition of Velocities Fall 2018 Prof. Sergio B. Mendes 53
54 Taking the differentials: tt = γγ vv oo cc 2 xx + γγ tt ddtt = γγ vv oo cc 2 ddxx + γγ dddd xx = γγ xx γγ vv oo tt ddxx = γγ dddd γγ vv oo ddtt yy = yy zz = zz ddyy = ddyy ddzz = dddd Fall 2018 Prof. Sergio B. Mendes 54
55 Along x-axis ddxx = γγ dddd γγ vv oo ddtt ddtt = γγ vv oo cc 2 ddxx + γγ dddd ddxx ddtt = γγ dddd γγ vv oo dddd γγ vv oo dddd + γγ dddd cc 2 uuu xx ddxx ddtt = uu xx vv oo 1 vv oo uu xx cc 2 Fall 2018 Prof. Sergio B. Mendes 55
56 ddyy Along y-axis ddyy = ddyy ddtt = γγ vv oo cc 2 ddtt = γγ vv oo cc 2 ddxx + γγ dddd dddd dddd + γγ dddd uuu yy ddyy ddtt = uu yy γγ 1 vv oo uu xx cc 2 Fall 2018 Prof. Sergio B. Mendes 56
57 Along z-axis ddzz = dddd ddzz ddtt = γγ vv oo cc 2 ddtt = γγ vv oo cc 2 ddxx + γγ dddd dddd dddd + γγ dddd uuu zz ddzz ddtt = uu zz γγ 1 vv oo uu xx cc 2 Fall 2018 Prof. Sergio B. Mendes 57
58 In Summary, Addition of Velocities: uuu xx = uu xx vv oo 1 vv oo uu xx cc 2 uuu yy = uu yy γγ 1 vv oo uu xx cc 2 uuu zz = uu zz γγ 1 vv oo uu xx cc 2 Fall 2018 Prof. Sergio B. Mendes 58
59 Inverted Relations: uu xx = uuu xx + vv oo 1 + vv oo uuu xx cc 2 uu yy = uuu yy γγ 1 + vv oo uuu xx cc 2 uu zz = uuu zz γγ 1 + vv oo uuu xx cc 2 Fall 2018 Prof. Sergio B. Mendes 59
60 Example uu xx = cc vv oo = cc uuu xx = = uu xx vv oo 1 vv oo uu xx cc cc cc cc cc 1 cc 2 = cc Fall 2018 Prof. Sergio B. Mendes 60
61 2.7 Experimental Verification of Special Relativity Fall 2018 Prof. Sergio B. Mendes 61
62 Cosmic Rays and Muon Decay NN NN oo = ee llll 2 tt ττ ττ = 1.52 μμμμ Fall 2018 Prof. Sergio B. Mendes 62
63 (Incorrect) Classical Calculation h = 2000 mm vv = 0.98 cc tt = h vv = 6.80 μμμμ NN NN oo = ee llll 2 tt ττ ττ = 1.52 μμμμ llll μμμμ = ee 1.52 μμμμ = 4.5% Don t agree with experiment Fall 2018 Prof. Sergio B. Mendes 63
64 Relativistic Calculation vv = 0.98 cc γγ = tt = tt γγ 6.80 μμμμ = μμμμ NN NN oo = ee llll 2 tt ττ llll μμμμ = ee 1.52 μμμμ = 54% Agrees with experiment Fall 2018 Prof. Sergio B. Mendes 64
65 Atomic Clock Measurement Fall 2018 Prof. Sergio B. Mendes 65
66 Velocity Addition ππ 0 γγ + γγ vv oo = cc uu xx = uuu xx + vv oo 1 + vv oo uuu xx cc 2 uuu xx = cc vv xx = cc Fall 2018 Prof. Sergio B. Mendes 66
67 2.10 Relativistic Doppler effect for light waves in vacuum Although light velocity in vacuum is always constant c, the frequency will change for a relative motion between source and observer Source- Observer approaching Higher Frequency Source- Observer receding Lower Frequency Fall 2018 Prof. Sergio B. Mendes 67
68 Relativistic Doppler effect for light waves in vacuum cc vv vv TT cc TT lllllllllll oooo ttttt wwwwwwww tttttttttt = cc TT vv TT nn = cc TT vv TT λλ ff = cc λλ = cc nn cc TT vv TT Fall 2018 Prof. Sergio B. Mendes 68
69 nn = pppppppppppp tttttttt oooo ttttt ssssssssssss tttttttt pppppppppppp oooo ttttt ssssssssssss = TTT oo 1 fffoo TT = γγ TTT oo nn = TT fff oo γγ ff = cc nn cc TT vv TT = cc γγ cc vv fff oo = 1 + ββ 1 ββ fff oo Fall 2018 Prof. Sergio B. Mendes 69
70 Relativistic Doppler effect for light propagating in vacuum Source- Observer approaching Source- Observer receding ff = 1 + ββ 1 ββ fff oo ff = 1 ββ 1 + ββ fff oo blueshifted redshifted Fall 2018 Prof. Sergio B. Mendes 70
71 Rotation of Venus Fall 2018 Prof. Sergio B. Mendes 71
72 Laser Cooling Fall 2018 Prof. Sergio B. Mendes 72
73 Laser Radar Technology Fall 2018 Prof. Sergio B. Mendes 73
74 2.11 Relativistic Linear Momentum Fall 2018 Prof. Sergio B. Mendes 74
75 Classical Expressions from Galileo and Newton: pp mm uu Linear Momentum dd pp FF = dddd Newton s law Newton s law is invariant under Galilean transformations, but not under Lorentz transformations. Fall 2018 Prof. Sergio B. Mendes 75
76 Relativistic Linear Momentum Invariance of Newton s law under Lorentz transformations will be shown to lead to: pp = ΓΓ mm uu ΓΓ 1 1 uu cc 2 Fall 2018 Prof. Sergio B. Mendes 76
77 Conservation of Linear Momentum in an Elastic Collision: uu xx = 0 uu xx = uuu xx + vv 1 + vv uuu xx cc 2 uu xx = vv uu xx = 0 uu yy = uu oo uu yy = uuu yy γγ 1 + vv uuu xx cc 2 uu yy = uu oo γγ uu yy = 2 uu oo γγ mm mm pp = mm uu + uu 0 mm uu xx = 0 uu yy = ± uu oo uu xx = 0 uu yy = 2 uu oo Fall 2018 Prof. Sergio B. Mendes 77
