Question

The angle of a prism is A. One of its refracting surfaces is silvered. Light rays falling at an angle of incidence 2A on the first surface returns back through the same path after suffering reflection at the silvered surface. The refractive index p, of the prism is

• A. $ 2 \sin A$
• B. $ 2 \cos A$
• C. $ \frac{1}{2} \cos A$
• D. $\tan A$

Answer 1

The Correct Option is B

To solve this problem, we need to understand the behavior of light in a prism with one refracting surface silvered.

Given:

• Angle of prism = A
• Light is incident at an angle 2A and returns back through the same path.
• The refractive index of the prism is p.

In this situation, we have a prism with an angle A and one of the surfaces is silvered.

Let's analyze the path of light:

1. The light ray is incident on the first surface at an angle 2A.
2. It refracts into the prism and strikes the silvered surface, getting reflected back within the prism.
3. Since the silvered surface causes the reflected ray to retrace the path of incidence, the ray hits the first surface again at the same angle (inside the prism).
4. For the ray to exit normally, the angle of refraction r must equal 0 (since the path is retraced).

From the angle of prism property, for the first refraction, Snell's law states:

p \sin r = \sin 2A

Given r = A (because it allows for the path to be retraced back after internal reflection), we substitute:

p \sin A = \sin 2A

Using the trigonometric identity \sin 2A = 2 \sin A \cos A, we can rewrite the equation:

p \sin A = 2 \sin A \cos A

Assuming \sin A \neq 0, we can safely divide both sides by \sin A to find:

p = 2 \cos A

Therefore, the refractive index p of the prism is 2 \cos A.