Fringe-Shift in Michelson Interferometer with a Moving Source

In summary, the Michelson interferometer can be used to study properties of light emitted by distant sources. By analyzing the fringe shift, the wavelength of light emitted by a moving source can be determined in its own rest frame. In this case, the interferometer is used to measure the wavelength of light emitted by source ##S_2##, which is moving at a speed of ##1.5\times {10}^7~\rm {ms^{-1}}## towards the interferometer, and the result is found to be approximately 628.83 nanometers.
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VSayantan
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Homework Statement


The Michelson interferometer in the figure below can be used to study properties of light emitted by distant sources
interferometer.jpg

A source ##S_1##, when at rest, is known to emit light at wavelength ##632.8~ \rm nm##. In this case, if the movable mirror is translated through a distance ##d##, it is seen that ##99,565## interference fringes pass across the photo-detector. For another source ##S_2##, moving at an uniform speed ##1.5\times {10}^7~ \rm {ms^{-1}}## towards the interferometer along the straight line joining it to the beam splitter, one sees ##100,068## interference fringes pass across the photo-detector for the same displacement ##d## of the movable mirror. What is the wavelength of light emitted by the source ##S_2##, in its own rest frame?

Homework Equations


Fringe shift $$n=\frac {2 d v^2}{\lambda c^2}$$

The Attempt at a Solution


When the whole set up is at rest (in the laboratory frame) there should be no fringe shift. But there will be a shift if the apparatus moves, as is the case in the frame of ##S_2##.

For the first source ##S_1## the total path difference is $$\Delta = 2\cdot d \cdot \cos \theta~ + ~\frac {\lambda}{2}$$
So, for one fringe to appear or disappear $$d= \frac {n \lambda}{2}$$

When the source ##S_2## moves towards the interferometer, along the specified direction, in its own reference frame the interferometer moves away from the source ##S_2##.
The distance ##d## through which the movable mirror moves also moves away from the source, with speed ##v = 1.5\times {10}^7~\rm {ms^{-1}}## - along the direction of motion.
Therefore this distance ##d## is contracted by a factor of ##\sqrt {1-{(\frac {v}{c})}^2}##.
So, for the second source $$d\cdot \sqrt {1-{(\frac {v}{c})}^2}=\frac {n' {\lambda}'}{2}$$

Eliminating ##d## from the two expressions, one obtains
$$n\lambda=\frac {n' {\lambda}'}{\sqrt {1-{(\frac {v}{c})}^2}}$$
Simplifying
$${\lambda}'=\frac {n {\lambda}{\sqrt {1-{(\frac {v}{c})}^2}}}{n'}$$
Substituting values,
$${\lambda}'= \frac {99,565 \times 632.8 ~\rm {nm} \times {\sqrt {1-{(\frac {1.5\times {10}^7~\rm {ms^{-1}}}{3\times {10}^8~\rm {ms^{-1}} } )}^2}}}{100,068}$$
Which gives a value $${\lambda}'\approx 628.83~\rm {nm}$$.

Is this right?
 

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  • #2


Yes, your solution is correct. You have correctly used the formula for fringe shift and accounted for the Lorentz contraction of distance in the frame of the moving source. Your final answer for the wavelength of light emitted by source ##S_2## in its own rest frame is also consistent with the given data. Well done!
 

Related to Fringe-Shift in Michelson Interferometer with a Moving Source

What is a fringe shift in a Michelson interferometer with a moving source?

A fringe shift is a change in the interference pattern produced by a Michelson interferometer when the source of light is in motion. This shift occurs because the path length of the light waves from the moving source to the mirrors and back is different for each mirror, causing a change in the interference pattern.

What causes a fringe shift in a Michelson interferometer with a moving source?

The fringe shift is caused by the relative motion between the source of light and the interferometer. When the source is in motion, the path length of the light waves from the source to the mirrors and back changes, resulting in a shift in the interference pattern.

How is the fringe shift in a Michelson interferometer with a moving source calculated?

The fringe shift can be calculated using the formula: Δx = 2vTλ/c, where Δx is the fringe shift, v is the velocity of the moving source, T is the time taken for the light to travel from the source to the mirrors and back, λ is the wavelength of the light, and c is the speed of light.

What is the significance of the fringe shift in a Michelson interferometer with a moving source?

The fringe shift can be used to measure the velocity of the moving source. By measuring the amount of shift and using the formula mentioned above, the velocity of the source can be calculated. This technique is used in various fields such as astronomy, physics, and engineering.

What are some applications of the fringe shift in a Michelson interferometer with a moving source?

The fringe shift is used in various applications such as measuring the velocity of stars and other celestial objects, determining the speed of moving objects in physics experiments, and in the development of precision instruments such as laser interferometers used in gravitational wave detection.

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