Problem 32 - University Physics with Modern Physics

Question

What is the speed of light in crown glass, whose index of refraction is \(1.52 ?\)

Step-by-Step Solution

Answer

Answer: The speed of light in crown glass is approximately \(1.97 \times 10^8 m/s\).

Step 1: Remember the speed of light in vacuum

The speed of light in vacuum is \(c = 3 \times 10^8 m/s\).

Step 2: Write down the index of refraction

The index of refraction for crown glass, \(n\), is given as \(1.52\).

Step 3: Apply the formula to find the speed of light in crown glass

Use the formula \(v = \frac{c}{n}\) to find the speed of light in crown glass. Plug in the given values: \(v = \frac{3 \times 10^8 m/s}{1.52}\).

Step 4: Calculate the speed of light in crown glass

Divide \(3 \times 10^8 m/s\) by \(1.52\): \(v \approx 1.97 \times 10^8 m/s\).

Step 5: State the answer

The speed of light in crown glass is approximately \(1.97 \times 10^8 m/s\).

Key Concepts

Index of Refraction

The index of refraction, often denoted by the letter \( n \), is a measure that describes how light propagates through a material. It is a dimensionless number that indicates how much slower the speed of light is in the material compared to a vacuum. In essence, it tells us how much the light bends when it enters the material from another medium, like air or vacuum.

The index of refraction is calculated using the formula:

• \( n = \frac{c}{v} \)

where \( c \) is the speed of light in a vacuum (approximately \( 3 \times 10^8 \) m/s) and \( v \) is the speed of light in the material. When light passes from one medium to another, such as from air into crown glass, the change in speed results in light bending or refracting. This property makes the index of refraction extremely important in understanding optical applications and phenomena, such as lenses, prisms, and fiber optics.

Crown Glass

Crown glass is a type of optical glass with relatively low dispersion and a moderate refractive index. It has been used historically for lenses and optical instruments due to its clarity and the way it bends light. The specific value of the index of refraction for crown glass is around 1.52, which means that light travels 1.52 times slower in crown glass than in a vacuum.

This property of crown glass makes it a common material choice in optical engineering because it provides a good balance of light refraction and clarity, minimizing distortion while focusing images. Crown glass also plays a significant role in the manufacture of eyeglasses, microscopes, and telescopes, allowing these devices to focus light efficiently and effectively onto a precise point.

Physics Formula

The formula used to calculate the speed of light in a material is derived from the relationship between the speed of light in a vacuum and the index of refraction of the material. The expression is given by:

• \( v = \frac{c}{n} \)

where \( v \) is the speed of light in the material, \( c \) is the speed of light in a vacuum, and \( n \) is the index of refraction of the material. This formula helps us understand how materials affect the velocity of light passing through them.

In the context of our problem, this formula was used to compute the speed of light in crown glass. By substituting \( c = 3 \times 10^8 \) m/s and \( n = 1.52 \), we get the speed of light in crown glass as \( v \approx 1.97 \times 10^8 \) m/s. This result affirms the principle that light slows down as it travels through different materials depending on their optical density.

Light in Materials

The behavior of light as it travels through different materials is a key aspect of optics and physics. As light enters a material, its speed decreases, depending on the material's index of refraction. Since the speed reduces, the wavelength of light also becomes shorter, while the frequency remains constant.

Understanding how light interacts with materials is crucial for designing various optical devices. When designers know the refractive properties of materials like crown glass, they can predict how light will bend, focus, and disperse, thus enabling the creation of efficient lenses and optical instruments.

• Applications involve lens crafting for eyeglasses, cameras, and microscopes.
• The principles are applied in fiber optic cables, where light is used to transmit information over long distances.

With these insights, we can craft practical applications and technologies that enhance our abilities to capture, use, and manipulate light to suit various needs.