A laser beam with wavelength \(633 \mathrm{nm}\) is split into two beams by a beam splitter. One beam goes to Mirror \(1,\) a distance \(L\) from the beam splitter, and returns to the beam splitter, while the other beam goes to Mirror \(2,\) a distance \(L+\Delta x\) from the beam splitter, and returns to the same beam splitter. The beams then recombine and go to a detector together. If \(L=1.00000 \mathrm{~m}\) and \(\Delta x=1.00 \mathrm{~mm},\) which best describes the kind of interference at the detector? (Hint: To doublecheck your answer, you may need to use a formula that was originally intended for combining two beams in a different geometry, but which still is applicable here.) a) purely constructive b) purely destructive c) mostly constructive d) mostly destructive e) neither constructive nor destructive

Step by step solution

01

Understand the Concepts of Constructive and Destructive Interference

Constructive interference occurs when two waves add up to create a larger wave. It occurs when the difference in the path lengths is an integer number of wavelengths. Destructive interference occurs when two waves cancel each other out, resulting in a smaller wave or zero amplitude. It happens when the difference in the path lengths is an odd-integer (odd multiple) of half wavelengths.

02

Calculate the Path Length Difference

Calculate the path length difference ΔL to determine the difference in distance between the two beams. ΔL = 2Δx = 2 × 1.00 mm = 2.00 mm

03

Convert All Distances to the Same Unit and Calculate Number of Wavelengths in ΔL

Convert all distances to the same unit (meters) and calculate the number of wavelengths in the path length difference. lambda = 633 nm = 633 x 10^{-9} m ΔL = 2.00 mm = 2.00 x 10^{-3} m Number of wavelengths in ΔL = ΔL / lambda = (2.00 × 10^{-3}) / (633 × 10^{-9}) = 3162.39

04

Determine Interference Type

Now, we'll determine the interference type based on the number of wavelengths in the path length difference. Since 3162.39 is very close to an integer value (3162), the interference will be mostly constructive. Therefore, the correct answer is (c) mostly constructive.

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