Question

At the Long-baseline Interferometer Gravitational-wave Observatory (LIGO) facilities in Hanford, Washington, and Livingston, Louisiana, laser beams of wavelength \(550.0 \mathrm{nm}\) travel along perpendicular paths \(4.000 \mathrm{~km}\) long. Each beam is reflected along its path and back 100 times before the beams are combined and compared. If a gravitational wave increases the length of one path and decreases the other, each by 1.000 part in \(10^{21}\), what is the resulting phase difference between the two beams?

Short answer

Question: Determine the phase difference between two laser beams after they are reflected 100 times and travel along paths affected by a gravitational wave, considering the path length is 4km, the gravitational wave causes a change of 1.000 part in \(10^{21}\) in both path lengths, and the wavelength of the laser beams is given. Answer: To find the phase difference between the two beams, you should follow these steps: 1. Calculate the total distance traveled by each laser beam without the gravitational wave, considering they are reflected 100 times along a 4km path. 2. Find the change in path length due to the gravitational wave. 3. Calculate the total distance traveled for the changed paths by adding and subtracting the change in path length for paths 1 and 2, respectively, and multiplying it by the number of reflections. 4. Determine the number of wavelengths in the new path lengths by dividing the total distance traveled by the wavelength of the laser beams. 5. Calculate the phase difference by finding the difference in the number of wavelengths in each new path length and multiplying this value by \(2\pi\).

Step-by-step solution

Find the total distance traveled by each laser beam

As the beams are reflected along their path and back 100 times, we need to find the total distance traveled by each beam. Since the path is 4km long and they travel back and forth, the total distance traveled for each reflection would be 2 times the path length. Therefore, the total distance traveled will be the product of the number of reflections and the distance traveled per reflection.

Calculate change in path length due to the gravitational wave

The problem states that the gravitational wave causes a change of 1.000 part in \(10^{21}\) in both path lengths. We can find the absolute change in the path length by multiplying this factor with their respective path lengths.

Calculate the total distance traveled for the changed paths

Now that we have the change in path length for both paths, we can find the total distance traveled for each beam after the gravitational wave has affected them. We will add (or subtract) the change in path length to the original path length and multiply it by the number of reflections to obtain the updated total distance.

Calculate the number of wavelengths in the new path lengths

We can find the number of wavelengths (how many times a wave can "fit" into a certain distance) in each new path length by dividing the total distance traveled by the wavelength of the laser beams.

Find the phase difference between the two beams

The phase difference can be found by finding the difference between the number of wavelengths in each new path length. Once we have this difference, we can calculate the phase difference in radians by multiplying this value by \(2\pi\). Now, let's plug in the values to calculate the phase difference between the two beams: Step 1: Total distance traveled (without gravitational wave) = 2 × 4.0 km × 100 Step 2: Change in path length = 1.000 × \(10^{-21}\) × 4.0 km Step 3: Total distance traveled (with gravitational wave) for each path: - Path 1: (4.0 km + change in path length) × 100 - Path 2: (4.0 km - change in path length) × 100 Step 4: Number of wavelengths for each path: - Path 1: Total distance (path 1) / wavelength - Path 2: Total distance (path 2) / wavelength Step 5: Phase difference = (Number of wavelengths (path 1) - Number of wavelengths (path 2)) × \(2\pi\) After performing these calculations, you will get the phase difference between the two beams as a result.