Question

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

Short answer

Answer: Approximately 1696 fringes shift on the screen.

Step-by-step solution

Understand the formula for fringe shift

In a Michelson interferometer, the number of fringes shifted (n) on the screen when the movable mirror is moved can be found by using the formula: \(n = \frac{2d}{\lambda}\) Where \(n\) is the number of fringes shifted, \(d\) is the distance the movable mirror moved, and \(\lambda\) is the wavelength of the light used.

Convert the distance into meters

In the given problem, the distance the movable mirror moved (d) is 0.381 mm. We need to convert it to meters to make it consistent with the wavelength, which is given in nm. To convert mm to meters, multiply by 10^(-3): \(d = 0.381 \times 10^{-3} \mathrm{m}\)

Convert the wavelength into meters

Next, we need to convert the given wavelength from nm to meters. To do that, multiply by 10^(-9): \(\lambda = 449\times 10^{-9} \mathrm{m}\)

Calculate the number of fringes shifted

Now that we have both the distance and the wavelength in meters, we can use the formula from step 1 to find the number of fringes shifted: \(n = \frac{2d}{\lambda} = \frac{2(0.381 \times 10^{-3}\mathrm{m})}{449 \times 10^{-9}\mathrm{m}}\)

Solve the equation and find the fringes shifted

Now, solve the equation to find the number of fringes shifted: \(n = \frac{2(0.381 \times 10^{-3}\mathrm{m})}{449 \times 10^{-9}\mathrm{m}} \approx 1695.99\) Since the number of fringes shifted must be a whole number, we round it to the nearest whole number: \(n \approx 1696\) Therefore, there are approximately 1696 fringes shifted on the screen when the movable mirror is moved 0.381mm.