Second Lecture
Michelson-Morley Experiment
Michelson, who became the first American to win a Nobel prize
in physics in 1907 , invented a new instruments of
unprecedented sensitivity to determine the velocity of the
earth through the ether , i.e., a medium for it to move in , then
light should be dragged along by this ether as the earth moves
along through space .In principle , the ether drift test consists
simply of observing the entire instrument is turned through an
angle 90.
The experimental arrangement is shown in fig (2) . A
monochromatic beam of light falls on a semi-silvered glass
plate P placed at 450 to the beam and is partly reflected and
partly transmitted . The reflected portion travels in a direction
at right angles to that of the incident beam, falls normally at A
on the plane mirror M1 , and is reflected back to P . The
transmitted ray , travelling along the direction of the incident
ray, falls normally at B on the plane mirror M2 and is reflected
back to P . The two beams , thus turned to P , interfere on their
final journey towards the telescope T so that an interference
pattern can be observed and studied .The beam coming from S
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and entering into T after being reflected from M1 traverses the
plate P thrice while that reflected from M2 traverses the plate P
only once. Thus the optical paths of two interfering beams are
not equal. They are made equal by introducing a compensating
plate not shown in fig 2 of the same thickness and material as P
between P and M2 . whole of the apparatus is floated on
mercury, contained in a large vessel so that the interferometer
may be rotated in a desired direction.
If the whole apparatus were at rest in ether the two rays
would take the same time to return to P (distance PA and PB
are kept equal ). But in the actual experiment the whole
apparatus is moving with the earth . Let us assume that the
direction of motion of the earth coincides with the direction of
the incident beam of light ( in the short time of an experiment,
this velocity can be considered as uniform and linear , Since the
earth makes one complete rotation round the sun in one year ).
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On account of motion at the apparatus along with the earth
direction PM2 with velocity v relative to the ether the paths of
the two rays and the positions of their reflections from the
mirrors will be as shown in fig 3 .
After reflection at P the ray proceeds towards the mirror M 1. In
the time the ray reaches the mirror, it moves to the position
M`1 so that the reflection occurs at this position of mirror M1 . If
t1 is time taken by the path PM`1P`1 , in the meantime plate P
will move distance PP`=vt1 . From fig 3 we have
2
 vt 
PM 1  PN  M 1N   1   L2
 2 


2
Dis tan ce
2
PM 1P1  2 PM 1  2 L2 
v 2 t12
4
This distance PM`1P`1 has been traversed by the beam in the
ether or absolute or inertial frame , in which the velocity of
light is c in all directions . Hence
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ct1  2 L2 
vt12
4
or
c 2t12  4L2  v 2t12
or
t1 
2L
c2  v2
2L  v 2 
1  

c  c 2 
2L

c 1
1 2

v2
c2
2L 
v2 
1  2  v  c 
c  2c 
(1)
The beam transmitted through P and moving longitudinally
towards M2 has velocity (c+v) relative to the mirror M2 because
the mirror is moving with the velocity of the earth (v). This
beam after reflection at mirror M2 will move with a velocity (cv) relative to the plate P, because now the plate is moving
opposite to the beam . If t2 be the total time taken by this ray
to return to the plate P , then
t2 
L
L
2L

 2
c  v c  v c  v2
1
2L  v 2 
2L  v 2 
1  2  
1   v  c 

c  c 
c  c 2 
(2)
Thus the difference between the times of travel of the
longitudinal and transverse beam is
2 L  v 2  
v 2 

t2  t1 
 1     1 
c  c 2   2c 2 
(3)
L v 2 Lv 2
t   2  3
c c
c
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This will introduce an optical path difference Δ between the
two interfering waves , on account of velocity v of the
laboratory frame , given by
  ct2  t1   ct  L
v2
c2
(4)
The interference between pattern produced by the two
components when they recombine will thus shift from the
position it would have occupied if v were equal to zero . this is
the principle of Michelson Morley experiment . If Δ
corresponds to the shifting on n fringes , then
  n
(5)
Where λ is the wavelength of the light used . Equation (4), (5)
for Δ , one obtains
n
ct