78 Relativistic Linear Momentum pp = mm ΓΓ uu pp = mm ΓΓ uu + uu = 0 ΓΓ 1 1 uu cc 2 uu xx = vv uu yy = uu oo γγ ΓΓ uu xx = 0 ΓΓ uu yy = ΓΓ 2 uu oo γγ = 2 uu oo 1 uu oo 2 cc 2 ΓΓ = 1 1 vv 2 + uu oo 2 γγ 2 cc 2 uu xx = 0 ΓΓ uu xx = 0 uu yy = ± uu oo ΓΓ uu yy = 2 uu oo 1 uu oo 2 cc 2 ΓΓ = 1 1 uu oo 2 cc 2 Fall 2018 Prof. Sergio B. Mendes 78
79 Relativistic Linear Momentum pp = mm ΓΓ uu ΓΓ 1 1 uu cc 2 Fall 2018 Prof. Sergio B. Mendes 79
80 2.12 Relativistic Energy FF = dd pp dddd 2 2 dd pp WW 12 = KK 2 KK 1 = FF dd rr = 1 1 dddd dd rr Fall 2018 Prof. Sergio B. Mendes 80
81 pp = mm ΓΓ uu dd pp = mm dd ΓΓ uu uu = dd rr dddd dd rr = uu dddd 2 dd pp KK 2 KK 1 = 1 dddd dd rr 2 dd ΓΓ uu = mm 1 dddd uu dddd 2 = mm uu dd ΓΓ uu 1 Fall 2018 Prof. Sergio B. Mendes 81
82 Solving the integral: ΓΓ 1 1 uu cc 2 2 KK = mm uu dd ΓΓ uu 1 uu = mm 0 uu 1 uu cc dddd = mm cc uu cc 2 1 Fall 2018 Prof. Sergio B. Mendes 82
83 Kinetic Energy KK = mm cc uu cc 2 1 uu cc uu cc 2 1 uu cc KK mm uu2 2 Fall 2018 Prof. Sergio B. Mendes 83
84 Comparison of Classical and Relativistic Kinetic Energy Fall 2018 Prof. Sergio B. Mendes 84
85 Experimental Results Fall 2018 Prof. Sergio B. Mendes 85
86 Total Relativistic Energy KK = mm cc uu cc 2 1 KK + mm cc 2 mm cc 2 Total Energy: EE tttttttttt = = = ΓΓ mm cc 2 1 uu cc 2 Rest Energy: EE 0 = mm cc 2 Fall 2018 Prof. Sergio B. Mendes 86
87 Hydrogen Fusion and Solar Energy 1 HH + 1 HH 2 HH + ee + + νν ee MeV 2 HH + 1 HH 3 HHee + γγ MeV 3 HHHH + 3 HHHH 4 HHee HH + γγ MeV Fall 2018 Prof. Sergio B. Mendes 87
88 Total Energy and Linear Momentum EE tttttttttt = mm cc 2 1 uu cc 2 EE tttttttttt 2 = mm2 cc 4 1 uu cc 2 pp = mm uu 1 uu cc 2 pp 2 cc 2 = mm2 uu 2 cc 2 1 uu cc 2 EE tttttttttt 2 pp 2 cc 2 = mm 2 cc 4 Fall 2018 Prof. Sergio B. Mendes 88
89 Massless Particles: EE tttttttttt 2 pp 2 cc 2 = mm 2 cc 4 mm = 0 EE tttttttttt = pp cc uu = cc Fall 2018 Prof. Sergio B. Mendes 89
90 Units of Energy: WW = qq VV ee = CC VV = 1 VV WW = ee VV = JJ 1 eeee Fall 2018 Prof. Sergio B. Mendes 90
91 Units of Mass: EE 0 = mm cc 2 mm = EE 0 cc 2 1 kkkk = JJ/cc 2 1 uu 1 12 mmmmmmmm oooo nnnnnnnnnnnnnn 12 CC = kkkk = MMMMMM/cc 2 Particle Mass (MeV) / c 2 Mass (u) electron proton neutron Higgs boson 125, Fall 2018 Prof. Sergio B. Mendes 91
92 Units of Linear Momentum EE tttttttttt 2 pp 2 cc 2 = mm 2 cc 4 lliiiiiiiiii mmmmmmmmmmmmmmmm = eeeeeeeeeeee cc Fall 2018 Prof. Sergio B. Mendes 92
93 Binding Energy: EE 0, pp + 2 EE 0, nn EE 0, HHHH = EE BB,HHHH uu uu u cc 2 = uu cc 2 = 28.3 MMMMMM = EE BB,HHHH Fall 2018 Prof. Sergio B. Mendes 93
94 2.9 Spacetime Representation Fall 2018 Prof. Sergio B. Mendes 94
95 Conventional Representation: (tt, xx) Fall 2018 Prof. Sergio B. Mendes 95
96 Spacetime Representation (xx, cccc) When describing events in relativity, it is convenient to represent events on a spacetime diagram. In this diagram one spatial coordinate x, to specify position, is used and instead of time t, ct is used as the other coordinate so that both coordinates will have dimensions of length. Spacetime diagrams were first used by H. Minkowski in 1908 and are often called Minkowski diagrams. Paths in Minkowski spacetime are called worldlines. Fall 2018 Prof. Sergio B. Mendes 96
97 Worldline Fall 2018 Prof. Sergio B. Mendes 97
98 tan θθ = xx cc tt = uu cc = ββ xx cc tt uu tan θθ = uu cc = ββ 1 θθ θθ 45 Fall 2018 Prof. Sergio B. Mendes 98
99 Light Cone Fall 2018 Prof. Sergio B. Mendes 99
100 Spacetime and Inertial Frames of Reference xx = γγ xx γγ vv oo tt tt = γγ vv oo cc 2 xx + γγ tt cc tt θθ cc tt θθ xx xx tan θθ = vv oo cc Fall 2018 Prof. Sergio B. Mendes 100
101 Back to the Lorentz Transformations: ddtt = γγ vv oo cc 2 ddxx = γγ dddd γγ vv oo ddtt ddyy = ddyy ddzz = dddd ddxx + γγ dddd xx yy zz cccc four-vector ddxx 2 + ddyy 2 + ddzz 2 cc 2 ddtt 2 = ddxxx 2 + ddyy 2 + ddzz 2 cc 2 ddttt 2 Invariant: its value does not change among inertial frames of reference Fall 2018 Prof. Sergio B. Mendes 101
