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Chapter 34: Problem 35

34.35 Monochromatic blue light \((\lambda=449 \mathrm{nm})\) is beamed into a Michelson interferometer. How many fringes move by the screen when the movable mirror is moved a distance \(d=\) \(0.381 \mathrm{~mm} ?\)

Short Answer

Expert verified
Answer: Approximately 1696 fringes move across the screen.

Step by step solution

01

Convert the given wavelength and distance to the same units

We need to make sure that the values for the wavelength and distance are in the same units. The given wavelength is 449 nm, and the distance the mirror is moved is 0.381 mm. Convert the distance to nanometers, as follows: \(d = 0.381\,\text{mm} \times \frac{1000\,\text{nm}}{1\,\text{mm}} = 381000\,\text{nm}\) Now, both wavelength and distance values are in nanometers.
02

Calculate the path length difference between the interferometer's arms

Remember that the path length difference between the two arms of the interferometer is given by \(2d\). So, we can calculate it as follows: \(\Delta L = 2d = 2 \times 381000\,\text{nm} = 762000\,\text{nm}\)
03

Calculate the number of fringes that move across the screen

To find the number of fringes that pass the screen, divide the path length difference by the wavelength of the light. Then, round the result to the nearest whole number to get the number of fringes: \(N = \frac{\Delta L}{\lambda} = \frac{762000\,\text{nm}}{449\,\text{nm}} \approx 1696\) So, when the movable mirror is moved a distance \(d = 0.381\,\text{mm}\), approximately 1696 fringes move across the screen in the Michelson interferometer.

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