Problem 3 - University Physics with Modern Physics

Question

A beam of light has a wavelength of 650 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

(a) The speed of light in the liquid is approximately \(2.04 \times 10^8\) m/s. (b) The wavelength in the liquid is approximately 442 nm.

Step 1: Identify Given Values

First, identify the values provided. The wavelength in a vacuum is 650 nm, which we convert to meters: \( 650 \text{ nm} = 650 \times 10^{-9} \text{ m} \). The index of refraction for the liquid is given as 1.47.

Step 2: Use the Speed of Light Equation

The speed of light in a medium is related to its speed in a vacuum by the equation \( v = \frac{c}{n} \), where \( v \) is the speed in the medium, \( c = 3 \times 10^8 \text{ m/s} \) is the speed in vacuum, and \( n \) is the index of refraction. Substituting the given values, \( v = \frac{3 \times 10^8}{1.47} \approx 2.04 \times 10^8 \text{ m/s} \).

Step 3: Calculate Wavelength in Liquid

The wavelength of light in a medium is found using the equation \( \lambda = \frac{\lambda_0}{n} \), where \( \lambda_0 = 650 \times 10^{-9} \text{ m} \) is the wavelength in vacuum and \( n = 1.47 \). Therefore, \( \lambda \approx \frac{650 \times 10^{-9}}{1.47} \approx 442 \times 10^{-9} \text{ m} \).

Key Concepts

Speed of Light

The speed of light is a fundamental constant of nature. In a vacuum, light always travels at a staggering speed of approximately \( 3 \times 10^8 \text{ m/s} \). This speed is often denoted by the symbol \( c \). However, when light enters a material medium, such as water or glass, it slows down.

The degree to which light slows down is determined by the medium's index of refraction, \( n \), a dimensionless number that quantifies how much the light bends as it enters the material. The index is unique to each material and is related to the speed of light in that medium, \( v \), by the simple equation: \[ v = \frac{c}{n} \]

For instance, in a liquid where the index of refraction is 1.47, the speed of light becomes: \[ v = \frac{3 \times 10^8 \text{ m/s}}{1.47} \approx 2.04 \times 10^8 \text{ m/s} \]

This equation highlights that in denser mediums, where the index of refraction is higher, the speed of light is slower. Similarly, in less dense mediums, light retains more of its speed.

Wavelength in a Medium

When light travels from one medium to another, its speed changes and so does its wavelength. Wavelength refers to the distance between successive peaks of a wave, and in the vacuum, light has a longer wavelength due to its higher speed.

As light slows down in a medium such as a liquid, its wavelength decreases. The new wavelength, \( \lambda \), can be calculated using the original wavelength in a vacuum, \( \lambda_0 \), and the index of refraction, \( n \), of the medium: \[ \lambda = \frac{\lambda_0}{n} \]

For example, if the wavelength of light in a vacuum is 650 nm and it enters a medium with an index of refraction of 1.47, its new wavelength can be computed as: \[ \lambda \approx \frac{650 \times 10^{-9} \text{ m}}{1.47} \approx 442 \times 10^{-9} \text{ m} \]

This reduction in wavelength is a vital concept as it affects how we perceive light, influencing colors and optical properties in various mediums. As a rule of thumb, the higher the index of refraction, the more compressed the wavelength will be.

Light in Vacuum

In a vacuum, light behaves uniquely due to the absence of any materials to slow it down. This environment allows it to travel at its absolute maximum speed, \( c \), which is about \( 3 \times 10^8 \text{ m/s} \).

A vacuum is considered the baseline for many optical properties, including measurements of wavelength and frequency. This is because the properties of light are most straightforward in a vacuum, where it does not experience any dispersion or slowing down.

When we describe light in a vacuum, it's important to remember:

• Its speed is constant and uninfluenced by external factors.
• The wavelength of light in vacuum is its natural, undistorted length.
• Light's characteristics, like frequency, remain unchanged as it moves from place to place, unless it enters a different medium.

Light's interaction with various media, altering its speed and wavelength, reminds us of the unique and constant nature it holds in a vacuum, which becomes a fundamental reference in the field of optics.