102 Example 2.13 KK = 2.00 GGGGGG KK = 2.00 GGGGGG EE tttttttttt =?? = KK + mm cc 2 = 2.00 GGGGGG GeV = 2.94 GGGGGG Fall 2018 Prof. Sergio B. Mendes 102
103 Example 2.13 KK = 2.00 GGGGGG KK = 2.00 GGGGGG pp =?? EE tttttttttt 2 pp 2 cc 2 = mm 2 cc 4 pp = 1 cc EE tttttttttt 2 mm cc 2 2 = 1 cc GGGGGG GGGGGG 2 = 2.78 GGGGGG cc Fall 2018 Prof. Sergio B. Mendes 103
104 Example 2.13 KK = 2.00 GGGGGG KK = 2.00 GGGGGG uu =?? ββ =?? ΓΓ =?? EE tttttttttt = KK + mm cc 2 = mm cc 2 2 = ΓΓ mm cc2 1 uu cc ΓΓ = EE tttttttttt mm cc 2 = GGGGGG GGGGGG = 3.13 ββ = ΓΓ2 1 ΓΓ 2 = Fall 2018 Prof. Sergio B. Mendes 104
105 Topics Inertial Frames of Reference Conceptual and Experimental Inconsistencies The Michelson-Morley Experiment Einstein s Postulates The Lorentz Transformation Time Dilation and Length Contraction Addition of Velocities Experimental Verification Twin Paradox Space-Time Doppler Effect Relativistic Momentum Relativistic Energy Computations in Modern Physics Electromagnetism and Relativity Fall 2018 Prof. Sergio B. Mendes 105
CHAPTER 2 Special Theory of Relativity-part 1
CHAPTER 2 Special Theory of Relativity-part 1 2.1 The Apparent Need for Ether 2.2 The Michelson-Morley Experiment 2.3 Einstein s Postulates 2.4 The Lorentz Transformation 2.5 Time Dilation and Length Contraction
More informationCHAPTER 2 Special Theory of Relativity
CHAPTER 2 Special Theory of Relativity 2.1 The Apparent Need for Ether 2.2 The Michelson-Morley Experiment 2.3 Einstein s Postulates 2.4 The Lorentz Transformation 2.5 Time Dilation and Length Contraction
More informationUnit- 1 Theory of Relativity
Unit- 1 Theory of Relativity Frame of Reference The Michelson-Morley Experiment Einstein s Postulates The Lorentz Transformation Time Dilation and Length Contraction Addition of Velocities Experimental
More informationThornton & Rex, 4th ed. Fall 2018 Prof. Sergio B. Mendes 1
Modern Physics for Scientists and Engineers Thornton & Rex, 4th ed. Fall 2018 Prof. Sergio B. Mendes 1 CHAPTER 1 The Birth of Modern Physics Fall 2018 Prof. Sergio B. Mendes 2 Topics 1) Classical Physics
More informationRelativity. An explanation of Brownian motion in terms of atoms. An explanation of the photoelectric effect ==> Quantum Theory
Relativity Relativity In 1905 Albert Einstein published five articles in Annalen Der Physik that had a major effect upon our understanding of physics. They included:- An explanation of Brownian motion
More informationCHAPTER 4 Structure of the Atom
CHAPTER 4 Structure of the Atom Fall 2018 Prof. Sergio B. Mendes 1 Topics 4.1 The Atomic Models of Thomson and Rutherford 4.2 Rutherford Scattering 4.3 The Classic Atomic Model 4.4 The Bohr Model of the
More informationRotational Motion. Chapter 10 of Essential University Physics, Richard Wolfson, 3 rd Edition
Rotational Motion Chapter 10 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 We ll look for a way to describe the combined (rotational) motion 2 Angle Measurements θθ ss rr rrrrrrrrrrrrrr
More informationPhotons in the universe. Indian Institute of Technology Ropar
Photons in the universe Photons in the universe Element production on the sun Spectral lines of hydrogen absorption spectrum absorption hydrogen gas Hydrogen emission spectrum Element production on the
More informationENTER RELATIVITY THE HELIOCENTRISM VS GEOCENTRISM DEBATE ARISES FROM MATTER OF CHOOSING THE BEST REFERENCE POINT. GALILEAN TRANSFORMATION 8/19/2016
ENTER RELATIVITY RVBAUTISTA THE HELIOCENTRISM VS GEOCENTRISM DEBATE ARISES FROM MATTER OF CHOOSING THE BEST REFERENCE POINT. GALILEAN TRANSFORMATION The laws of mechanics must be the same in all inertial
More informationCHAPTER 5 Wave Properties of Matter and Quantum Mechanics I
CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I 1 5.1 X-Ray Scattering 5.2 De Broglie Waves 5.3 Electron Scattering 5.4 Wave Motion 5.5 Waves or Particles 5.6 Uncertainty Principle Topics 5.7
More informationQuantum Mechanics. An essential theory to understand properties of matter and light. Chemical Electronic Magnetic Thermal Optical Etc.
