Question

A Michelson interferometer is illuminated with a 600.-nm light source. How many fringes are observed to shift if one of the mirrors of the interferometer is moved a distance of \(200 . \mu \mathrm{m} ?\)

Short answer

Answer: Approximately 667 fringes would be observed to shift.

Step-by-step solution

Gather Information

In this problem, we are given the following information: - Wavelength of the light source, \(\lambda = 600 \, \text{nm}\) - Distance the mirror is moved, \(d = 200 \, \mu \text{m}\) We need to find the number of fringes shifted.

Convert Units

First, we need to make sure all units are consistent. Let's convert the wavelength and distance to meters: $$ \lambda = 600 \, \text{nm} \times \frac{1 \times 10^{-9} \, \text{m}}{1 \, \text{nm}} = 6 \times 10^{-7} \, \text{m} $$ $$ d = 200 \, \mu \text{m} \times \frac{1 \times 10^{-6} \, \text{m}}{1 \, \mu \text{m}} = 2 \times 10^{-4} \, \text{m} $$

Calculate Path Difference

In a Michelson interferometer, when one mirror is moved, the path difference between the two arms changes. The path difference is equal to twice the distance the mirror is moved (since the light goes back and forth between the mirror). So, the path difference is: $$ \Delta = 2d = 2 \times (2 \times 10^{-4} \text{m}) = 4 \times 10^{-4} \, \text{m} $$

Calculate Fringe Shift

The fringe shift is related to the path difference and the wavelength of the light source. The formula for the fringe shift is given by: $$ n = \frac{\Delta}{\lambda} $$ where \(n\) is the number of fringes shifted, \(\Delta\) is the path difference, and \(\lambda\) is the wavelength of the light source. We can now plug in the values for the path difference and the wavelength to calculate the number of fringes shifted: $$ n = \frac{4 \times 10^{-4} \, \text{m}}{6 \times 10^{-7} \, \text{m}} \approx 666.67 $$ Since the number of fringes shifted must be an integer, we can round it to the nearest whole number.

Round the Result and Report the Answer

We round the number of fringes shifted to the nearest integer: $$ n \approx 667 \, \text{fringes} $$ So, if one of the mirrors of the Michelson interferometer is moved a distance of \(200 . \mu \mathrm{m}\), approximately 667 fringes would be observed to shift.