Step 1: Understanding the Concept:
The critical angle ($\theta_c$) is the angle of incidence in an optically denser medium for which the angle of refraction in the less dense medium (usually air or vacuum) is $90^\circ$. The refractive index of a medium is related to its relative permittivity ($\epsilon_r$) and relative permeability ($\mu_r$).
Step 2: Key Formula or Approach:
The refractive index $n$ of a medium is given by:
\[ n = \sqrt{\epsilon_r \mu_r} \]
The critical angle $\theta_c$ is related to the refractive index $n$ by:
\[ \sin(\theta_c) = \frac{1}{n} \]
Step 3: Detailed Explanation:
Given values:
Relative permittivity, $\epsilon_r = 3$
Relative permeability, $\mu_r = \frac{4}{3}$
First, calculate the refractive index $n$ of the medium:
\[ n = \sqrt{3 \times \frac{4}{3}} \]
\[ n = \sqrt{4} = 2 \]
Now, use the critical angle formula:
\[ \sin(\theta_c) = \frac{1}{n} = \frac{1}{2} \]
Taking the inverse sine:
\[ \theta_c = \arcsin\left(\frac{1}{2}\right) = 30^\circ \]
Step 4: Final Answer:
The critical angle of the medium is $30^\circ$. The correct option is (B).