Michelson-Morley Experiment and its Outcome | MECHANICS | University Notes | Physics Notes | B.Sc. Physics Notes by Study Buddy Notes

Michelson-Morley Experiment and its Outcome

Michelson-Morley Experiment and its Outcome | MECHANICS | University Notes | Physics Notes | B.Sc. Physics Notes by Study Buddy Notes

The Michelson-Morley experiment was conducted in 1887 by Albert A. Michelson and Edward W. Morley to detect the existence of the "aether," a hypothetical medium through which light was believed to propagate. This experiment is famous for yielding a null result, providing crucial evidence against the aether theory and paving the way for Einstein's theory of relativity.

1. Background and Purpose

In the late 19th century, light was understood to be a wave, requiring a medium (like sound in air or waves in water) for its propagation. This hypothetical medium was called the "luminiferous aether". The Earth’s motion through this aether was expected to affect the speed of light, similar to how the speed of sound is affected by wind. The Michelson-Morley experiment aimed to measure the Earth's velocity relative to this aether.

2. Experimental Setup

Michelson-Morley Experiment and its Outcome | MECHANICS | University Notes | Physics Notes | B.Sc. Physics Notes by Study Buddy Notes

Michelson and Morley used an interferometer, a device that splits a light beam into two perpendicular paths and then recombines them to create an interference pattern.

1. Light Source: A monochromatic light source emits a beam.
2. Beam Splitter: This beam hits a half-silvered mirror (beam splitter) which splits it into two perpendicular paths:
   • One path runs parallel to the Earth's supposed motion through the aether.
   • The other path runs perpendicular to this motion.
3. Mirrors and Recombination: After traveling equal distances, the two beams are reflected back by mirrors and recombine at the detector.

3. Expected Result Based on Aether Theory

According to aether theory, the speed of light would vary depending on the direction of the Earth’s motion through the aether:

• Parallel Path: For the beam moving parallel to the aether "wind," the speed of light would be affected by the Earth’s velocity through the aether.
• Perpendicular Path: For the beam moving perpendicular to this motion, the light would be unaffected by the aether wind.

Let:

• $c$ = speed of light in a vacuum.
• $v$ = speed of the Earth through the aether.
• $L$ = length of each arm of the interferometer.

Time Calculation Along the Parallel Path

For the path parallel to the motion, light experiences two segments:

1. Outward Trip: Time taken = $\frac{L}{c + v}$
2. Return Trip: Time taken = $\frac{L}{c - v}$

The total time for the parallel trip is:

$$T_{\parallel }=\frac{L}{c+v}+\frac{L}{c-v}=\frac{L(c-v)+L(c+v)}{(c+v)(c-v)}=\frac{2Lc}{c^2-v^2}=\frac{2L}{c}\cdot \frac{1}{1-\frac{v^2}{c^{\mathrm{2}}}}$$

Using a binomial approximation for small $v/c$, this becomes:

$$T_{\parallel }\approx \frac{2L}{c}(1+\frac{v^2}{c^2})$$

Time Calculation Along the Perpendicular Path

For the perpendicular path, the effective speed of light relative to the motion can be derived using Pythagoras' theorem. The effective speed of light is $\sqrt{c^2 - v^2}$, so the time for a round trip is:

$$T_{\perp }=\frac{2L}{\sqrt{c^2-v^2}}=\frac{2L}{c}\cdot \frac{1}{\sqrt{1-\frac{v^2}{c^{\mathrm{2}}}}}$$

Using a similar binomial approximation, we get:

$$T_{\perp }\approx \frac{2L}{c}(1+\frac{v^2}{2c^2})$$

Expected Phase Shift

The difference in times between the two paths would lead to a phase shift in the interference pattern. The expected time difference $\Delta T$ between the parallel and perpendicular paths is:

$$\mathrm{\Delta }T=T_{\parallel }-T_{\perp }=\frac{2L}{c}\left(\frac{v^2}{c^2}\right)$$

This time difference should cause a measurable shift in the interference fringes when the apparatus is rotated.

4. Observations and Null Result

When Michelson and Morley performed the experiment, they observed no shift in the interference pattern. This null result meant that there was no detectable difference in the speed of light due to the Earth’s motion through the supposed aether.

5. Interpretation of the Null Result

The null result of the Michelson-Morley experiment had profound implications:

1. No Aether: It suggested that the aether did not exist, as there was no medium affecting the speed of light.
2. Constancy of Light Speed: The speed of light is constant in all directions, regardless of the motion of the source or observer.
3. Special Relativity: This result laid the foundation for Einstein’s Theory of Special Relativity (1905), which posits that the laws of physics, including the speed of light, are the same in all inertial frames.

6. Mathematical Implications in Special Relativity

Einstein’s interpretation removed the need for an aether and led to the famous equation for time dilation and length contraction. In particular, the idea that the speed of light is constant for all observers led to the Lorentz transformations, which describe how time and space coordinates change between inertial frames:

$$t' = \gamma \left( t - \frac{vx}{c^2} \right), \quad x' = \gamma (x - vt)$$

where $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$

Summary

The Michelson-Morley experiment is considered one of the most significant negative results in physics. It disproved the concept of the aether and established the constancy of the speed of light, a cornerstone of modern physics and the theory of relativity.