Understanding Electromagnetic Wave Propagation in a Medium

This question asks about the speed of an electromagnetic wave as it travels through a specific type of material. We need to find the ratio of its speed in this material (denoted by $v$) compared to its speed in a vacuum (denoted by $c$). The material is described as linear, isotropic, and homogeneous, and we are given its relative permittivity ($\varepsilon_r$) and relative permeability ($\mu_r$).

Speed of Light Formulas

Let's recall the fundamental formulas for the speed of light:

• The speed of light in a vacuum, $c$, is a fundamental constant related to the permeability of free space ($\mu_0$) and the permittivity of free space ($\varepsilon_0$) by the equation: $c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$
• The speed of an electromagnetic wave in a material medium, $v$, depends on the medium's absolute permeability ($\mu$) and absolute permittivity ($\varepsilon$): $v = \frac{1}{\sqrt{\mu \varepsilon}}$

Relating Medium Properties to Vacuum Properties

The question provides the relative properties of the medium:

• Relative permittivity ($\varepsilon_r$): This tells us how the electric field properties of the medium compare to a vacuum. It's defined as $\varepsilon_r = \frac{\varepsilon}{\varepsilon_0}$. Therefore, the permittivity of the medium is $\varepsilon = \varepsilon_r \varepsilon_0$.
• Relative permeability ($\mu_r$): This tells us how the magnetic field properties of the medium compare to a vacuum. It's defined as $\mu_r = \frac{\mu}{\mu_0}$. Therefore, the permeability of the medium is $\mu = \mu_r \mu_0$.

Since the medium is linear, isotropic, and homogeneous, these relative values are constant.

Calculating the Speed in the Medium ($v$)

We can substitute the expressions for $\mu$ and $\varepsilon$ into the formula for $v$:

Rearranging the terms under the square root:

We can separate this expression:

Recognizing that the second part is the speed of light in vacuum, $c$:

Finding the Ratio $v/c$

To find the required ratio, we divide the expression for $v$ by $c$:

The $c$ terms cancel out, leaving:

Conclusion

The ratio of the speed of the electromagnetic wave in the medium ($v$) to its speed in vacuum ($c$) is determined by the square root of the product of the relative permeability and relative permittivity of the medium. The calculation shows this ratio to be $\frac{1}{\sqrt{\varepsilon_r \mu_r}}$.

Therefore, the correct option representing the ratio $\frac{v}{c}$ is $\frac{1}{\sqrt{\varepsilon_r \mu_r}}$.