Quantum Mechanics An essential theory to understand properties of matter and light. Chemical Electronic Magnetic Thermal Optical Etc. Fall 2018 Prof. Sergio B. Mendes 1 CHAPTER 3 Experimental Basis of
More informationChapter 2: The Special Theory of Relativity. A reference fram is inertial if Newton s laws are valid in that frame.
Chapter 2: The Special Theory of Relativity What is a reference frame? A reference fram is inertial if Newton s laws are valid in that frame. If Newton s laws are valid in one reference frame, they are
More informationWave Motion. Chapter 14 of Essential University Physics, Richard Wolfson, 3 rd Edition
Wave Motion Chapter 14 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 Waves: propagation of energy, not particles 2 Longitudinal Waves: disturbance is along the direction of wave propagation
More informationWork, Energy, and Power. Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition
Work, Energy, and Power Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 With the knowledge we got so far, we can handle the situation on the left but not the one on the right.
More informationIntroduction. Classical vs Modern Physics. Classical Physics: High speeds Small (or very large) distances
Introduction Classical vs Modern Physics High speeds Small (or very large) distances Classical Physics: Conservation laws: energy, momentum (linear & angular), charge Mechanics Newton s laws Electromagnetism
More informationChapter 36 The Special Theory of Relativity. Copyright 2009 Pearson Education, Inc.
Chapter 36 The Special Theory of Relativity Units of Chapter 36 Galilean Newtonian Relativity The Michelson Morley Experiment Postulates of the Special Theory of Relativity Simultaneity Time Dilation and
More informationRelativity Albert Einstein: Brownian motion. fi atoms. Photoelectric effect. fi Quantum Theory On the Electrodynamics of Moving Bodies
Relativity 1905 - Albert Einstein: Brownian motion fi atoms. Photoelectric effect. fi Quantum Theory On the Electrodynamics of Moving Bodies fi The Special Theory of Relativity The Luminiferous Ether Hypothesis:
More informationAngular Momentum, Electromagnetic Waves
Angular Momentum, Electromagnetic Waves Lecture33: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay As before, we keep in view the four Maxwell s equations for all our discussions.
More informationSpecial Theory of Relativity (I) Newtonian (Classical) Relativity. Newtonian Principle of Relativity. Inertial Reference Frame.
Special Theory of Relativity (I) Newtonian (Classical) Relativity Einstein s Postulates The Lorentz Transformation Time Dilation and Length Contraction Addition of Velocities Assumption It is assumed that
More informationInteraction with matter
Interaction with matter accelerated motion: ss = bb 2 tt2 tt = 2 ss bb vv = vv 0 bb tt = vv 0 2 ss bb EE = 1 2 mmvv2 dddd dddd = mm vv 0 2 ss bb 1 bb eeeeeeeeeeee llllllll bbbbbbbbbbbbbb dddddddddddddddd
More informationRotational Mechanics and Relativity --- Summary sheet 1
Rotational Mechanics and Relativity --- Summary sheet 1 Centre of Mass 1 1 For discrete masses: R m r For continuous bodies: R dm i i M M r body i Static equilibrium: the two conditions for a body in static
More informationPhysics 371 Spring 2017 Prof. Anlage Review
Physics 71 Spring 2017 Prof. Anlage Review Special Relativity Inertial vs. non-inertial reference frames Galilean relativity: Galilean transformation for relative motion along the xx xx direction: xx =
More information8.20 MIT Introduction to Special Relativity IAP 2005 Tentative Outline
8.20 MIT Introduction to Special Relativity IAP 2005 Tentative Outline 1 Main Headings I Introduction and relativity pre Einstein II Einstein s principle of relativity and a new concept of spacetime III
More informationPHL424: Feynman diagrams
PHL424: Feynman diagrams In 1940s, R. Feynman developed a diagram technique to describe particle interactions in space-time. Feynman diagram example Richard Feynman time Particles are represented by lines
More informationTopics: Relativity: What s It All About? Galilean Relativity Einstein s s Principle of Relativity Events and Measurements
Chapter 37. Relativity Topics: Relativity: What s It All About? Galilean Relativity Einstein s s Principle of Relativity Events and Measurements The Relativity of Simultaneity Time Dilation Length g Contraction
More informationModern Physics. Third Edition RAYMOND A. SERWAY CLEMENT J. MOSES CURT A. MOYER
Modern Physics Third Edition RAYMOND A. SERWAY CLEMENT J. MOSES CURT A. MOYER 1 RELATIVITY 1.1 Special Relativity 1.2 The Principle of Relativity, The Speed of Light 1.3 The Michelson Morley Experiment,
More informationLecture 3 Transport in Semiconductors
