9.Michelson Interferometer

Introduction

The purpose of this experiment is to measure the wavelength λ of microwaves which are a form of electromagnetic radiation.  The experimental value of λ is determined by direct spatial measurement using the Michelson interferometer.  The uncertainty in the wavelength Δλ is determined using the sample standard deviation.
	The microwaves are known to have a frequency f=10.5 Giga Hz (Giga = (10^9)) and using the relationship λf=c where c=3.0×(10^8) m/sec is the speed of light, one can calculate a theoretical value for the wavelength λ for comparison.

The Interferometer

The basic apparatus is indicated in the diagram below:

Microwaves travel from the source S and reach the detector by two routes:
  
ROUTE #1:  The microwaves travel from the source S to O where the microwaves are reflected by the half-silvered mirror reaching mirror A.  At mirror A, the microwaves are reflected back to the half-silvered mirror at O.  This time the microwaves pass right through the mirror at O and eventually the microwaves reach the detector at B.
ROUTE #s2:  The microwaves travel from the source S to O where the microwaves pass through the  half-silvered mirror reaching mirror C.  Mirror C reflects the microwaves back to the half-silvered mirror at O.  This time the microwaves are reflected by the mirror at O and eventually the microwaves reach the detector at B.

The waves from both route #1 and route #2 combine or superpose at the detector B.  When the microwaves reach the detector they interfere either constructively (in phase) or destructively (out of phase).  

There are two extremes:  either complete destructive interference (if the waves from the two pathts are 180° out of phase) or complete constructive interference (if the waves from the two paths are in phase).  Moving the mirror A outward causes the waves at the detector to go in and out of phase.  Suppose moving the mirror A outward a distance W causes the waves at the detector B to go through two successive peaks of contructive interference.   Then 2W=λ where λ is the wavelength of the microwaves since the waves travel and additional (λ/2) to mirror A and an additional (λ/2) back to the original position of mirror A.

Experimental Procedure

1. Move mirror A inward observing the spatial location on the ruler where there is a maximum.  Record the location of about 10 t0 12 successive maxima and call these (Subscript[x, n]) where n=1,...,11. 
2. Calculate the differences (Subscript[W, 1])=(Subscript[x, 2])-(Subscript[x, 1]),  (Subscript[W, 2])=(Subscript[x, 3])-(Subscript[x, 2]), ... ,  (Subscript[W, 10])=(Subscript[x, 10])-(Subscript[x, 9]) and the corrresponding wavelengths by multiplying by two.
3. Calculate the average (Subscript[λ, Ave]) and sample standard deviation σ from your data.  Your best estimate of the wavelength according to this method is

(Subscript[λ, Ave])±σ

4. Calculate the theoretical value of (Subscript[λ, theory]) using

λf = c

The microwaves are known to have a frequency f=10.5 Giga Hz (Giga = (10^9)) and  c=3.0×(10^8) m/sec is the speed of light.   Does  (Subscript[λ, theory]) lie in the range (Subscript[λ, Ave])-σ<(Subscript[λ, theory])<(Subscript[λ, Ave])+σ?   A sketch of an ideal situation is below: