Sagnac effect- the maths

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In summary, the light can make four loops around the interferometer before its intensity is reduced by 99%.
  • #1
MathematicalPhysicist
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in this webpage: http://www.mathpages.com/rr/s2-07/2-07.htm we have an explanation to the saganc effect and in page 2 and 3 i got stumbled:
i can't get to the quadratic equation in page 3 i.e:
[c^2-R^2*w^2*cos(theta)]*T^2+-[2R^2*w*sin(theta)]*T-2R^2[1-cos(thata)]=0
from this:
c^2*T^2=2R^2*[1-cos(w*T)*cos(theta)+-sin(wT)sin(theta)]
in page 2.

i tried to use limits because in the paper the said that w*T is extremely small, but what i got is
c^2*T^2=2R^2*[1-cos(theta)]
(i took limits to both the sin and cos functions with w*T as the angle), but as you see it doesn't match, then what have idone wrong?
 
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  • #2
no one knows?
 
  • #3
for those too much lazy or don't understnad the equations i wrote here they are in the attachment.
 
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  • #4
sorry but the other picture is too much big (anyway you can see it in the link i first gave :eek: i know :yuck: ).
 
  • #5
loop quantum gravity said:
in this webpage: http://www.mathpages.com/rr/s2-07/2-07.htm we have an explanation to the saganc effect and in page 2 and 3 i got stumbled:
i can't get to the quadratic equation in page 3 i.e:
[c^2-R^2*w^2*cos(theta)]*T^2+-[2R^2*w*sin(theta)]*T-2R^2[1-cos(thata)]=0
from this:
c^2*T^2=2R^2*[1-cos(w*T)*cos(theta)+-sin(wT)sin(theta)]
in page 2.

i tried to use limits because in the paper the said that w*T is extremely small, but what i got is
c^2*T^2=2R^2*[1-cos(theta)]
(i took limits to both the sin and cos functions with w*T as the angle), but as you see it doesn't match, then what have idone wrong?
Its just math. Perhaps no one cares, but why assume no one knows? That is "your" speculation. Try again, but this time as it said, keep powers of omega*T up to second order, NOT JUST FIRST ORDER.
 
  • #6
w is the angular velocity of the device and T is the time required for the light pulse to travel from one mirror to the other in the foward and reverse direction. w*T therefore is a measure of distance, not angle. You need to plug in the radius of the ring to derive the angle of rotation, which is microscopic.
 
  • #7
DW said:
Its just math. Perhaps no one cares, but why assume no one knows? That is "your" speculation. Try again, but this time as it said, keep powers of omega*T up to second order, NOT JUST FIRST ORDER.

i hoped that with asking: "no one knows?" some users will be incentived to answer, anyway i don't understand neither your answer nor chronos answer.
when you mean omega*T up to second order do you mean that the deravartive of the cos and sin of w*t will be of 2degree?

and to chronos when you multiply omega with T you get the unit of rad which can be translated to degrees with which you measure the length of the arc.
anyway chronos, when you mean the radius of the ring are you reffering to capital R in this paper?
 
  • #8
loop quantum gravity said:
when you mean omega*T up to second order do you mean that the deravartive of the cos and sin of w*t will be of 2degree?

No, I mean that you have to carry out the taylor series expansion of the cos function to second order in omega*t, not just to first. The sin function's second order term is zero so even though you are really to carry them both out to second order be sure to pick up the second order term from the cos expansion.
In other words use:
[tex]sin(\omega T) \approx \omega T[/tex]
and
[tex]cos(\omega T) \approx 1 - \frac{\omega ^{2}T^{2}}{2}[/tex]
 
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  • #9
DW said:
No, I mean that you have to carry out the taylor series expansion of the cos function to second order in omega*t, not just to first. The sign function's second order term is zero so even though you are really to carry them both out to second order be sure to pick up the second order term from the cos expansion.
In other words use:
[tex]sin(\omega T) \approx \omega T[/tex]
and
[tex]cos(\omega T) \approx 1 - \frac{\omega ^{2}T^{2}}{2}[/tex]
now that's more clear, thanks.
 
  • #10
can anyone answer this question
In Sagnac interferometer replace the beam splitter by a fourth mirror and calculate how many loops around the interferometer can the light make before its intensity is reduced by 99% ?
 

What is the Sagnac effect?

The Sagnac effect, also known as the Sagnac interferometer, is a phenomenon in which the rotation of a closed loop interferometer causes a phase shift in the light waves travelling through it.

How is the Sagnac effect measured?

The Sagnac effect can be measured by using an interferometer, which splits a beam of light into two paths that are then recombined to produce an interference pattern. The rotation of the interferometer causes a change in the interference pattern, which can be measured and used to calculate the rotation rate.

What is the mathematical equation for the Sagnac effect?

The mathematical equation for the Sagnac effect is ΔΦ = (4πAω)/c², where ΔΦ is the phase shift, A is the area enclosed by the interferometer, ω is the angular velocity, and c is the speed of light.

What is the significance of the Sagnac effect?

The Sagnac effect is significant because it is the basis for many modern navigation systems, such as gyroscopes and ring laser gyroscopes, which use the rotation of the Earth to determine direction and location.

How does the Sagnac effect relate to Einstein's theory of relativity?

The Sagnac effect is a result of the principles of special relativity, which state that the speed of light is constant in all inertial frames of reference. It also demonstrates the concept of frame dragging, in which the rotation of a massive object affects the space and time around it.

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