ทฤษฎีสัมพัทธภาพพิเศษ (special theory of relativity)

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บทที่ 30 ทฤษฎีสัมพัทธภาพพิเศษ (Special Theory of Relativity)วิชา ฟิสิกส์ 2 (Physics 2)

TRANSCRIPT

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    30 (Special Theory of Relativity)

    : 3 .. 2558

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    1 2 3 4 5 6 7

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    , & &

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    , & (Event) (Space) (Universe) 3

    (Classical Physics) (Euclidean space) 3 E3( R3) (homogeneous) (isotropic)

    + / ( . ) 30 : 3 .. 2558 4 / 68

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    (Time) (dimension) (identical events) 2

    1 13th CGPM, 1967/68

    `` The second is the duration of 9 192 631 770 periods of theradiation corresponding to the transition between the twohyperne levels of the ground state of the caesium 133 atom.'' 9 192 631 770 133

    ( . ) 30 : 3 .. 2558 5 / 68

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    1 17th CGPM, 1983

    `` The metre is the length of the path travelled by light invacuum during a time interval of 1/299 792 458 of a second.'' 1/299 792 458

    (Speed of Light in Vacuum)c = 299 792 458 ms1 () (1)

    + c = 3 108 ms1 ( . ) 30 : 3 .. 2558 6 / 68

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    & (Frame of Reference) (coordinate system) (set of axes)

    XY

    Z

    Oxy

    z

    x,y,z( )

    1: 3 (E3) ( . ) 30 : 3 .. 2558 7 / 68

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    (Inertial Frame of Reference)

    1 2

    (uniform motion) (fixed star)A : (Newton's laws of motion)!

    ( . ) 30 : 3 .. 2558 8 / 68

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    O0 S0 ~u x O S( x ) t = 0 t0 = 0 ( S S0 )

    2: 2 S S0 (event)

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    3: 2 S S0 P (a) ( S0) P P(b) ( S) P

    ( . ) 30 : 3 .. 2558 10 / 68

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    4: P (x0; y0; z0) t0 S0 (x; y; z) t S ( 14 )

    4 x0 = x ut ; y0 = y z0 = z

    ( . ) 30 : 3 .. 2558 11 / 68

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    S S0 (time scale) t = t0

    P (Galilean coordinate transformation) (Galilean Coordinate Transformation)

    x0 = x uty0 = y

    z0 = z

    t0 = t

    9>>>>>=>>>>>;()

    8>>>>>>>>>:

    x = x0 + ut0

    y = y0

    z = z0

    t = t0

    ( u c) (2)

    ( . ) 30 : 3 .. 2558 12 / 68

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    P x, y z S S0

    vx =dxdt

    , vy = dydt vz =dzdt

    Sv0x =

    dx0

    dt0, v0y = dy

    0

    dt0 v0z = dz

    0

    dt0 S0

    (Galilean velocity transformation) (Galilean Velocity Transformation)

    v0x = vx uv0y = vy

    v0z = vz

    9>>=>>; ()8>>>:vx = v

    0x + u

    vy = v0y

    vz = v0z

    ( u c) (3)

    ( . ) 30 : 3 .. 2558 13 / 68

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    P x

    vS0/S = u ; vP/S = vx vP/S0 = v0x (4)

    5: 2 S S0 P x

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    P x x

    vP/S0 = vP/S vS0/S vP/S = vP/S0 + vS0/S (5)( vS0/S; vP/S; vP/S0 c)

    vS0/S S0 SvP/S0 P x S0vP/S P x S

    ( . ) 30 : 3 .. 2558 15 / 68

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    ( S0) ( S) vS0/S = 1000 ms1 (missile) P vP/S0 = 2000 ms1 ( S0) 6

    (5) vP/S = vP/S0 + vS0/S = 2000 ms1 + 1000 ms1

    6: ( . ) 30 : 3 .. 2558 16 / 68

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    P x, y z S S0

    ax =dvxdt

    , ay = dvydt az =dvzdt

    Sa0x =

    dv0xdt0

    , a0y = dv0y

    dt0 a0z = dv

    0z

    dt0 S0

    (Galilean acceleration transformation) (Galilean Acceleration Transformation)

    a0x = ax

    a0y = ay

    a0z = az

    9>>=>>; ()8>>>:ax = a

    0x

    ay = a0y

    az = a0z

    ( u c) (6)

    ( . ) 30 : 3 .. 2558 17 / 68

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    m ~v ( v = j~vj) (Classical Momentum)

    ~p = m~v ( v c) (7) (Classical Kinetic Energy)

    Ek =1

    2mv2 ( v c) (8)

