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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.

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.

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.