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Measure Light Velocity

Michelson and Morley performed their famous experiment to measure velocity of light using an instrument of incredible sensitivity (devised earlier by Michelson), known as Michelson interferometer. As shown in fig. in this instrument light beam from a source S is incident at 45˚ on a glass plate P whose front face is semi-silvered. The plate splits the beam into two parts. The transmitted ray passes through the plate P to strike the mirror M₁, from where it is reflected back to P. again getting partially reflected at semi-silvered face, it travels through P to the telescope T. A compensating glass plate P’ of same thickness as P is placed on the path of second ray so that both the rays travel same thickness through the glass plates before they finally join on way to telescope. The superposition of two rays produce interference fringes which are seen through telescope.

The two rays travel different optical paths, l₁ = PM₁ and l₂ = PM₂ (to and fro). If a bright fringe is seen on the cross-wire of T, the path difference between two rays must be an integer multiple of λ (the wavelength of light used), i.e. we have

2 (l₁ – l₂) = m λ

where m is some integer.

Now suppose that the entire apparatus is kept in the laboratory in such a way that path PM₁ is parallel to the direction of motion of the earth through ether. That is, the apparatus is moving at speed v in direction PM₁ relative to ether, where v is velocity of earth. The time taken by first light ray from P to M₁ and back, therefore is,

In the above expression, c’ = c – v is the velocity of light relative to M₁when light moves towards M₁, and c’ = c + v when light moves back from M₁.

In order to calculate the time taken by ‘second’ light ray, note that M₂ moves towards the direction PM₁ with velocity v, as the light starting from P travels towards M₂. Hence, if light takes time T to reach M₂, we have

Time for return trip is, therefore,

The two rays travel through different times and consequently interfere with a phase difference such Ø that

where,

Change of phase from Ø to Ø’ must lead to a shift of the interference pattern. One fringe shifts for a phase difference of 2 π. Hence, when phase change of (Ø’ + Ø) occurs, the number of fringes that must shift on the telescope are given by,

Michelson and Morley took l₁ = l₂ = 1 l m and λ = 5.6 × 10^-7 m. If one takes speed v of earth relative to ether nearly the same as its speed around the Sun, v ≃ 30 km/s, then v/c = 10^-4. This would give δ ≃ 0.4 fringe.

The experiment was performed very carefully, day, night and at different times of the year. It was sensitive enough to measure a shift of one hundredth of a fringe. However, no such shift was observed. Similar experiments were repeated later by several other people, but the result was always the same δ = 0, or v = 0, i.e. there is no observable velocity of earth relative to ether.

The null result Michelson-Morley experiments was totally unexpected according to Galilean relativity and Maxwell’s theory of light. Several attempts were made to account for it, but without much success. The most notable was the ad hoc hypothesis of H A Lorentz and GP Fitzgerald contraction. Even this hypothesis could not give null shift if l₁ ≠ l₂, as it was observed in Kennedy-Thorndike experiment in 1932. The problem was finally resolved only when A. Einstein propounded his special theory of relativity.