The index of refraction of a glass rod is 1.48 at T=20C and varies linearly with temperature, with a coefficient of linear expansion of the glass of 2.5*10^-5/C. The coefficient of linear expansion of the glass is 5*10^-5/C. At 20C the length of the rod is 3 cm. A Michelson interferometer has this glass rod in one arm, and the rod is being heated so that its temperature increases at a rate of 5C/min. The light source has wavelength 589 nm, and the rod is initially at T=20C. How many fringes cross the field of view each minute?

To determine how many fringes cross the field of view each minute in a Michelson interferometer with a glass rod that is being heated, we need to analyze how the changing temperature affects both the length of the rod and the index of refraction. Let's break this down step by step.

Understanding the Problem

We have a glass rod with an initial length of 3 cm at a temperature of 20°C. The index of refraction of the glass is 1.48 at this temperature, and it changes linearly with temperature. The rod is heated at a rate of 5°C per minute. The wavelength of the light source is 589 nm.

Calculating the Change in Length

The coefficient of linear expansion for the glass is given as 5 × 10^-5/°C. The change in length (ΔL) of the rod due to temperature change can be calculated using the formula:

• ΔL = L₀ × α × ΔT

Where:

• L₀ = initial length of the rod = 3 cm = 0.03 m
• α = coefficient of linear expansion = 5 × 10^-5/°C
• ΔT = change in temperature = 5°C/min

Substituting the values:

• ΔL = 0.03 m × (5 × 10^-5/°C) × 5°C = 0.03 m × 2.5 × 10^-4 = 7.5 × 10^-6 m

Calculating the Change in Index of Refraction

The index of refraction varies linearly with temperature. We need to find the change in the index of refraction (Δn) as the temperature increases. The change in index can be approximated as:

• Δn = (dn/dT) × ΔT

Where:

• dn/dT = change in index of refraction per degree Celsius (not provided, but we can assume a small value for glass, typically around -1.0 × 10^-5/°C)
• ΔT = 5°C

Assuming dn/dT = -1.0 × 10^-5/°C:

• Δn = (-1.0 × 10^-5/°C) × 5°C = -5.0 × 10^-5

Calculating the Change in Optical Path Length

The optical path length (OPL) in the interferometer is affected by both the length of the rod and the index of refraction. The change in optical path length (ΔOPL) can be expressed as:

• ΔOPL = n × ΔL + L × Δn

Substituting the values:

• ΔOPL = (1.48) × (7.5 × 10^-6) + (0.03) × (-5.0 × 10^-5)
• ΔOPL = 1.11 × 10^-6 m - 1.5 × 10^-6 m = -3.9 × 10^-7 m

Calculating the Number of Fringes

The number of fringes (N) that cross the field of view can be calculated using the formula:

• N = (ΔOPL / λ)

Where λ is the wavelength of the light source (589 nm = 589 × 10^-9 m):

• N = (-3.9 × 10^-7 m) / (589 × 10^-9 m) ≈ -663.5

Since we are interested in the absolute value of the number of fringes, we take the positive value. However, we need to consider that this is the change per minute due to the heating rate of 5°C/min.

Final Calculation of Fringes per Minute

Since the optical path length change is negative, it indicates a shift in the fringe pattern. The number of fringes crossing the field of view each minute is approximately:

• Number of fringes per minute ≈ 663.5 fringes/min

Thus, approximately 664 fringes will cross the field of view each minute as the glass rod is heated. This result illustrates how temperature changes can significantly affect optical devices like interferometers.