Step 1: Understanding the Concept:
For a light ray to return back through the same path (retrace its path), it must strike the silvered reflecting surface normally. This means the angle of refraction at the second surface is zero.
Step 2: Key Formula or Approach:
1. For a prism: \( r_1 + r_2 = A \).
2. Snell's Law: \( \mu = \frac{\sin i}{\sin r_1} \).
Step 3: Detailed Explanation:
Since the ray retraces its path after striking the second silvered surface, it must have hit it normally.
Therefore, the angle of incidence at the second surface, \( r_2 = 0^\circ \).
Using the prism formula:
\[ r_1 + r_2 = A \]
\[ r_1 + 0 = A \implies r_1 = A \]
Given the angle of incidence at the first surface is \( i = 2A \).
Applying Snell's Law at the first surface:
\[ \mu = \frac{\sin i}{\sin r_1} = \frac{\sin(2A)}{\sin A} \]
Using the trigonometric identity \( \sin(2A) = 2 \sin A \cos A \):
\[ \mu = \frac{2 \sin A \cos A}{\sin A} \]
\[ \mu = 2 \cos A \]
Step 4: Final Answer:
The refractive index of the prism is \( 2 \cos A \).