EE 471: Transport Phenomena in Solid State Devices Spring 2018 Lecture 3 Transport in Semiconductors Bryan Ackland Department of Electrical and Computer Engineering Stevens Institute of Technology Hoboken,
More informationJF Theoretical Physics PY1T10 Special Relativity
JF Theoretical Physics PY1T10 Special Relativity 12 Lectures (plus problem classes) Prof. James Lunney Room: SMIAM 1.23, jlunney@tcd.ie Books Special Relativity French University Physics Young and Freedman
More informationCHAPTER 2 Special Theory of Relativity Part 2
CHAPTER 2 Special Theory of Relativity Part 2 2.1 The Apparent Need for Ether 2.2 The Michelson-Morley Experiment 2.3 Einstein s Postulates 2.4 The Lorentz Transformation 2.5 Time Dilation and Length Contraction
More informationThe Theory of Relativity
At end of 20th century, scientists knew from Maxwell s E/M equations that light traveled as a wave. What medium does light travel through? There can be no doubt that the interplanetary and interstellar
More informationPHL424: Nuclear Shell Model. Indian Institute of Technology Ropar
PHL424: Nuclear Shell Model Themes and challenges in modern science Complexity out of simplicity Microscopic How the world, with all its apparent complexity and diversity can be constructed out of a few
More informationChapter 1. Relativity 1
Chapter 1 Relativity 1 Classical Relativity inertial vs noninertial reference frames Inertial Reference Frames Galilean transformation: x = x vt; y = y; z = z; t = t u x = u x v; u y = u y ; u z = u z
More informationMidterm Solutions. 1 1 = 0.999c (0.2)
Midterm Solutions 1. (0) The detected muon is seen km away from the beam dump. It carries a kinetic energy of 4 GeV. Here we neglect the energy loss and angular scattering of the muon for simplicity. a.
More informationAnnouncements. Muon Lifetime. Lecture 4 Chapter. 2 Special Relativity. SUMMARY Einstein s Postulates of Relativity: EXPERIMENT
Announcements HW1: Ch.2-20, 26, 36, 41, 46, 50, 51, 55, 58, 63, 65 Lab start-up meeting with TA tomorrow (1/26) at 2:00pm at room 301 Lab manual is posted on the course web *** Course Web Page *** http://highenergy.phys.ttu.edu/~slee/2402/
More informationReview for Exam Hyunse Yoon, Ph.D. Adjunct Assistant Professor Department of Mechanical Engineering, University of Iowa
Review for Exam2 11. 13. 2015 Hyunse Yoon, Ph.D. Adjunct Assistant Professor Department of Mechanical Engineering, University of Iowa Assistant Research Scientist IIHR-Hydroscience & Engineering, University
More informationReview for Exam Hyunse Yoon, Ph.D. Assistant Research Scientist IIHR-Hydroscience & Engineering University of Iowa
57:020 Fluids Mechanics Fall2013 1 Review for Exam3 12. 11. 2013 Hyunse Yoon, Ph.D. Assistant Research Scientist IIHR-Hydroscience & Engineering University of Iowa 57:020 Fluids Mechanics Fall2013 2 Chapter
More informationCharge carrier density in metals and semiconductors
Charge carrier density in metals and semiconductors 1. Introduction The Hall Effect Particles must overlap for the permutation symmetry to be relevant. We saw examples of this in the exchange energy in
More informationBig Bang Planck Era. This theory: cosmological model of the universe that is best supported by several aspects of scientific evidence and observation
Big Bang Planck Era Source: http://www.crystalinks.com/bigbang.html Source: http://www.odec.ca/index.htm This theory: cosmological model of the universe that is best supported by several aspects of scientific
More informationName the object labelled B and explain its purpose.
PhysicsAndMathsTutor.com 1 1. The diagram represents the Michelson-Morley interferometer. surface-silvered mirror M 1 l 1 extended source of monochromatic light B surface-silvered mirror M 2 A l 2 viewing
More information2.1 The Ether and the Michelson-Morley Experiment
Chapter. Special Relativity Notes: Some material presented in this chapter is taken The Feynman Lectures on Physics, Vol. I by R. P. Feynman, R. B. Leighton, and M. Sands, Chap. 15 (1963, Addison-Wesley)..1
More informationPostulate 2: Light propagates through empty space with a definite speed (c) independent of the speed of the source or of the observer.
Einstein s Special Theory of Relativity 1 m E = mv E =m*c m* = KE =m*c - m c 1- v p=mv p=m*v c 9-1 Postulate 1: The laws of physics have the same form in all inertial reference frames. Postulate : Light
More informationAnnouncement. Einstein s Postulates of Relativity: PHYS-3301 Lecture 3. Chapter 2. Sep. 5, Special Relativity
Announcement PHYS-3301 Lecture 3 Sep. 5, 2017 2 Einstein s Postulates of Relativity: Chapter 2 Special Relativity 1. Basic Ideas 6. Velocity Transformation 2. Consequences of Einstein s Postulates 7. Momentum
More informationProblem 3.1 (Verdeyen 5.13) First, I calculate the ABCD matrix for beam traveling through the lens and space.
