Problem 33 - College Physics

Question

\(\bullet\) The speed of light with a wavelength of 656 \(\mathrm{nm}\) in heavy flint glass is \(1.82 \times 10^{8} \mathrm{m} / \mathrm{s} .\) What is the index of refraction of the glass at this wavelength?

Step-by-Step Solution

Answer

The index of refraction of the glass is approximately 1.65.

Step 1: Understand the Formula for Index of Refraction

The index of refraction, often denoted by the letter 'n', can be determined by the formula \( n = \frac{c}{v} \) where \( c \) is the speed of light in a vacuum, approximately \( 3 \times 10^8 \text{ m/s} \), and \( v \) is the speed of light in the medium.

Step 2: Identify Given Values

From the problem, we know that the speed of light in heavy flint glass, \( v \), is \( 1.82 \times 10^8 \text{ m/s} \). The speed of light in vacuum, \( c \), is a constant \( 3 \times 10^8 \text{ m/s} \).

Step 3: Plug Values into the Formula

Substitute the known values into the formula for the index of refraction: \[ n = \frac{3 \times 10^8 \text{ m/s}}{1.82 \times 10^8 \text{ m/s}} \].

Step 4: Calculate the Index of Refraction

Perform the division: \( n = \frac{3}{1.82} \). Calculating this gives \( n \approx 1.65 \).

Step 5: Review and Finalize

Ensure the calculation is correct and aligns with the physical meaning; the index of refraction is a unitless number that is a ratio of light speeds.

Key Concepts

Speed of Light

The speed of light is a fundamental constant in physics and is denoted by the letter 'c'. In a vacuum, it travels at approximately \( 3 imes 10^8 \) meters per second. This impressive speed serves as a benchmark for measuring how light slows down when passing through various materials.
Light travels at different speeds depending on the medium it occupies, such as air, water, or glass.
This discrepancy is crucial when calculating the index of refraction, as it provides insight into how much slower light moves through a given material compared to a vacuum.

• Standard Speed: Approximately \( 3 imes 10^8 \) m/s in a vacuum.
• Importance: Acts as a reference point for the index of refraction.
• Variability: Speed decreases as light enters denser materials.

Understanding the speed of light is essential to optical physics, influencing technologies like cameras, glasses, and even fiber optics.

Wavelength

Wavelength is the distance between two consecutive peaks of a wave, usually measured in nanometers (nm) for light waves.
In our exercise, we focus on light with a wavelength of 656 nm.
This parameter significantly influences how light interacts with different materials, impacting color perception and energy.

• Definition: The length between successive wave crests.
• Measurement: Commonly measured in nanometers.
• Impact: Affects the speed of light in different media, as materials may scatter different wavelengths uniquely.

Different wavelengths correspond to different types of electromagnetic waves, such as radio waves, microwaves, and visible light, each playing a role in various technological applications.

Optical Physics

Optical physics is a fascinating field that studies how light behaves and interacts with matter.
It covers topics ranging from the fundamental properties of light to the complex behaviors seen in lenses and optical fibers.
Understanding these principles is vital for applications such as telescopes, microscopes, and lasers.

• Core Focus: Investigates light's nature and manipulation.
• Applications: Ubiquitous in scientific instruments and practical technology.
• Importance: Drives innovation in communication, medical imaging, and more.

Through this branch of physics, we gain insights into how natural phenomena like rainbows occur and how technologies like holography are developed.

Heavy Flint Glass

Heavy flint glass is a type of optical glass known for its high refractive index and significant light dispersion.
It is commonly used in lenses where the ability to bend light dramatically is advantageous, such as in corrective glasses or camera lenses.
The higher index of refraction indicates that heavy flint glass slows down light more than many other materials.

• Properties: High refractive index, notable dispersion.
• Uses: Common in lenses and other optical instruments.
• Relevance: The choice of glass affects how devices like eyeglasses and binoculars operate.

Because of these characteristics, heavy flint glass is essential in optical engineering, helping to correct aberrations and refine the focus of light in sophisticated optical systems.