Problem 13 - University Physics with Modern Physics

Question

In a material having an index of refraction \(n,\) a light ray has frequency \(f,\) wavelength \(\lambda,\) and speed \(v .\) What are the frequency, wavelength, and speed of this light (a) in vacuum and (b) in a material having refractive index \(n^{\prime} ?\) In each case, express your answers in terms of only \(f, \lambda, v, n,\) and \(n^{\prime} .\)

Step-by-Step Solution

Answer

(a) In vacuum: \(f_v=f, \lambda_v=\lambda n, c=vn\). (b) In material with \(n'\): \(f_{n'}=f, \lambda_{n'}=\lambda \frac{n}{n'}, v_{n'}=\frac{vn}{n'}\).

Step 1: Understanding Light in a Medium

When light travels through a medium with an index of refraction, its speed is reduced compared to its speed in a vacuum. The relationship is given by the formula \( v = \frac{c}{n} \), where \( c \) is the speed of light in a vacuum, \( v \) is the speed of light in the medium, and \( n \) is the refractive index of the material. The frequency of light remains unchanged.

Step 2: Frequency and Speed in Vacuum

In vacuum, the speed of light is a constant \( c \), and the frequency \( f \) remains the same as in any medium. Therefore, in vacuum: - Frequency: \( f_v = f \), - Speed: \( c = vn \).

Step 3: Wavelength in Vacuum

The wavelength in a medium and a vacuum are related by the equation \( \lambda_{v} = \frac{v}{f} \). In a vacuum, the speed is \( c \), so the wavelength is: \( \lambda_v = \frac{c}{f} = \frac{v n}{f} = \lambda n \).

Step 4: Frequency and Speed in a Different Material

When light enters another material with refractive index \( n' \), its frequency remains unchanged, and the speed in the new material is given by: - Frequency: \( f_{n'} = f \), - Speed: \( v_{n'} = \frac{c}{n'} = \frac{vn}{n'} \).

Step 5: Wavelength in a Different Material

To find the wavelength in the new material, use the relation \( \lambda_{n'} = \frac{v_{n'}}{f} \). Substituting for \( v_{n'} \), we get: \( \lambda_{n'} = \frac{\left(\frac{vn}{n'}\right)}{f} = \lambda \frac{n}{n'} \).

Key Concepts

Light Speed in Medium

When light travels through any medium other than vacuum, its speed decreases. This decrease in speed happens because light interacts with the particles within the medium. The relationship between the speed of light in a medium and its speed in a vacuum is given by the formula: \[v = \frac{c}{n} \]where:

• \( v \) is the speed of light in the medium,
• \( c \) is the speed of light in a vacuum \((3 \times 10^8 \text{ m/s})\),
• \( n \) is the refractive index of the medium.

This formula tells us that as the refractive index \( n \) increases, the speed \( v \) of light decreases. However, it's important to note that despite the reduction in speed, the frequency of light remains the same in any medium.

Light Frequency

Frequency is a fundamental property of light that does not change when light moves between different media. Frequency, denoted as \( f \), is the number of oscillations or waves that pass a point per second.The formula for frequency in any medium can be expressed as:\[ f = \frac{v}{\lambda} \]where:

• \( f \) is the frequency of light,
• \( v \) is the speed of light in the medium,
• \( \lambda \) is the wavelength of light in the medium.

Since the speed \( v \) changes with the medium while \( f \) remains constant, the wavelength \( \lambda \) adjusts in response, maintaining a constant relationship between speed, frequency, and wavelength.

Wavelength in Vacuum

The wavelength of light changes with the medium it travels through. In a vacuum, the wavelength of light can be calculated using its speed and frequency. The formula for finding the wavelength in a vacuum is:\[ \lambda_v = \frac{c}{f} \]where:

• \( \lambda_v \) is the wavelength in a vacuum,
• \( c \) is the speed of light in a vacuum,
• \( f \) is the frequency of light.

Thus, the wavelength in a vacuum is straightforward to calculate since \( c \) is a constant \((3 \times 10^8 \text{ m/s})\) and the light's frequency \( f \) remains unchanged from the medium it was in before entering the vacuum.

Refractive Index

The refractive index is a dimensionless number that describes how fast light travels through a material. It is typically denoted as \( n \). This index is crucial because it determines the extent to which light rays are bent, or "refracted," when entering a material.The refractive index is defined as:\[ n = \frac{c}{v} \]where:

• \( c \) is the speed of light in a vacuum,
• \( v \) is the speed of light in the material.

The refractive index alters the speed and wavelength of light within the material but does not affect its frequency. Knowing the refractive indices helps us calculate how light will behave when it transitions between different media.