Question

One side of an equilateral prism is painted by a transparent material of refractive index \( n_2 \). The refractive index of prism is 1.6. The minimum value of \( n_2 \) required for total internal reflection from the painted face is _______.

• A. \( \frac{\sqrt{3}}{1.6} \)
• B. \( \sqrt{3} \)
• C. \( \frac{3.2}{\sqrt{3}} \)
• D. \( \frac{4\sqrt{3}}{5} \)

Answer 1

The Correct Option is B

Step 1: Total Internal Reflection Condition.
For total internal reflection to occur, the refractive index \( n_2 \) of the medium outside the prism must satisfy the critical angle condition: \[ n_2 \geq \frac{n_{\text{prism}}}{\sin \theta_c} \] where \( \theta_c \) is the critical angle, and \( n_{\text{prism}} = 1.6 \) is the refractive index of the prism.
Step 2: Critical Angle for the Prism.
The critical angle for an equilateral prism is \( \theta_c = 30^\circ \), since the angle between the sides of the equilateral triangle is \( 60^\circ \) and the critical angle is \( 30^\circ \).
Step 3: Solve for \( n_2 \).
Now, using the relation: \[ n_2 = \frac{1.6}{\sin 30^\circ} = \frac{1.6}{0.5} = \sqrt{3} \]
Final Answer: \( \sqrt{3} \)