Speed of Electromagnetic Wave in a Medium

This explanation covers how to calculate the speed of an electromagnetic wave as it travels through a specific medium. We will use the properties of the medium, namely its relative permittivity and relative permeability, along with the speed of light in a vacuum.

The Wave Speed Formula

The speed of an electromagnetic wave in a medium ($v$) is related to the speed of light in a vacuum ($c$) by the following formula:

$ v = \frac{c}{\sqrt{\epsilon_r \mu_r}} $

Here:

• $v$ represents the speed of the electromagnetic wave in the medium (measured in meters per second, m/s).
• $c$ is the constant speed of light in vacuum, given as $3 \times 10^8$ m/s.
• $\epsilon_r$ is the relative permittivity of the medium, which indicates how the electric field is affected by the medium.
• $\mu_r$ is the relative permeability of the medium, indicating how the magnetic field is affected. Both $\epsilon_r$ and $\mu_r$ are dimensionless quantities.

Analyzing Given Medium Properties

The question provides the following details about the medium:

• Relative Permittivity: $\epsilon_r = 4$
• Relative Permeability: $\mu_r = 1$
• Speed of Light in Vacuum: $c = 3 \times 10^8$ m/s

The fact that the medium is described as non-magnetic means its relative permeability ($\mu_r$) is close to 1, as given.

Step-by-Step Calculation of Wave Speed

To find the speed of the wave in this specific medium, we substitute the given values into the formula:

1. Write down the formula:

   $ v = \frac{c}{\sqrt{\epsilon_r \mu_r}} $

2. Insert the known values:

   $ v = \frac{3 \times 10^8 \text{ m/s}}{\sqrt{4 \times 1}} $

3. Simplify the term inside the square root:

   $ \sqrt{4 \times 1} = \sqrt{4} = 2 $

4. Perform the division:

   $ v = \frac{3 \times 10^8 \text{ m/s}}{2} $

5. Calculate the final speed:

   $ v = 1.5 \times 10^8 \text{ m/s} $

Result Summary

Based on the calculation, the speed of the electromagnetic wave propagating through the medium with $\epsilon_r = 4$ and $\mu_r = 1$ is $1.5 \times 10^8$ m/s.