Elements of Modern Physics
4
and the magnitude of c–v is (c
2
–v
2
)
½
, so that the time taken for light to travel
from A to C and back, is given by
2
2
2
2 1 2
2
/
l
t ( c v )
=
−
(1.7)
Thus the difference in the two times is
Fig. 1.1 Schematic diagram of the Michelson-Morley experiment.
1
2
1
2
2
2
2
2 1/2
2
2
(
)
l c
l
t t c v
c v
∆ = − =
−
−
−
(1.8)
If the apparatus is turned through 90°, the roles of l 1 and l 2 are interchanged,
and the difference in the times becomes
1
2
1
2
2
2 1 /2
2
2
2
2
(
)
l
lc
t t
c v
c v
′
′
′
∆ = − =
−
−
−
(1.9)
The expected shift in the interference fringe at D, is
1
2
2
2 1/2
2
2
(
)
2(
)
1
1
(1
/ )
1
/
c
l l
v c
v c
′
∆ − ∆
=


+
=
−


−
−


δ
λ
λ
2
1
2
2
(
)
for
 
+
≈ −
>>
 
 
l l
v
v c
c
λ
(1.10)
In the experiment of Michelson and Morley, l 2 + l 2 was 22 m, and
λ = 5.9 × 10
–7
m. The value of v is at least of the order v ≈ 30 km/s corresponding
to the velocity of the earth’s motion around the sun, even if the motion of the
solar system around the galactic centre is ignored. For these values
δ ≈ 0.37
(1.11)