Question

The prism has refracting angle A. The second refracting surface of the prism is silvered. Light ray falling on first refracting surface with angle of incidence \(2A\), reaches the second surface and returns back through same path due to reflection at the silvered surface. The refractive index of the material of prism is

• A. \(2sinA\)
• B. \(2cosA\)
• C. \(\frac{1}{2}sinA\)
• D. \(\frac{1}{2}cosA\)

Hint:

The ray retraces its path only if it hits the silvered face normally, so r1 = A.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The ray returns along the same path when it strikes the silvered second surface at normal incidence, so that it is reflected back on itself.

Step 2: Angle conditions:
For a prism, \(r_1 + r_2 = A\). If the ray falls normally on the second surface, \(r_2 = 0\), so \(r_1 = A\).

Step 3: Apply Snell's law at the first surface:
\[ \mu = \frac{\sin i}{\sin r_1} = \frac{\sin 2A}{\sin A} = \frac{2\sin A\cos A}{\sin A} = 2\cos A \]

Step 4: Why the other options are wrong.
\(2\sin A\) would result from \(\sin 2A\) divided by \(\cos A\). Options with a factor \(\frac12\) cannot be greater than 1 for typical prism angles, whereas the refractive index must exceed 1.

Final Answer:
The refractive index is \(2\cos A\), option (B). \[ \boxed{2\cos A} \]