    ( . ) 30 : 3 .. 2558 18 / 68

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    ~F m ~v ~a = d~v

    dt ~F ~p

    2 (Newtons 2nd law of motion) 2 (Newtons 2nd Law of Motion)

    ~F = d~pdt

    ( ~F 6= m~a) (9)

    m v c ~F = d(m~v)dt

    = md~vdt

    2

    ~F = m~a ( v c m ) (10) ( . ) 30 : 3 .. 2558 19 / 68

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    (10) m v c ~F ~a

    (6) (S0 S)

    ~a 0 = ~a (11) m ()

    ~F0 = ~F (12)

    ( . ) 30 : 3 .. 2558 20 / 68

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    m A ( ~vA) B ( ~vB) ~F (Work)

    WA!B =Z BA

    ~F d~r (13)

    ( . ) 30 : 3 .. 2558 21 / 68

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    d~r (infinitesimal displacement) (infinitesimal time interval) dt ~v t

    d~r = ~vdt (14) m v c

    WA!B =Z BA

    m~a ~vdt = mZ BA

    d~vdt

    ~vdt = mZ BA~v d~v

    = m

    Z vBvA

    v dv = m v2

    2

    vBv=vA

    ( . ) 30 : 3 .. 2558 22 / 68

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    (WorkEnergy Theorem)

    WA!B =1

    2mv2B

    1

    2mv2A (15)

    ( vA; vB c m )

    WA!B = EkB EkA (16)

    = (17)

    ( . ) 30 : 3 .. 2558 23 / 68

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    (postulates) 2

    1: (Principle of Relativity)

    `` ''

    2:

    `` ''

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    7 1 (Electromotive Force emf)

    7: :(a) (b)

    ( . ) 30 : 3 .. 2558 25 / 68

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    ( S0) ( S) vS0/S = 1000 ms1 (light beam) P vP/S0 = c ( S0) 8

    (5) vP/S = vP/S0 + vS0/S = c+ 1000 ms1 6= c

    8: 2

    ( . ) 30 : 3 .. 2558 26 / 68

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    2 `` ''

    9: (Einstein's speed limit) ( . ) 30 : 3 .. 2558 27 / 68

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    (proper time) (proper length)

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    10: (a) S0 (source) (mirror) 2d t0 ( S0)(b) S 2` t ( S)

    ( . ) 30 : 3 .. 2558 29 / 68

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    S0 S

    t0 =2d

    c t = 2`

    c(18)

    ` =

    sd2 +

    ut

    2

    2 (19) (19) (18)

    t =2

    c

    sd2 +

    ut

    2

    2 (20) ( . ) 30 : 3 .. 2558 30 / 68

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    (18) d = ct02

    (20)

    t =2

    c

    sct02

    2+

    ut

    2

    2 (21) (Time Dilation)

    t =t0r1 u

    2

    c2

    (22)

    + u < c t > t0 ! (Time dilation) t0

    ( . ) 30 : 3 .. 2558 31 / 68

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    1r1 u

    2

    c2

    (Lorentz factor) (Lorentz Factor)

    =1r

    1 u2

    c2

    (23)

    (22)

    t = t0 (24)

    ( . ) 30 : 3 .. 2558 32 / 68

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    1: (muon) 2:2 0:99 S S0 S0 t0 = 2:2 s u = 0:99c (22)

    t =t0r1 u

    2

    c2

    =2:2 sr

    1 (0:99c)2

    c2

    = 16 s

    ) 16

    ( . ) 30 : 3 .. 2558 33 / 68

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    11: (a) `0 S0 (b) S `

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    (source) (mirror)

    S0

    t0 =2`0c

    (25)

    S ` u t1 ( S) ut1 `

    d = `+ ut1 (26) ( . ) 30 : 3 .. 2558 35 / 68

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    d = ct1 (26)

    ct1 = `+ ut1 (27) () S

    t1 =`

    c u (28)

    A : ` c u c u S `

    ( . ) 30 : 3 .. 2558 36 / 68

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    () S

    t2 =`

    c+ u(29)

    S t = t1 +t2

    t =`

    c u +`

    c+ u=

    2`

    c

    11 u

    2

    c2

    (30)

    S S0 (22) t = t0r

    1 u2

    c2

    ( . ) 30 : 3 .. 2558 37 / 68

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    (25) (30) (22)

    2`

    c

    11 u

    2

    c2

    =2`0c

    r1 u

    2

    c2

    (Length Contraction)

    ` = `0

    r1 u

    2

    c2(31)

    + u < c ` < `0 ! ! S S0 (Length contraction) `0

    ( . ) 30 : 3 .. 2558 38 / 68

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    (31)

