Question

The critical angle for total internal reflection of a ray from any medium to vacuum is 30\(^{\circ}\). Then velocity of light in the medium will be

• A. 3 x 10\(^8\) m/s
• B. 1.5 x 10\(^5\) m/s
• C. 6 x 10\(^8\) m/s
• D. 4.5 x 10\(^8\) m/s

Hint:

Before starting calculations, check the options. The speed of light in any medium cannot exceed the speed of light in a vacuum (\(c \approx 3 \times 10^8\) m/s). This immediately eliminates options (C) and (D) as physically impossible, simplifying your problem.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The critical angle is related to the refractive index of the medium. The refractive index, in turn, relates the speed of light in a vacuum to the speed of light in the medium.
Step 2: Key Formula or Approach:
1. The relationship between the refractive index (\(n\)) of a medium and the critical angle (\(C\)) for light going into a vacuum is:
\[ n = \frac{1}{\sin(C)} \] 2. The definition of refractive index is:
\[ n = \frac{\text{speed of light in vacuum (c)}}{\text{speed of light in medium (v)}} \] Step 3: Detailed Explanation:
Part 1: Calculate the refractive index (n)
Given the critical angle \(C = 30^\circ\).
\[ n = \frac{1}{\sin(30^\circ)} \] We know that \(\sin(30^\circ) = 0.5\) or \(1/2\).
\[ n = \frac{1}{0.5} = 2 \] The refractive index of the medium is 2.
Part 2: Calculate the velocity of light in the medium (v)
We use the formula \(n = c/v\), where \(c\) is the speed of light in vacuum, \(c \approx 3 \times 10^8\) m/s.
Rearranging the formula to solve for \(v\):
\[ v = \frac{c}{n} \] Substituting the values:
\[ v = \frac{3 \times 10^8 \text{ m/s}}{2} \] \[ v = 1.5 \times 10^8 \text{ m/s} \] Part 3: Compare with options
The calculated velocity is \(1.5 \times 10^8\) m/s.
Looking at the options:
(A) 3 x 10\(^8\) m/s (This is the speed in vacuum)
(B) 1.5 x 10\(^5\) m/s (The numerical value is correct, but the power of 10 is wrong, likely a typo)
(C) 6 x 10\(^8\) m/s (Faster than light in vacuum, impossible)
(D) 4.5 x 10\(^8\) m/s (Faster than light in vacuum, impossible)
The value \(1.5 \times 10^8\) m/s is the correct answer.
Step 4: Final Answer:
The calculated velocity of light in the medium is \(1.5 \times 10^8\) m/s. Option (B) is the intended answer, despite the typo.