Problem 3 - University Physics with Modern Physics

Question

A beam of light has a wavelength of 650 \(\mathrm{nm}\) in vacuum. (a) What is the speed of this light in a liquid whose index of refraction at this wavelength is 1.47\(?\) (b) What is the wavelength of these waves in the liquid?

Step-by-Step Solution

Answer

Speed: \(2.04 \times 10^8\) m/s; Wavelength: 442 nm.

Step 1: Understanding the Problem

We are given a wavelength of 650 nm for light in a vacuum and need to find the speed and wavelength of this light in a liquid with an index of refraction of 1.47.

Step 2: Speed of Light in Vacuum

The speed of light in a vacuum, denoted as \( c \), is constant and approximately equal to \( 3 \times 10^8 \) meters per second.

Step 3: Formula for Speed of Light in a Medium

The speed of light in a medium can be calculated using the formula: \(v = \frac{c}{n}\), where \(n\) is the index of refraction and \(c\) is the speed of light in vacuum.

Step 4: Calculate Speed of Light in Liquid

Using the formula \( v = \frac{c}{n} = \frac{3 \times 10^8}{1.47} \), we find the speed of light in the liquid. Calculate this to find \( v \approx 2.04 \times 10^8 \) m/s.

Step 5: Formula for Wavelength in a Medium

The wavelength of light in a medium is given by \(\lambda_{medium} = \frac{\lambda_{vacuum}}{n}\), where \( \lambda_{vacuum} \) is the wavelength in vacuum and \( n \) is the index of refraction.

Step 6: Calculate Wavelength in Liquid

The original wavelength \( \lambda_{vacuum} \) is 650 nm. Using \(\lambda_{medium} = \frac{650 \text{ nm}}{1.47}\), calculate \(\lambda_{medium} \approx 442 nm\).

Key Concepts

Wavelength in Medium

When light travels through a medium such as a liquid, its wavelength changes. This is because the medium affects how light waves propagate. To find the wavelength of light within a medium, we use the equation:

• \( \lambda_{medium} = \frac{\lambda_{vacuum}}{n} \)

Here, \( \lambda_{medium} \) represents the wavelength of the light in the medium, \( \lambda_{vacuum} \) is the original wavelength in a vacuum, and \( n \) is the index of refraction of the medium.

In our exercise, the light's wavelength in a vacuum is given as 650 nm, and the medium's index of refraction is 1.47. By applying the formula, we calculate the wavelength in the medium as approximately 442 nm. This reduction in wavelength occurs because the light waves slow down when they enter the medium, leading to a compression of the waves.

Remember, while the speed and wavelength of light can change in a medium, its frequency remains constant.

Index of Refraction

The index of refraction is a crucial concept in optics that explains how light behaves as it moves through different materials. Symbolized by \( n \), the index of refraction quantifies how much a medium can bend the path of light passing through it. The index is given by the ratio:

• \( n = \frac{c}{v} \)

In this formula, \( c \) represents the speed of light in a vacuum, while \( v \) is the speed of light in the medium.

In practical terms, the index of refraction tells us how much slower light travels in the medium compared to a vacuum. For instance, an index of refraction of 1.47, as in the liquid from our exercise, suggests that light travels about 1.47 times slower in the liquid than in a vacuum. This decrease in speed also causes the wavelength of light to shorten, while its frequency remains the same.

Understanding the index of refraction helps explain why objects appear bent at the surface of water or why lenses can focus light to a point.

Speed of Light in Medium

The speed of light in a medium depends on the medium's index of refraction. Light begins its journey at the universal speed of approximately \( 3 \times 10^8 \) meters per second in a vacuum.

When it enters a medium, we calculate its new speed using the formula:

• \( v = \frac{c}{n} \)

In this equation, \( v \) is the speed of light in the medium, \( c \) is the speed of light in a vacuum, and \( n \) is the index of refraction for the medium.

For our exercise, with an index of refraction of 1.47, the speed of light in the liquid can be calculated as \( 2.04 \times 10^8 \) meters per second. This reduction in speed is why light bends as it passes through different materials, like water or glass, affecting how we perceive images through them.

Understanding the speed of light in different media is fundamental for technologies such as fiber optics and lenses, which rely heavily on the properties of light behavior.