Problem 3. (Verdeyen 5.3) First, I calculate the ABCD matrix for beam traveling through the lens and space. T = dd 0 0 dd 2 ff 0 = dd 2 dd ff 2 + dd ( dd 2 ff ) dd ff ff Aording to ABCD law, we can have
More informationRelativity SPECIAL, GENERAL, AND COSMOLOGICAL SECOND EDITION. Wolfgang Rindler. Professor of Physics The University of Texas at Dallas
Relativity SPECIAL, GENERAL, AND COSMOLOGICAL SECOND EDITION Wolfgang Rindler Professor of Physics The University of Texas at Dallas OXPORD UNIVERSITY PRESS Contents Introduction l 1 From absolute space
More informationTwo postulates Relativity of simultaneity Time dilation; length contraction Lorentz transformations Doppler effect Relativistic kinematics
Two postulates Relativity of simultaneity Time dilation; length contraction Lorentz transformations Doppler effect Relativistic kinematics Phys 2435: Chap. 37, Pg 1 Two postulates New Topic Phys 2435:
More informationRELATIVITY. The End of Physics? A. Special Relativity. 3. Einstein. 2. Michelson-Morley Experiment 5
1 The End of Physics? RELATIVITY Updated 01Aug30 Dr. Bill Pezzaglia The following statement made by a Nobel prize winning physicist: The most important fundamental laws and facts of physical science have
More informationGravitation. Chapter 8 of Essential University Physics, Richard Wolfson, 3 rd Edition
Gravitation Chapter 8 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 What you are about to learn: Newton's law of universal gravitation About motion in circular and other orbits How to
More informationMassachusetts Institute of Technology Physics Department Physics 8.20 IAP 2005 Introduction to Special Relativity
Massachusetts Institute of Technology Physics Department Physics 8.20 IAP 2005 Introduction to Special Relativity Problem Set 1 1. Speeds What fraction of the speed of light does each of the following
More informationIntroduction to Relativity & Time Dilation
Introduction to Relativity & Time Dilation The Principle of Newtonian Relativity Galilean Transformations The Michelson-Morley Experiment Einstein s Postulates of Relativity Relativity of Simultaneity
More informationHeat, Work, and the First Law of Thermodynamics. Chapter 18 of Essential University Physics, Richard Wolfson, 3 rd Edition
Heat, Work, and the First Law of Thermodynamics Chapter 18 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 Different ways to increase the internal energy of system: 2 Joule s apparatus
More informationPhysics 2D Lecture Slides Sept 29. Vivek Sharma UCSD Physics
Physics 2D Lecture Slides Sept 29 Vivek Sharma UCSD Physics Galilean Relativity Describing a Physical Phenomenon Event ( and a series of them) Observer (and many of them) Frame of reference (& an Observer
More information(1) Correspondence of the density matrix to traditional method
(1) Correspondence of the density matrix to traditional method New method (with the density matrix) Traditional method (from thermal physics courses) ZZ = TTTT ρρ = EE ρρ EE = dddd xx ρρ xx ii FF = UU
More informationNewtonian or Galilean Relativity
Relativity Eamples 1. What is the velocity of an electron in a 400 kv transmission electron microscope? What is the velocity in the 6 GeV CESR particle accelerator?. If one million muons enter the atmosphere
More information2.1 Einstein s postulates of Special Relativity. (i) There is no ether (there is no absolute system of reference).
Chapter 2 Special Relativity The contradiction brought about by the development of Electromagnetism gave rise to a crisis in the 19th century that Special Relativity resolved. 2.1 Einstein s postulates
More informationYang-Hwan Ahn Based on arxiv:
Yang-Hwan Ahn (CTPU@IBS) Based on arxiv: 1611.08359 1 Introduction Now that the Higgs boson has been discovered at 126 GeV, assuming that it is indeed exactly the one predicted by the SM, there are several
More informationLecture 8 : Special Theory of Relativity
Lecture 8 : Special Theory of Relativity The speed of light problem Einstein s postulates Time dilation 9/23/10 1 Sidney Harris I: THE SPEED OF LIGHT PROBLEM Recap Relativity tells us how to relate measurements
More informationChemical Engineering 412
Chemical Engineering 412 Introductory Nuclear Engineering Lecture 12 Radiation/Matter Interactions II 1 Neutron Flux The collisions of neutrons of all energies is given by FF = ΣΣ ii 0 EE φφ EE dddd All
More informationElastic light scattering
Elastic light scattering 1. Introduction Elastic light scattering in quantum mechanics Elastic scattering is described in quantum mechanics by the Kramers Heisenberg formula for the differential cross
More informationA. B. Lahanas University of Athens, Physics Department, Nuclear and Particle Physics Section, Athens , Greece
SPECIAL RELATIVITY A. B. Lahanas University of Athens, Physics Department, Nuclear and Particle Physics Section, Athens 157 71, Greece Abstract We give an introduction to Einstein s Special Theory of Relativity.
