Problem 55 - University Physics with Modern Physics

Question

The index of refraction of a glass rod is 1.48 at \(T\) =20.0\(^\circ\)C and varies linearly with temperature, with a coefficient of 2.50 \(\times\) 10\(^{-5}\)/C\(^\circ\). The coefficient of linear expansion of the glass is 5.00 \(\times\) 10\(^{-6}\)/C\(^\circ\). At 20.0\(^\circ\)C the length of the rod is 3.00 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 5.00 C\(^\circ\)/min. The light source has wavelength \(\lambda\) = 589 nm, and the rod initially is at \(T\) = 20.0\(^\circ\)C. How many fringes cross the field of view each minute?

Step-by-Step Solution

Answer

Approximately 8 fringes per minute.

Step 1: Calculate the change in refractive index

The refractive index \( n(T) \) at temperature \( T \) can be calculated using the equation: \( n(T) = n_0 + \alpha_T \Delta T \), where \( n_0 = 1.48 \) is the initial refractive index, \( \alpha_T = 2.50 \times 10^{-5}/\text{C}^\circ \) is the temperature coefficient of the refractive index, and \( \Delta T = 5 \)C\(^\circ\) is the change in temperature per minute. So, we find \( \Delta n = \alpha_T \Delta T = 2.50 \times 10^{-5} \times 5 = 1.25 \times 10^{-4} \).

Step 2: Calculate the change in length of the rod

The change in length \( \Delta L \) of the rod is given by the formula: \( \Delta L = L_0 \alpha_L \Delta T \), where \( L_0 = 3.00 \) cm is the initial length and \( \alpha_L = 5.00 \times 10^{-6}/\text{C}^\circ \) is the coefficient of linear expansion. This gives \( \Delta L = 3.00 \times 5.00 \times 10^{-6} \times 5 = 7.5 \times 10^{-5} \) cm.

Step 3: Calculate optical path length change

The change in optical path length \( \Delta OPL \) is calculated as: \( \Delta OPL = \Delta (nL) = L_0 \Delta n + n_0 \Delta L \). Substituting the known values, \( \Delta OPL = 3.00 \times 1.25 \times 10^{-4} + 1.48 \times 7.5 \times 10^{-5} = 3.75 \times 10^{-4} + 1.11 \times 10^{-4} = 4.86 \times 10^{-4} \) cm.

Step 4: Calculate number of fringes

The number of fringes \( m \) crossing the field of view is given by: \( m = \frac{\Delta OPL}{\lambda} \), where \( \lambda = 589 \) nm = \( 589 \times 10^{-7} \) cm. Thus, \( m = \frac{4.86 \times 10^{-4}}{589 \times 10^{-7}} \approx 8.25 \). Therefore, approximately 8 fringes cross per minute.

Key Concepts

Optical Path Length

When we talk about optical path length (OPL), we refer to the product of the physical length of a medium and its refractive index. This concept is crucial in understanding light behavior in materials, as it considers both the distance light travels and how fast it travels through a material.
In a Michelson interferometer, the OPL difference between the arms can change the interference pattern observed. If the OPL increases or decreases, the number of fringes seen changes. This directly affects the interference pattern, causing the fringes to move.
To calculate the change in optical path length, consider both the change in refractive index and the change in the physical length of the material as temperatures fluctuate. The formula used is \( \Delta \text{OPL} = L_0 \Delta n + n_0 \Delta L \), where \( \Delta n \) is the change in refractive index, \( L_0 \) is the original length, and \( \Delta L \) is the change in length.

Refractive Index

The refractive index of a material defines how much it can bend or refract light as it passes through. The higher the refractive index, the slower light travels through that medium. In our exercise, the refractive index of the glass rod changes with temperature—this is called thermo-optic effect.
It's given that the refractive index of the glass rod at 20°C is 1.48, and changes linearly with a temperature coefficient of \( 2.50 \times 10^{-5}/\text{C}^\circ \). As the temperature rises, the refractive index increases, making the optical path longer, even if the physical length of the material doesn't change much.
Calculating the change in refractive index involves the formula \( n(T) = n_0 + \alpha_T \Delta T \), where \( \alpha_T \) is the temperature coefficient for refractive index. In practice, this results in the refractive properties adapting slightly with temperature, slightly adjusting how many fringes we observe.

Linear Expansion

Linear expansion describes how the length of an object changes with temperature. For any solid material, heating generally causes it to expand. The coefficient of linear expansion tells us how much change in length per unit length per degree of temperature change to expect.
In our scenario, the glass rod expands with a coefficient of \( 5.00 \times 10^{-6}/\text{C}^\circ \). The formula for calculating the change in length is \( \Delta L = L_0 \alpha_L \Delta T \), where \( \alpha_L \) is the linear expansion coefficient. Even a small change in temperature will adjust the physical dimensions of the rod, impacting the optical path length in the interferometer.
By determining this expansion, we can find how the length increment affects the overall optical path, contributing to the movement of fringes across the view.

Fringe Pattern

When light beams are split in a Michelson interferometer, they travel different paths and recombine. This creates an interference pattern called fringes. These fringes are the dark or bright bands seen due to constructive and destructive interference of light.
The shift in fringe pattern results from changes in optical path lengths of the two arms. More specifically, the number of fringes crossing the field of view in a minute is directly related to these changes. Each fringe corresponds to a shift of one wavelength in the optical path.

• The formula \( m = \frac{\Delta \text{OPL}}{\lambda} \) is used to determine the number of fringes, \(m\), where \( \Delta \text{OPL} \) is the change in optical path length and \( \lambda \) is the wavelength of the light source.
• In our exercise, this calculation showed approximately 8 fringes per minute, indicating the sensitivity of the system to small changes in temperature.

These fringes' movement provides critical data for understanding the materials and conditions they reflect, making fringe patterns an essential component in precision measurements.