Question

Find velocity of em wave in terms of C if $\epsilon_r$ = 4 and $\mu_r$ = 1.

• A. C
• B. $\frac{C}{2}$
• C. 2C
• D. $\frac{C}{3}$
• E. $\frac{C}{4}$

Hint:

The refractive index $n$ of a medium is defined as $n = c/v$. From the derivation, we also know $n = \sqrt{\mu_r \epsilon_r}$. For most non-magnetic optical materials, $\mu_r \approx 1$, so $n \approx \sqrt{\epsilon_r}$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The velocity of an electromagnetic (EM) wave through a specific medium depends on the electric and magnetic properties of that medium. Specifically, it depends on its permittivity (\(\epsilon\)) and permeability (\(\mu\)). In a vacuum, the velocity is \(c\).
Step 2: Key Formula or Approach:
The speed of light in a vacuum is:
\[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \]
The speed of an EM wave in a medium is:
\[ v = \frac{1}{\sqrt{\mu \epsilon}} \]
Where \(\mu = \mu_r \mu_0\) and \(\epsilon = \epsilon_r \epsilon_0\).
Therefore, the velocity in the medium can be expressed in terms of \(c\):
\[ v = \frac{c}{\sqrt{\mu_r \epsilon_r}} \]
Step 3: Detailed Explanation:
Given values for the medium:
Relative permittivity (dielectric constant), \(\epsilon_r = 4\)
Relative permeability, \(\mu_r = 1\)
Substitute these values into the formula relating \(v\) and \(c\):
\[ v = \frac{c}{\sqrt{\mu_r \cdot \epsilon_r}} \]
\[ v = \frac{c}{\sqrt{1 \cdot 4}} \]
\[ v = \frac{c}{\sqrt{4}} \]
\[ v = \frac{c}{2} \]
Step 4: Final Answer:
The velocity of the em wave in the medium is \(\frac{C}{2}\).