    ` =`0

    (32)

    12:

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    13: (a) (b)

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    2: 60 0:8

    S S0 S0 `0 = 60 m u = 0:8c (31)

    ` = `0

    r1 u

    2

    c2= (60 m)

    r1 (0:8c)

    2

    c2= 36 m

    ) 36

    ( . ) 30 : 3 .. 2558 41 / 68

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    x0 P S0 ( 4)

    S x0

    14

    14: P (x0; y0; z0) t0 S0 (x; y; z) t S ( 4 )

    ( . ) 30 : 3 .. 2558 42 / 68

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    14 x0

    = x ut ; y0 = y z0 = z

    x0 =

    x ut ; y0 = y z0 = z (33)

    1 x =

    x0 + ut0

    ; y = y0 z = z0 (34)

    x0 (33) (34) x =

    h

    x ut+ ut0i

    ( . ) 30 : 3 .. 2558 43 / 68

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    t0 =

    t ux

    c2

    (35) x (34) (33)

    x0 = h

    x0 + ut0

    uti

    t =

    t0 +

    ux0

    c2

    (36)

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    P (Lorentz coordinate transformation) (Lorentz Coordinate Transformation)

    x0 = x ut

    y0 = y

    z0 = z

    t0 = t ux

    c2

    9>>>>>>=>>>>>>;()

    8>>>>>>>>>>>>>:

    x = x0 + ut0

    y = y0

    z = z0

    t =

    t0 +

    ux0

    c2

    (37)

    ( . ) 30 : 3 .. 2558 45 / 68

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    dx0 =

    dx udt = vx udt

    dy0 = dy

    dz0 = dz

    dt0 = dt udx

    c2

    =

    1 uvx

    c2

    dt

    9>>>>>>>=>>>>>>>;(38)

    dx =

    dx0 + udt0

    =

    v0x + u

    dt0

    dy = dy0

    dz = dz0

    dt = dt0 + udx

    0

    c2

    =

    1 +

    uv0xc2

    dt0

    9>>>>>>>=>>>>>>>;(39)

    ( . ) 30 : 3 .. 2558 46 / 68

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    P x, y z S S0 (Lorentz velocity transformation) (Lorentz Velocity Transformation)

    v0x =vx u1 uvx

    c2

    v0y =

    vy

    1 uvx

    c2

    v0z =

    vz

    1 uvx

    c2

    9>>>>>>>>>>>>=>>>>>>>>>>>>;()

    8>>>>>>>>>>>>>>>>>>>>>>>>>>>:

    vx =v0x + u1 +

    uv0xc2

    vy =

    v0y

    1 +

    uv0xc2

    vz =

    v0z

    1 +

    uv0xc2

    (40)

    ( . ) 30 : 3 .. 2558 47 / 68

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    P x x

    vP/S0 =vP/S vS0/S 1 vS0/S vP/S

    c2

    ! vP/S = vP/S0 + vS0/S 1 +

    vS0/S vP/S0

    c2

    ! (41)

    vS0/S S0 SvP/S0 P x S0vP/S P x S

    ( . ) 30 : 3 .. 2558 48 / 68

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    8 x (41)

    vP/S =vP/S0 + vS0/S 1 +

    vS0/S vP/S0

    c2

    ! = c+ 1000 ms1 1 +

    1000 ms1(c)

    c2

    ! = c

    15: 2

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    3: 0:8c 0:9c() S, S0 P

    vS0/S = 0:8c vP/S0 = 0:9c

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    (41) vP/S =

    vP/S0 + vS0/S 1 +

    vS0/S vP/S0

    c2

    ! = (0:9c) + (0:8c)1 +

    (0:8c)(0:9c)

    c2

    = 0:988c < c

    ) 0:988c

    A : (5)

    vP/S = vP/S0 + vS0/S = 0:9c+ 0:8c = 1:7c > c

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    m ~v ( v = j~vj) (Relativistic Momentum)

    ~p = m~vr1 v

    2

    c2

    (42)

    + v c ~p = m~v ()

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    S0 m ( S0) ~v S( S0 )

    =1r

    1 v2

    c2

    (43)

    (42)

    ~p = m~v (44) p = j~pj

    p = mv (45)

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    Helios SpacecraftHelios spacecraft () 252 792 (0:000 234 )

    v = 252 792 kmh1 0:000 234c

    1:000 000 027

    + v c 1 http://en.wikipedia.org/wiki/

    Helios (spacecraft)

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    ~F m ~F ~p 2 (9) ~F = d~p

    dt

    m ~F 6= m~a( ~F ~a )