More informationPhysics 2D Lecture Slides Lecture 2. Jan. 5, 2010
Physics 2D Lecture Slides Lecture 2 Jan. 5, 2010 Lecture 1: Relativity Describing a Physical Phenomenon Event (s) Observer (s) Frame(s) of reference (the point of View! ) Inertial Frame of Reference Accelerated
More information2.4 The Lorentz Transformation
Announcement Course webpage http://highenergy.phys.ttu.edu/~slee/2402/ Textbook PHYS-2402 Lecture 4 Jan. 27, 2015 Lecture Notes, HW Assignments, Physics Colloquium, etc.. 2.4 The Lorentz Transformation
More informationThe Constancy of the Speed of Light
The Constancy of the Speed of Light Also, recall the Michelson-Morley experiment: c-u c+u u Presumed ether wind direction u is the relative speed between the frames (water & shore) Result: Similar There
More informationChapter 26 Special Theory of Relativity
Chapter 26 Special Theory of Relativity Classical Physics: At the end of the 19 th century, classical physics was well established. It seems that the natural world was very well explained. Newtonian mechanics
More informationVariations. ECE 6540, Lecture 02 Multivariate Random Variables & Linear Algebra
Variations ECE 6540, Lecture 02 Multivariate Random Variables & Linear Algebra Last Time Probability Density Functions Normal Distribution Expectation / Expectation of a function Independence Uncorrelated
More information10.4 The Cross Product
Math 172 Chapter 10B notes Page 1 of 9 10.4 The Cross Product The cross product, or vector product, is defined in 3 dimensions only. Let aa = aa 1, aa 2, aa 3 bb = bb 1, bb 2, bb 3 then aa bb = aa 2 bb
More informationPhoton Interactions in Matter
Radiation Dosimetry Attix 7 Photon Interactions in Matter Ho Kyung Kim hokyung@pusan.ac.kr Pusan National University References F. H. Attix, Introduction to Radiological Physics and Radiation Dosimetry,
More informationSECTION 7: FAULT ANALYSIS. ESE 470 Energy Distribution Systems
SECTION 7: FAULT ANALYSIS ESE 470 Energy Distribution Systems 2 Introduction Power System Faults 3 Faults in three-phase power systems are short circuits Line-to-ground Line-to-line Result in the flow
More informationRelativistic Kinetic Energy Simplified. Copyright Joseph A. Rybczyk
Relativistic Kinetic Energy Simplified Copyright 207 Joseph A. Rybczyk Abstract The relativistic form of the kinetic energy formula is derived directly from the relativized principles of the classical
More informationCollege Physics B - PHY2054C. Special Relativity 11/10/2014. My Office Hours: Tuesday 10:00 AM - Noon 206 Keen Building.
College - PHY2054C 11/10/2014 My Office Hours: Tuesday 10:00 AM - Noon 206 Keen Building Outline 1 2 3 1 The speed of light is the maximum possible speed, and it is always measured to have the same value
More informationPhysics 2D Lecture Slides Lecture 2. March 31, 2009
Physics 2D Lecture Slides Lecture 2 March 31, 2009 Newton s Laws and Galilean Transformation! But Newton s Laws of Mechanics remain the same in All frames of references!! 2 2 d x' d x' dv = " dt 2 dt 2
More informationRelativity. Physics April 2002 Lecture 8. Einstein at 112 Mercer St. 11 Apr 02 Physics 102 Lecture 8 1
Relativity Physics 102 11 April 2002 Lecture 8 Einstein at 112 Mercer St. 11 Apr 02 Physics 102 Lecture 8 1 Physics around 1900 Newtonian Mechanics Kinetic theory and thermodynamics Maxwell s equations
More informationModule 2: Special Theory of Relativity - Basics
Lecture 01 PH101: Physics 1 Module 2: Special Theory of Relativity - Basics Girish Setlur & Poulose Poulose gsetlur@iitg.ac.in Department of Physics, IIT Guwahati poulose@iitg.ac.in ( 22 October 2018 )
More informationDressing up for length gauge: Aspects of a debate in quantum optics
Dressing up for length gauge: Aspects of a debate in quantum optics Rainer Dick Department of Physics & Engineering Physics University of Saskatchewan rainer.dick@usask.ca 1 Agenda: Attosecond spectroscopy
More informationExam 2 Fall 2015
1 95.144 Exam 2 Fall 2015 Section instructor Section number Last/First name Last 3 Digits of Student ID Number: Show all work. Show all formulas used for each problem prior to substitution of numbers.
More informationChapter 12. Electrodynamics and Relativity. Does the principle of relativity apply to the laws of electrodynamics?
Chapter 12. Electrodynamics and Relativity Does the principle of relativity apply to the laws of electrodynamics? 12.1 The Special Theory of Relativity Does the principle of relativity apply to the laws
More informationPostulates of Special Relativity
Relativity Relativity - Seen as an intricate theory that is necessary when dealing with really high speeds - Two charged initially stationary particles: Electrostatic force - In another, moving reference
More informationCollege Physics B - PHY2054C. Special & General Relativity 11/12/2014. My Office Hours: Tuesday 10:00 AM - Noon 206 Keen Building.
