Question

A ray of light passing through an equilateral prism is having velocity $2.12 \times 10^8 \text{ m/s}$ in the prism material, then the minimum angle of deviation is _______ degrees.

• A. 45
• B. 30
• C. 28
• D. 58

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The topic of this question is Ray Optics, focusing on refraction through a prism.
We need to determine the refractive index of the prism using the speed of light, and then compute the minimum angle of deviation.
Step 2: Key Formula or Approach:
1. Refractive index: $\mu = \frac{c}{v}$.
2. Prism formula for minimum deviation: $\mu = \frac{\sin[(A+\delta_m)/2]}{\sin(A/2)}$.
Step 3: Detailed Explanation:
First, calculate the refractive index $\mu$:
With $c = 3 \times 10^8 \text{ m/s}$ and $v = 2.12 \times 10^8 \text{ m/s}$:
\[ \mu = \frac{3 \times 10^8}{2.12 \times 10^8} \approx 1.414 = \sqrt{2} \]
An equilateral prism implies the prism angle is $A = 60^\circ$.
Using the prism formula:
\[ \sqrt{2} = \frac{\sin \left( \frac{60^\circ + \delta_m}{2} \right)}{\sin(30^\circ)} = \frac{\sin \left( \frac{60^\circ + \delta_m}{2} \right)}{0.5} \]
\[ \sin \left( \frac{60^\circ + \delta_m}{2} \right) = \sqrt{2} \times 0.5 = \frac{1}{\sqrt{2}} \]
Since $\sin(45^\circ) = \frac{1}{\sqrt{2}}$:
\[ \frac{60^\circ + \delta_m}{2} = 45^\circ \implies 60^\circ + \delta_m = 90^\circ \implies \delta_m = 30^\circ \]
Step 4: Final Answer:
The minimum angle of deviation is $30^\circ$.