Problem 42 - University Physics with Modern Physics

Question

In a physics lab, light with wavelength 490 nm travels in air from a laser to a photocell in 17.0 ns. When a slab of glass 0.840 m thick is placed in the light beam, with the beam incident along the normal to the parallel faces of the slab, it takes the light 21.2 \(\mathrm{ns}\) to travel from the laser to the photocell. What is the wavelength of the light in the glass?

Step-by-Step Solution

Answer

The wavelength of the light in glass is approximately 326.7 nm.

Step 1: Understand the problem

We need to find the wavelength of light in glass. We know the time it takes for light to travel in air and in glass, the thickness of the glass slab, and the wavelength of light in air. We will use the relationship between speed, wavelength, and refractive index to solve the problem.

Step 2: Find the speed of light in air

The speed of light in air is close to the speed in vacuum, \( c = 3.00 \times 10^8 \text{ m/s} \). We use the time it takes for the light to travel from the laser to the photocell in air (17 ns) to calculate the distance traveled without the glass. The distance \( d = c \times t \).

Step 3: Calculate distance light travels without glass

Convert 17 ns to seconds: \( 17 \text{ ns} = 17 \times 10^{-9} \text{ s} \). The distance traveled in air: \[ d = 3.00 \times 10^8 \text{ m/s} \times 17 \times 10^{-9} \text{ s} \approx 5.10 \text{ m} \].

Step 4: Calculate the speed of light in glass

Use the total time with glass (21.2 ns) to find the time it takes to travel through glass. Convert 21.2 ns to seconds: \( 21.2 \times 10^{-9} \text{ s} \). Subtract the time in air to find time in glass: \[ t_{glass} = 21.2 \times 10^{-9} \text{ s} - 17 \times 10^{-9} \text{ s} = 4.2 \times 10^{-9} \text{ s} \]. The speed of light in the glass: \[ v_{glass} = \frac{0.840 \text{ m}}{4.2 \times 10^{-9} \text{ s}} \approx 2.00 \times 10^8 \text{ m/s} \].

Step 5: Calculate refractive index of the glass

The refractive index \( n \) is defined by the ratio of the speed of light in air to the speed of light in the material: \[ n = \frac{c}{v_{glass}} = \frac{3.00 \times 10^8 \text{ m/s}}{2.00 \times 10^8 \text{ m/s}} = 1.50 \].

Step 6: Calculate wavelength of light in glass

The wavelength of light in a medium is given by: \[ \lambda_{glass} = \frac{\lambda_{air}}{n} = \frac{490 \text{ nm}}{1.50} \approx 326.7 \text{ nm} \].

Key Concepts

Speed of Light

The speed of light is one of the most fundamental constants in physics. It is represented by the symbol \( c \) and is approximately equal to \( 3.00 \times 10^8 \text{ m/s} \) in a vacuum. Light travels extremely fast, but not infinitely fast, allowing us to measure and calculate its speed with high precision.
In air, light travels at nearly the same speed as in a vacuum, which is why we often use \( c \) for calculations involving air unless high precision is needed. However, when light enters other mediums, such as glass or water, its speed decreases.
This decrease in speed is caused by the interaction of light with the medium’s molecules. It is important in determining how light behaves when it enters new materials and results in phenomena such as refraction. Knowing the speed of light in different mediums helps us understand how light travels, which is essential in fields like optics.

Wavelength in Mediums

Wavelength refers to the distance between successive peaks of a wave. For light, wavelengths determine their color and range from shorter wavelengths (blue light) to longer ones (red light).
As light enters a medium other than air or vacuum, its wavelength changes. This is due to the slowdown in speed of light as it passes through the medium. Although the frequency of the light wave remains constant, the wavelength becomes shorter compared to its original value in air.
This relationship is expressed mathematically as \( \lambda_{medium} = \frac{\lambda_{air}}{n} \), where \( \lambda_{medium} \) is the wavelength in the medium, \( \lambda_{air} \) is the wavelength in air, and \( n \) is the refractive index of the medium. Understanding how to calculate this change is key to solving many optics-related problems, like in the exercise stated.

Optics

Optics is the branch of physics that studies light and its interactions with different materials. It deals with the behavior of visible, ultraviolet, and infrared light as it interacts with objects.
Key concepts in optics include reflection, refraction, and diffraction. Refraction is particularly important as it describes how light bends when entering a different medium, governed by the refractive index.
The refractive index is a measure indicating how much the speed of light is reduced inside a medium compared to its speed in a vacuum. From the exercise, we can extend our understanding by noting that when light slows down, it also bends, altering its wavelength. This bending of light is critical in the functioning of lenses and optical devices, affecting how images are focused and transmitted.

• The laws of refraction, described by Snell's Law, allow us to predict the direction of light as it crosses into another medium.
• Optical instruments like microscopes and telescopes depend heavily on these principles to operate effectively.

So, optics not only aids in understanding natural phenomena but is crucial in technological advancements.