Special College - PHY2054C Special & 11/12/2014 My Office Hours: Tuesday 10:00 AM - Noon 206 Keen Building Outline Special 1 Special 2 3 4 Special Galilean and Light Galilean and electromagnetism do predict
More informationME5286 Robotics Spring 2017 Quiz 2
Page 1 of 5 ME5286 Robotics Spring 2017 Quiz 2 Total Points: 30 You are responsible for following these instructions. Please take a minute and read them completely. 1. Put your name on this page, any other
More informationGeneral Relativity I
General Relativity I presented by John T. Whelan The University of Texas at Brownsville whelan@phys.utb.edu LIGO Livingston SURF Lecture 2002 July 5 General Relativity Lectures I. Today (JTW): Special
More information(1) Introduction: a new basis set
() Introduction: a new basis set In scattering, we are solving the S eq. for arbitrary VV in integral form We look for solutions to unbound states: certain boundary conditions (EE > 0, plane and spherical
More informationThe Theory of Relativity
The Theory of Relativity Lee Chul Hoon chulhoon@hanyang.ac.kr Copyright 2001 by Lee Chul Hoon Table of Contents 1. Introduction 2. The Special Theory of Relativity The Galilean Transformation and the Newtonian
More informationChapter 22 : Electric potential
Chapter 22 : Electric potential What is electric potential? How does it relate to potential energy? How does it relate to electric field? Some simple applications What does it mean when it says 1.5 Volts
More informationPhotons in the universe. Indian Institute of Technology Ropar
Photons in the universe Photons in the universe Element production on the sun Spectral lines of hydrogen absorption spectrum absorption hydrogen gas Hydrogen emission spectrum Element production on the
More informationVarying accelerating fields
Varying accelerating fields Two approaches for accelerating with time-varying fields Linear Accelerators Circular Accelerators Use many accelerating cavities through which the particle beam passes once.
More informationModule 7 (Lecture 27) RETAINING WALLS
Module 7 (Lecture 27) RETAINING WALLS Topics 1.1 RETAINING WALLS WITH METALLIC STRIP REINFORCEMENT Calculation of Active Horizontal and vertical Pressure Tie Force Factor of Safety Against Tie Failure
More informationTherefore F = ma = ma = F So both observers will not only agree on Newton s Laws, but will agree on the value of F.
Classical Physics Inertial Reference Frame (Section 5.2): a reference frame in which an object obeys Newton s Laws, i.e. F = ma and if F = 0 (object does not interact with other objects), its velocity
More informationSpecial Theory of Relativity. A Brief introduction
Special Theory of Relativity A Brief introduction Classical Physics At the end of the 19th century it looked as if Physics was pretty well wrapped up. Newtonian mechanics and the law of Gravitation had
More informationProblem 4.1 (Verdeyen Problem #8.7) (a) From (7.4.7), simulated emission cross section is defined as following.
Problem 4.1 (Verdeyen Problem #8.7) (a) From (7.4.7), simulated emission cross section is defined as following. σσ(νν) = AA 21 λλ 2 8ππnn 2 gg(νν) AA 21 = 6 10 6 ssssss 1 From the figure, the emission
More informationExtra notes on rela,vity. Wade Naylor
Extra notes on rela,vity Wade Naylor Over 105 years since Einstein s Special theory of relativity A. Einstein, 1879-1955 The postulates of special relativity 1. The principle of relativity (Galileo) states
More informationSpecial Theory of Relativity. PH101 Lec-2
Special Theory of Relativity PH101 Lec-2 Newtonian Relativity! The transformation laws are essential if we are to compare the mathematical statements of the laws of physics in different inertial reference
More informationThe Foundations of Special Relativity
The Foundations of Special Relativity 1 Einstein's postulates of SR: 1. The laws of physics are identical in all inertial reference frames (IFs). 2. The speed of light in vacuum, c, is the same in all
More informationChapter-1 Relativity Part I RADIATION
Chapter-1 Relativity Part I RADIATION Radiation implies the transfer of energy from one place to another. - Electromagnetic Radiation - Light - Particle and Cosmic Radiation photons, protons, neutrons,
More informationRevision : Thermodynamics
Revision : Thermodynamics Formula sheet Formula sheet Formula sheet Thermodynamics key facts (1/9) Heat is an energy [measured in JJ] which flows from high to low temperature When two bodies are in thermal
More informationRelativity. Overview & Postulates Events Relativity of Simultaneity. Relativity of Time. Relativity of Length Relativistic momentum and energy
Relativity Overview & Postulates Events Relativity of Simultaneity Simultaneity is not absolute Relativity of Time Time is not absolute Relativity of Length Relativistic momentum and energy Relativity
More informationString Theory in the LHC Era
String Theory in the LHC Era J Marsano (marsano@uchicago.edu) 1 String Theory in the LHC Era 1. Electromagnetism and Special Relativity 2. The Quantum World 3. Why do we need the Higgs? 4. The Standard
More informationPHY103A: Lecture # 4
Semester II, 2017-18 Department of Physics, IIT Kanpur PHY103A: Lecture # 4 (Text Book: Intro to Electrodynamics by Griffiths, 3 rd Ed.) Anand Kumar Jha 10-Jan-2018 Notes The Solutions to HW # 1 have been
More informationINTRODUCTION TO QUANTUM MECHANICS
4 CHAPTER INTRODUCTION TO QUANTUM MECHANICS 4.1 Preliminaries: Wave Motion and Light 4.2 Evidence for Energy Quantization in Atoms 4.3 The Bohr Model: Predicting Discrete Energy Levels in Atoms 4.4 Evidence
More informationPrinciple of Relativity
Principle of Relativity Physical laws are the same in all inertial frames. 1) The same processes occur. But 2) the description of some instance depends on frame of reference. Inertial Frames An inertial
More informationThe impact of hot charge carrier mobility on photocurrent losses
Supplementary Information for: The impact of hot charge carrier mobility on photocurrent losses in polymer-based solar cells Bronson Philippa 1, Martin Stolterfoht 2, Paul L. Burn 2, Gytis Juška 3, Paul
More informationAcceleration to higher energies
Acceleration to higher energies While terminal voltages of 20 MV provide sufficient beam energy for nuclear structure research, most applications nowadays require beam energies > 1 GeV How do we attain
More information