Question

Two transparent media. A and B are separated by a plane boundary. The speed of light in those media are 1.5 × 10⁸ m/s and 2.0 × 10⁸ m/s, respectively. The critical angle for a ray of light for these two media is:

• A. sin^-1(0.5000)
• B. sin^-1(0.750)
• C. tan^-1(0.500)
• D. tan^-1(0.750)

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Critical angle is the angle of incidence in a denser medium for which the angle of refraction in the rarer medium is \(90^\circ\).
Total internal reflection can only occur when light travels from a denser medium to a rarer medium.
A medium is optically denser if the speed of light in it is lower.
Key Formula or Approach:
The relationship between critical angle \(\theta_c\) and refractive indices is:
\[ \sin \theta_c = \frac{n_{\text{rarer}}}{n_{\text{denser}}} \]
Since \(n = \frac{c}{v}\), we can express this in terms of speeds:
\[ \sin \theta_c = \frac{v_{\text{denser}}}{v_{\text{rarer}}} \]
Step 2: Detailed Explanation:
1. Identify denser and rarer media:
Speed in \(A\), \(v_A = 1.5 \times 10^8 \text{ m/s}\).
Speed in \(B\), \(v_B = 2.0 \times 10^8 \text{ m/s}\).
Since \(v_A<v_B\), medium \(A\) is the denser medium and medium \(B\) is the rarer medium.
2. Calculate \(\sin \theta_c\):
\[ \sin \theta_c = \frac{v_A}{v_B} \]
\[ \sin \theta_c = \frac{1.5 \times 10^8}{2.0 \times 10^8} = \frac{1.5}{2.0} \]
\[ \sin \theta_c = 0.75 \]
3. Find the critical angle:
\[ \theta_c = \sin^{-1}(0.750) \]
Step 3: Final Answer:
The critical angle for these two media is \(\sin^{-1}(0.750)\).