(Solved): Michelson interferometer. The diagram above shows a Michelson interferometer. A monochromatic (sin ...

Michelson interferometer. The diagram above shows a Michelson interferometer. A monochromatic (single colour) light source from A is directed at a beam splitter (a partially silvered mirror) \( \mathrm{C} \) where half the light gets transmitted and half the light gets reflected. The two waves then travel towards mirror \( M_{1} \) and \( M_{2} \). Half of the wave reflected from \( M_{1} \) is transmitted through the beam splitter, where it recombines with the reflected half of the wave returning from \( M_{2} \). The superposed waves travel on to the photodetector or the eye of the observer shown here. Mirror \( M_{2} \) can be moved backward and forward by turning a precision screw. (a) In terms of variables provided, write down the path-length difference of the two waves. (b) A typical interference pattern observed is as shown in the figure. The point that is of interest to us is the center of the pattern. The alternate bright and dark rings appear because some of the light waves enter the interferometer at slightly different angles in the actual experiment. Using your answer in part (a), write down the condition for constructive interference at the center of the pattern. Do the same for destructive interference (a dark spot in the middle). (c) As the mirror \( M_{2} \) is moved by turning the screw, the central spot of the circular pattern alternate between bright and dark. Derive the expression for the number \( \Delta m \) of maxima appearing as \( M_{2} \) moves through distance \( \Delta L_{2} \).