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Magazine Chapter 3: Problem 65
Crystal quartz has refractive indexes of 1.557 and 1.547 at wavelengths of \(410.0 \mathrm{nm}\) and \(550.0 \mathrm{nm}\), respectively. Using only the first two terms in Cauchy's Equation, calculate \(C_{1}\) and \(C_{2}\) and determine the index of refraction of quartz at \(610.0 \mathrm{nm}\)
Short Answer
Expert verified
Refractive index at 610 nm: approximately 1.544.
Step by step solution
01
Understand Cauchy's Equation
Cauchy's Equation is given by \( n = C_{1} + \frac{C_{2}}{\lambda^2} \), where \( n \) is the refractive index at wavelength \( \lambda \). We need to find \( C_{1} \) and \( C_{2} \) using given refractive indexes at two different wavelengths.
02
Setup Equations for Unknowns \( C_{1} \) and \( C_{2} \)
For wavelength \( \lambda_1 = 410.0 \text{ nm} \, ( \text{or } 410.0 \times 10^{-9} \text{ m}) \) with \( n_1 = 1.557 \):\[ 1.557 = C_1 + \frac{C_2}{(410.0 \times 10^{-9})^2} \]For wavelength \( \lambda_2 = 550.0 \text{ nm} \, (\text{or } 550.0 \times 10^{-9} \text{ m}) \) with \( n_2 = 1.547 \):\[ 1.547 = C_1 + \frac{C_2}{(550.0 \times 10^{-9})^2} \]
03
Solve the System of Equations
To find \( C_1 \) and \( C_2 \), subtract the second equation from the first:\[ 1.557 - 1.547 = \frac{C_2}{(410.0 \times 10^{-9})^2} - \frac{C_2}{(550.0 \times 10^{-9})^2} \]Calculate each term to solve for \( C_2 \), and then substitute \( C_2 \) back into one of the original equations to find \( C_1 \).
04
Calculate the Refractive Index at 610.0 nm
Use the found \( C_1 \) and \( C_2 \) values in Cauchy's equation to determine \( n \) at \( \lambda = 610.0 \text{ nm} \, (610.0 \times 10^{-9} \text{ m}) \):\[ n = C_1 + \frac{C_2}{(610.0 \times 10^{-9})^2} \]Calculate this to find the refractive index at this wavelength.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Refractive Index
The refractive index is a measure of how much light bends, or refracts, when entering a different medium. In optics, it is a crucial parameter demonstrating how light interacts with materials like crystal quartz. The refractive index, denoted by \( n \), is defined by the ratio of the speed of light in a vacuum to its speed in a given medium.
In the context of Cauchy's Equation, the refractive index can vary with the wavelength of light, which is why it's often calculated for different wavelengths in materials. Here, you are asked to determine refractive indices for quartz at specified wavelengths using Cauchy's Equation. Understanding that this quantity does not have a unit (being a ratio), it tells you how much slower light travels in the medium compared to a vacuum.
In the context of Cauchy's Equation, the refractive index can vary with the wavelength of light, which is why it's often calculated for different wavelengths in materials. Here, you are asked to determine refractive indices for quartz at specified wavelengths using Cauchy's Equation. Understanding that this quantity does not have a unit (being a ratio), it tells you how much slower light travels in the medium compared to a vacuum.
- Low refractive indices (closer to 1) indicate minimal slowing of light.
- Higher refractive indices mean more bending and slower light speed.
Wavelength
Wavelength is the distance between successive peaks of a wave and is typically measured in nanometers (nm) for optical wavelengths. It plays a central role in determining various optical properties, including the refractive index through formulas like Cauchy's Equation.
For crystal quartz, refractive indices were provided at 410.0 nm and 550.0 nm, being characteristic of how light interacts with this material at these chosen wavelengths.
Since the refractive index depends on wavelength, knowing how to find its value at various wavelengths is important for applications in optics, such as lens design and understanding optical fibers.
For crystal quartz, refractive indices were provided at 410.0 nm and 550.0 nm, being characteristic of how light interacts with this material at these chosen wavelengths.
Since the refractive index depends on wavelength, knowing how to find its value at various wavelengths is important for applications in optics, such as lens design and understanding optical fibers.
Crystal Quartz
Crystal quartz is a mineral often used in optical applications due to its transparency and consistency in refractive index measurement across a range of wavelengths.
In the context of this exercise, quartz's refractive indices are known at certain wavelengths (410.0 nm and 550.0 nm), allowing for the calculation of parameters in Cauchy's Equation. This material's consistent optical properties make it a common example in optical calculations to demonstrate theoretical concepts, such as how light behaves when passing through different materials.
In the context of this exercise, quartz's refractive indices are known at certain wavelengths (410.0 nm and 550.0 nm), allowing for the calculation of parameters in Cauchy's Equation. This material's consistent optical properties make it a common example in optical calculations to demonstrate theoretical concepts, such as how light behaves when passing through different materials.
- Highly stable and durable with clear optical properties.
- Great for lessons on light transmission and bending in physics.
Optics
Optics is the branch of physics that studies light and its interactions with different materials. This scientific field involves understanding phenomena such as reflection, refraction, and diffraction. In optics, materials like crystal quartz are analyzed for how efficiently they guide or focus light.
Within this field, equations like Cauchy's Equation provide powerful tools for predicting how light behaves under varying conditions. Principles of optics allow for innovations and technologies across many applications, from simple lenses to complex optical communication systems.
Within this field, equations like Cauchy's Equation provide powerful tools for predicting how light behaves under varying conditions. Principles of optics allow for innovations and technologies across many applications, from simple lenses to complex optical communication systems.
- Analyzes the behavior of light across different materials.
- Plays a crucial role in designing optical instruments.
System of Equations
A system of equations is a set of two or more equations with common variables. In this exercise, we solve a system comprising equations derived from Cauchy's Equation for different wavelengths.
The system helps find unknown parameters, such as \( C_1 \) and \( C_2 \), whose values depend on refractive indices at specified wavelengths. By solving the system, you effectively determine how these constants influence the refractive index for a given wavelength of light.
The system helps find unknown parameters, such as \( C_1 \) and \( C_2 \), whose values depend on refractive indices at specified wavelengths. By solving the system, you effectively determine how these constants influence the refractive index for a given wavelength of light.
- Used to solve for multiple unknowns simultaneously.
- Essential in determining how environmental variables change light behavior.
