Question

A Michelson interferometer is set up using white light. The arms are adjusted so that a bright white spot appears on the screen (constructive interference for all wavelengths). A slab of glass \((n=1.46)\) is inserted into one of the arms. To return to the white spot, the mirror in the other arm is moved $6.73 \mathrm{cm} .$ (a) Is the mirror moved in or out? Explain. (b) What is the thickness of the slab of glass?

Short answer

Answer: The mirror is moved out, and the thickness of the glass slab is 35.92 cm.

Step-by-step solution

Determine the path difference for the initial setup

For the initial setup of the interferometer, the two arms are arranged such that constructive interference occurs. This means that the path difference is equal to a multiple of the wavelength.

Calculate the optical path difference created by the glass slab

When a glass slab with a refractive index \(n\) (1.46 for this problem) is inserted in one of the arms, it will change the optical path length in that arm. The optical path difference \(d\) can be calculated using the formula: \(d = (n - 1) * t\) Here \(n\) is the refractive index of the glass slab, \(t\) is the thickness of the slab, and \(d\) is the optical path difference.

Determine the optical path difference due to the movement of the mirror

Since the mirror moves by 6.73 cm to return to the white spot, we can calculate the optical path difference created. The light travels twice this distance (reflection), so the path difference is \(2 * 6.73 cm = 13.46 cm\).

Decide the direction of the mirror movement

To return to the original white spot, the optical path length must be the same as before. If the glass slab introduces extra path length, the mirror should be moved out to compensate for that. If the glass slab reduces the path length, the mirror should be moved in. Since the refractive index of the glass slab is greater than 1, the glass increases the optical path length (slows down the light) compared to free space. Therefore, to keep the path lengths equal, we need to move the mirror out. So, the mirror is moved out.

Calculate the thickness of the slab of glass

Since we know the optical path difference due to the movement of the mirror is 13.46 cm and we have the formula for the optical path difference created by the glass slab, we can calculate the thickness of the glass slab. From Step 2, we have: \(d = (n - 1) * t\) We know that \(d = 13.46 cm\) and \(n = 1.46\). We want to find \(t\). Reorganize the formula to solve for \(t\): \(t = \frac{d}{(n - 1)}\) Insert the known values and solve for \(t\): \(t = \frac{13.46 \thinspace cm}{(1.46 - 1)} = 35.92 cm\) So, the thickness of the glass slab is 35.92 cm.