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Lecture 04
Special Theory of
Relativity II
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Maxwells Equations
Maxwells equations DO NOT have the sameform under a Galilean transformation.
In Maxwells theory the speed of light, in termsof the permeability and permittivity of free
space, was given by
Thus the velocity of light between movingsystems must be a constant.
1
v = c = 1 / 00
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Light Speed Contradiction
Based on Maxwells theory,
But classical mechanics tells us that
Which one is correct?
c = c '
c = c '+ v
2
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A Preferred Reference System
Ether was proposed as an absolute referencesystem in which the speed of light was this
constant and from which other measurementscould be made.
The Michelson-Morley experiment was anattempt to show the existence of ether.
Earth
Ether
v
v 10-4c
3
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The Michelson Interferometer
AC is parallel to the motionof the Earth.
Light from source S is splitby mirror A
and travels
to mirrors Cand D in mutually
perpendiculardirections.
After reflection the beamsrecombine at A slightly outof phase as viewed by
telescope E.
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Interferometer Fringe Pattern
Beam-splitter
Inputbeam
Mirror
Mirror
z
x
Earth
Drift
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Wave
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(x,t) =Asin(kxt+ 0)
whereAis amplitude,
=kx t+ 0 is phase, and 0 is the initial phase,
kis wave number: k=2
, where is wave length,
is angular velocity: = 2T
, where Tis period,
f =1
Tis frequency, the phase speed v =f =
T=
k.
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Wave Interference
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Wave Interference
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Sources S1 and S2 are coherent wave sources
Constructive Interferences: r2 r
1= m (m = 0,1,2,3,...)
Destructive Interferences: r2 r
1= (m + 0.5) (m = 0,1,2,3,...)
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Wave Interference
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Sources S1 and S2 are coherent wave sources
Constructive Interferences: = 2m (m = 0,1,2,3,...)
Destructive Interferences: = (2m +1) (m = 0,1,2,3,...)
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Wave Interference: Double-Slit
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At the center, = 0
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Interferometer Fringe Pattern
Beam-splitter
Inputbeam
Mirror
Mirror
z
x
Earth
Drift
The fringe pattern should shift if
the apparatus is rotated by 90.
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At the center, = t = t2f
n=
'
2=
(
t'
t)f
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Galilean Velocity Transformation
P
v
vP =
v 'P+
v
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vP
v
v 'P
vP
vv 'P
vP
v
v 'P
v '
P = v
p v
v 'P = v
p+ v
v 'P = v
P
2 v
2
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The Analysis
Time t1 from A to C and back:
Assuming the Galilean Transformation
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v
l1
l2
AC
D
Parallel and
anti-parallelpropagation
Time t2from A to D and back:
Perpendicular
propagation
t2 =
2l2
c2v
2
=
2l2
c
1
1v2/ c
2
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The Analysis
So that the change in time is:
Assuming the Galilean Transformation
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t' = t'2 t'
1 =
2
c
l2
1 v2 / c2 l1
1 v2 / c2
Upon rotating the apparatus, the optical path lengths 1and 2are interchanged producing a different change in
time: (note the change in denominators)
t = t2 t
1 =
2
c
l2
1 v2 / c2 l11 v2 / c2
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The Analysis
and upon a binomial expansion, assumingv
/c
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Michelson-Morley Experiment
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Michelson and Morley's results
from A. A. Michelson, Studies in
Optics
Interference fringes showed nochangeas the interferometer
was rotated.
The Michelson interferometer
shouldve revealed a fringe shift as
it was rotated with respect to the
ether velocity. MM expected 0.4periods of shift (11 m for l1and l2) and
could resolve 0.005 periods.
They saw none!Their apparatus
Nobel Prize in 1907
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In several repeats andrefinements with
assistance fromEdward Morley, he
always saw a null
result.
He concluded that thehypothesis of the
stationary ether must
be incorrect.
Thus, ether seemsnot to exist!
Michelsons Conclusion
Edward Morley
(1838-1923)
Albert Michelson
(1852-1931)
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Nobel Prize in 1907: One of the most famous failed expt.
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The Lorentz-FitzGerald Contraction
Another hypothesis proposedindependently by both H. A. Lorentzand G. F. FitzGerald suggested thatthe length in the direction of the
motion was contracted by a factorof
Therefore
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t =t'=
2
c
l2
1 v2 / c2
l1
1 v2 / c2
George F. FitzGerald
(1851-1901)
Hendrik A. Lorentz
(1853-1928)
This, however, was an ad hoc assumption that could not be
experimentally tested.
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Assignments
Reading: Ch. 2.3, 2.4.
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