    ~F = d(m~v)dt

    = md~vdt

    +ddt

    m~v = m~a+ ddt

    m~v

    ddt

    =

    3

    c2~a ~v ~F,

    ~v ~a 2

    ~F = m~a+

    2

    c2~a ~v~v ( m ) (46)

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    ddt

    =1

    mc2~F ~v

    ~a = 1

    m

    ~F 1

    c2~F ~v~v ( m ) (47)

    (46) (47) ~F ~a

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    m A ( ~vA)

    B ( ~vB) ~F (13)

    WA!B =Z BA

    ~F d~r

    2 (9) ~F = d~pdt

    WA!B =Z BA

    d~pdt

    d~r =Z BA

    d~rdt

    d~p =Z BA~v dm~v

    m

    WA!B = mZ BA~v d~v (48)

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    WA!B =mc2r1 v

    2Bc2

    mc2s

    1 v2Ac2

    ( m ) (49)

    m ~v ( v = j~vj) (Relativistic Energy)

    E =mc2r1 v

    2

    c2

    E = mc2 (50)

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    (rest energy) E0

    (Rest Energy)E0 = mc

    2 (51)

    Ek = E E0 (52)

    (Relativistic Kinetic Energy)Ek = ( 1)mc2 (53)

    ( . ) 30 : 3 .. 2558 59 / 68

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    (Maclaurin series) (1 + x)n jxj < 1

    (1 + x)n = 1 + nx+n(n 1)

    2!x2 +

    n(n 1)(n 2)3!

    x3 + (54)

    v < c

    = 1 +1

    2

    v2

    c2+

    3

    8

    v4

    c4+

    5

    16

    v6

    c6+ (55)

    (53)

    Ek =1

    2mv2 +

    3

    8

    mv4

    c2+

    5

    16

    mv6

    c4+ (56)

    + v c Ek = 12mv2 (

    ) ( . ) 30 : 3 .. 2558 60 / 68

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    (50) E2 = m2c41 v

    2

    c2

    E21 v

    2

    c2

    = m2c4

    E2 E2 v2

    c2= m2c4

    E2 mc22 v2c2

    = m2c4

    E2 mv2c2 = m2c4E2 p2c2 = m2c4

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    (EnergyMomentum Relation)

    E =qp2c2 +m2c4 (57)

    (massless particle) (photon) (gluon) (stronginteraction)

    E = pc (m = 0) (58)

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    4: ( 9:109 1031 kg) MeV 1 MeV = 1:602 1013 J m = 9:109 1031 kg (51)

    E0 = mc2 =

    9:109 1031 kg3 108 ms12 = 8:198 1014 J

    MeV 1:602 1013 JMeV1 E0 =

    8:198 1014 J

    1:602 1013 JMeV1 = 0:511 MeV) 0:511 MeV

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    [1] Hugh D. Young and Roger A. Freedman, (),

    3 , 1, , (2551).

    [2] , (: ), 1, ., (2548).

    [3] Hugh D. Young and Roger A. Freedman, Sears and ZemanskysUniversity Physics: With Modern Physics, 13th ed.,Pearson/Addison-Wesley, San Francisco (2012).

    [4] Kenneth S. Krane, Modern Physics, 3rd ed., John Wiley & Sons,New Jersey (2012).

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    [5] Raymond A. Serway and John W. Jewett, Jr., Physics for Scientistsand Engineers with Modern Physics, 8th ed., Brooks/Cole,Belmont, California (2010).

    [6] Raymond A. Serway, Clement J. Moses and Curt A. Moyer, ModernPhysics, 3rd ed., Brooks/ColeThomson Learning, Belmont,California (2005).

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    1 :

    http://www.sciencecartoonsplus.com/gallery/einstein/ein15.gif

    2 1: http://en.wikipedia.org/wiki/File:Coord system CA 0.svg

    3 2: 2.1 26 Krane [4]4 3: 39.1 1146 Serway & Jewett [5]5 4: 2.2 27 Krane [4]6 5: 2.1 26 Krane [4]7 6: 37.2 (a) 1225 Young & Freedman [3]8 7: 37.1 1224 Young & Freedman [3]9 8: 37.2 (b) 1225 Young & Freedman [3]

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    10 9: http://www.cartoonsidrew.com/2011/05/einsteins-speed-limit.html

    11 10: 37.6 1229 Young & Freedman [3]12 11: 37.10 1234 Young & Freedman [3]13 12: 2.11 34 Krane [4]14 13: 2.12 35 Krane [4]15 14: 2.2 27 Krane [4]16 15: 37.2 (b) 1225 Young & Freedman [3]17 :

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    30

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    30