Problem 4 - University Physics with Modern Physics

Question

Light with a frequency of \(5.80 \times 10^{14}\) Hz travels in a block of glass that has an index of refraction of 1.52. What is the wavelength of the light (a) in vacuum and (b) in the glass?

Step-by-Step Solution

Answer

(a) 517 nm in vacuum, (b) 340 nm in glass.

Step 1: Identify Known Values

We know the frequency of the light is \(5.80 \times 10^{14}\) Hz and the index of refraction for the glass is 1.52. The speed of light in vacuum, \(c\), is a known constant, \(3.00 \times 10^8\) m/s.

Step 2: Calculate Wavelength in Vacuum

Use the formula \(c = \lambda_0 \cdot f\) to find the wavelength in vacuum \(\lambda_0\). Rearrange the formula to \(\lambda_0 = \frac{c}{f}\). Substitute the values: \(\lambda_0 = \frac{3.00 \times 10^8\, \text{m/s}}{5.80 \times 10^{14}\, \text{Hz}} = 5.17 \times 10^{-7}\, \text{m}\).

Step 3: Calculate Wavelength in Glass

Use the formula for the wavelength in a medium: \(\lambda = \frac{\lambda_0}{n}\), where \(n\) is the refractive index. Substitute the known values: \(\lambda = \frac{5.17 \times 10^{-7}\, \text{m}}{1.52} = 3.40 \times 10^{-7}\, \text{m}\).

Key Concepts

Index of Refraction

The index of refraction, often represented by the symbol \( n \), is a measure of how much light slows down as it passes through a medium compared to its speed in a vacuum. This is a crucial concept when studying how light behaves in various substances like glass, water, or air.

Key features of the index of refraction include:

• A higher index indicates that light travels slower in the medium than in a vacuum.
• The index of refraction is calculated using the formula \( n = \frac{c}{v} \), where \( c \) is the speed of light in a vacuum, and \( v \) is the speed of light in the medium.
• Common examples include the index of refraction for air, which is approximately 1, and glass, which is typically around 1.5.

By understanding the index of refraction, you can predict how light will change direction, a phenomenon known as refraction, and also calculate the wavelength of light as it travels through different materials.

Frequency of Light

The frequency of light is a fundamental concept in optics and other areas of physics. It is defined as the number of oscillations or cycles that the light wave completes in one second, measured in Hertz (Hz).

Important aspects of light frequency include:

• It remains constant regardless of the medium through which the light travels. So, even when light enters glass from the air, the frequency does not change.
• The frequency is inversely related to the wavelength when the speed of light is considered. This means that as frequency increases, the wavelength decreases, given a constant speed of light.
• You can determine the wavelength of light in a vacuum using the relationship \( c = \lambda_0 \cdot f \), where \( \lambda_0 \) is the wavelength in a vacuum, \( c \) is the speed of light in a vacuum, and \( f \) is the frequency.

Knowing the frequency enables you to calculate other properties of light, like energy and wavelength, essential for understanding how it behaves in different environments.

Speed of Light in Vacuum

The speed of light in a vacuum is an essential physical constant denoted by \( c \). Understanding this speed is foundational when learning about various optical and electromagnetic phenomena.

Key facts about the speed of light in a vacuum:

• It is precisely measured at \( 3.00 \times 10^8 \) meters per second (m/s).
• This speed is the ultimate speed limit in the universe; no information or matter can exceed this speed.
• In a vacuum, light travels without any interactions that can slow it down, making \( c \) one of the most critical constants in physics.

The speed of light in a vacuum is used in calculating the wavelength of light through the formula \( \lambda_0 = \frac{c}{f} \), where \( f \) is the frequency of the light. Understanding \( c \) allows us to derive how light behaves under different conditions, including when it enters materials with various indices of refraction.