Lv 2
c 2
(6)
The path difference A is made large by making the total path
traversed by the component beams long by multiple
reflections. This way of elongating the path is more convenient
from the point of view of temperature control and mechanical
rigidity than a direct increase in the length of the arms of the
interferometer . To minimize the error due to vibrations , the
whole apparatus is mounted on a block of stone resting on a
special wooden ring which floats on mercury.
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Now , if the whole Apparatus is turned through 900 so
that the other arm PM1 becomes coincident with v . this causes
the difference of path in the opposite direction and hence the
displacement of fringes should be
2 Lv 2 c 2
corresponding to a
path difference 2Δ . This procedure is adopted in experiments ,
since the different estimations Δ from Eq.(4) involves the
determination of the shift of the fringe system from its position
under the condition v=0 which cannot be realized .In actual
experiment , distance L was taken nearly 11 m, the wavelength
of light used was about
c  3  108 m / s ,
o
6000 A .
Taking
v  3  104 m / s and
one obtains,
2  11  3  104 
2
2 
6  10 7  3  108 
11 1
   0.37 of a
3 10
2
finge  0.4
fringe
Thus a shift of about 0.4 fringe is expected in the fringe
pattern on turning the apparatus by 900. Michelson - Morley
experiment was sensitive enough to detect a fringe shift of the
order of 0.01 fringe , but in spite of taking into consideration all
experimental errors no fringe shift was observed . After
Michelson and Morley the experiment has been repeated with
different wavelength of light , using leather beam , at different
places on the earth , at different altitudes below the surface of
the earth , etc., but the same negative result has been obtained
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every time .The negative results of Michelson-Morley
experiment has following two consequences:
1. The speed of light is the same in all directions.
2. The assumption of the luminiferous ether is not true as
motion relative to it is undetectable .
Michelson tried to explain the negative result by assuming that
the ether is dragged by the moving bodies , so that velocity of
light is same in all directions . This idea was in contradiction to
the original assumption of the ether as an all pervasive
frictionless medium . Aberration experiment also contradicts
this hypothesis.
Lorentz and Fitzgerald hypothesis
Lorentz and Fitzgerald independently put forward another
explanation to preserve the concept of ether frame . According
to this hypothesis , along the direction of motion through the
ether the size of all material bodies contracts (Lorentz and
Fitzgerald contraction ) . Thus if we suppose that the distance in
the direction of the motion of earth is shortened to

, such
that
v2
   1  2
c
(7)
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Thus
 v2 
2 L1  2 
c 
t2  
 v2 
c1  2 
 c 

2L
(8)
 v2 
c 1  2 
 c 
Thus t1 and t2 will have same value or
t  t1  t2  0.
No positive experimental confirmation could be provided for
Lorentz - Fitzgerald hypothesis. The idea of such a contraction
was later found to be correct , although the assumption of the
presence of luminiferous ether was not found correct.
Example (1):
In actual Michelson –Morley experiment, The total distance
from the partially silvered mirror to each of the two mirrors
was 10 meters. The wavelength of the light used was
o
5000 A
. If
the orbital velocity of the earth is taken as 30 km/s , calculate
the expected total fringes shift when the apparatus is rotated
through 900 .
Solution
Expected fringe shift
2
   2 Lv2
c
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L  10m
  
2  10  30  103 
0
2
  5000 A
5  107  3  108 
2
c  3  108 m / s
v  3  103 m / s
   2  0.4 fringe
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Example (2):
What will be expected fringe shift in Michelson Morley
experiment , if the effective length of each part is 6 meters and
wavelength of light is used
o
6000 A ?
Velocity of earth is
3  104 m / s
Solution :
Expected fringe shift
2
   2 Lv2
c
  
2  10  3  10
L  6m

3 2
6  107  3  108 
2
  6000  1010 m
c  3  108 m / s
v  3  104 m / s
   1  0.2
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of a fringe
Example (3):
In Michelson-Morley experiment the length of the paths of the
two beams is 11 meters each. The wavelength of the light used
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is
o
6000 A .
The expected fringe shift is 0.4 fringe , calculate the
velocity of earth relative to ether .
Solution:
Expected fringe shift
2
   2 Lv2
c
L  11m

0.4 
2  11  v
  6000  10 10 m
  0.4 fringe
2
6  10 7  3  108 
2
c  3  108 m / s
v  3  104 m / s
04  6  107  3  108 
2  11
2
v2 
or
 04  6  107  3  108 2 

v


2

11


12
 1.2  105 m / s
Einstein`s special theory of relativity
The failure of such efforts as the Michelson-Morley and Others
to discover a preferred frame for Maxwell`s electromagnetic
equations suggested that the latter must confirm to a principle
of relativity . However , the fact that the Galilean principle of
relativity – which was known to be valid for classical mechanics
failed for Maxwell`s equations was a considerable source of
frustration to physicists at the turn of century . After a critical
examination of concepts of space , time and simultaneity ,
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Einstein discarded the Galilean principle of relativity and
postulated instead a principle of relativity for all physical laws.
His postulates may be stated as follows:
i. The laws of physics have the same form in all
inertial frames of reference moving with a
constant velocity with respect to one another .
Obviously
,
all
physical
phenomena
(mechanical
,electromagnetic, etc.) proceed in exactly the same way , under
the same conditions , in all inertial frames of reference, or, in
other words , it is impossible to ascertain , by means of any
experiments whatsoever , conducted in a closed system of
bodies , whether the system is at rest or is travelling at uniform
velocity in a straight line with respect to some inertial frame of
reference .
ii. The speed of light in vacuum is the same for all
observers who are in uniform rectilinear , relative
motion and is independent of the motion of the
source . Its free space value is the universal
constant given by Maxwell`s equations.
This postulate follows directly from the results of the
Michelson –Morley experiment and many others.
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Of these two postulates , the second represents an
experimental fact, whereas the first is a generalization from a
wide range of physical experience. The first postulate is in no
way self evident ; it is a hypothesis to be tested by experiment .
The above mentioned postulates are applicable to only
inertial frames and due to this specialty the principle of
relativity as applied to inertial frames is called special relativity.
Einstein later generalized special relativity to include noninertial frames also and the generalized theory is called
General Relativity.
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