Problem 15 - University Physics with Modern Physics

Question

An electromagnetic wave with frequency 5.70 \(\times\) 10\(^{14}\) Hz propagates with a speed of 2.17 \(\times\) 10\(^8\) m/s in a certain piece of glass. Find (a) the wavelength of the wave in the glass; (b) the wavelength of a wave of the same frequency propagating in air; (c) the index of refraction \(n\) of the glass for an electromagnetic wave with this frequency; (d) the dielectric constant for glass at this frequency, assuming that the relative permeability is unity.

Step-by-Step Solution

Answer

(a) 3.81 × 10⁻⁷ m; (b) 5.26 × 10⁻⁷ m; (c) n = 1.38; (d) εᵣ = 1.90

Step 1: Understand the Wavelength in Glass

To find the wavelength of the electromagnetic wave in the glass, we use the equation \( \lambda = \frac{v}{f} \), where \( \lambda \) is the wavelength, \( v \) is the speed of the wave in the medium, and \( f \) is the frequency. Here, \( v = 2.17 \times 10^8 \ m/s \) and \( f = 5.70 \times 10^{14} \ Hz \). Substitute these values into the formula to find \( \lambda \).

Step 2: Calculate the Wavelength in Glass

Using the formula for wavelength: \[ \lambda = \frac{2.17 \times 10^8}{5.70 \times 10^{14}} \]Perform the division to find \( \lambda = 3.81 \times 10^{-7} \ m \).

Step 3: Wavelength of Wave in Air

The speed of electromagnetic waves in air (or vacuum) is \( c = 3.00 \times 10^8 \ m/s \). Use the same formula \( \lambda = \frac{v}{f} \) for air, substituting \( v = 3.00 \times 10^8 \ m/s \).

Step 4: Calculate the Wavelength in Air

Substitute the values into the formula:\[ \lambda = \frac{3.00 \times 10^8}{5.70 \times 10^{14}} \]This gives \( \lambda = 5.26 \times 10^{-7} \ m \).

Step 5: Find the Index of Refraction

The index of refraction \( n \) is given by \( n = \frac{c}{v} \), where \( c \) is the speed of light in a vacuum and \( v \) is the speed of light in the glass. Substitute \( c = 3.00 \times 10^8 \ m/s \) and \( v = 2.17 \times 10^8 \ m/s \).

Step 6: Calculate the Index of Refraction

Using the formula:\[ n = \frac{3.00 \times 10^8}{2.17 \times 10^8} \]Calculate to find \( n = 1.38 \).

Step 7: Understand the Dielectric Constant

The dielectric constant \( \varepsilon_r \) is related to the index of refraction \( n \) by the equation \( n = \sqrt{\varepsilon_r} \) if the relative permeability is unity (\( \mu_r = 1 \)). Solve for \( \varepsilon_r \).

Step 8: Calculate the Dielectric Constant

Since \( n = \sqrt{\varepsilon_r} \), square the index of refraction:\[ \varepsilon_r = n^2 \]Calculate \( \varepsilon_r = 1.38^2 = 1.90 \).

Key Concepts

Wavelength Calculation

When an electromagnetic wave propagates through different mediums, its speed and wavelength can change, while its frequency remains constant. This change in wavelength can be calculated using the relation \( \lambda = \frac{v}{f} \). Here, \( \lambda \) stands for the wavelength, \( v \) is the speed of the wave in the medium, and \( f \) is the frequency.
To find the wavelength of an electromagnetic wave traveling through a piece of glass, we are given that the wave's speed in the glass is \( 2.17 \times 10^8 \ \text{m/s} \) and its frequency is \( 5.70 \times 10^{14} \ \text{Hz} \). By substituting these into the formula:

• \( v = 2.17 \times 10^8 \ \text{m/s} \)
• \( f = 5.70 \times 10^{14} \ \text{Hz} \)

Performing the division, we find: \( \lambda = \frac{2.17 \times 10^8}{5.70 \times 10^{14}} = 3.81 \times 10^{-7} \ \text{m} \).
In contrast, the same wave in air, where it travels at \( 3.00 \times 10^8 \ \text{m/s} \), has a different wavelength. Using the formula again, we calculate:\[ \lambda = \frac{3.00 \times 10^8}{5.70 \times 10^{14}} = 5.26 \times 10^{-7} \ \text{m} \].
This calculation demonstrates how the wavelength of a wave changes based on the medium it travels through.

Index of Refraction

The index of refraction, also represented by \( n \), is a fundamental concept in understanding how light or electromagnetic waves interact with different media. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. This ratio is given by the formula \( n = \frac{c}{v} \), where \( c \) is the speed of light in a vacuum (approximately \( 3.00 \times 10^8 \ \text{m/s} \)), and \( v \) is the speed of light in the medium under consideration.
In our specific exercise, we are interested in determining the index of refraction for glass. Given:

• Speed of light in vacuum, \( c = 3.00 \times 10^8 \ \text{m/s} \)
• Speed of light in glass, \( v = 2.17 \times 10^8 \ \text{m/s} \)

By substituting these values into the equation \( n = \frac{c}{v} \), we calculate:\[ n = \frac{3.00 \times 10^8}{2.17 \times 10^8} = 1.38 \].
This index of refraction signifies how much the speed of light is reduced in the glass compared to a vacuum.

Dielectric Constant

The dielectric constant, denoted as \( \varepsilon_r \), is another crucial concept when analyzing electromagnetic wave propagation in various media. It represents the medium's ability to permit the passage of electric flux and is related to the index of refraction by the equation \( n = \sqrt{\varepsilon_r} \), assuming that the relative permeability is unity (\( \mu_r = 1 \)).
To determine the dielectric constant from the known index of refraction, \( n = 1.38 \), we rearrange the equation to solve for \( \varepsilon_r \):

• First, square the index of refraction: \( \varepsilon_r = n^2 \)
• Calculate \( \varepsilon_r = 1.38^2 = 1.90 \)

This dielectric constant informs us about the electric properties of the glass and its ability to store electrical energy. Knowing \( \varepsilon_r \) is essential for designing and understanding applications involving capacitors and other electromagnetic devices in glass or similar materials.
This understanding of the dielectric constant helps in predicting how electromagnetic waves interact with materials, which is vital in fields such as optics and telecommunications.