AIPMT PRELIMS 2004 Physics Prisms MCQ Question

To solve the problem, we need to analyze the behavior of light as it travels through a prism, particularly when one of its refracting surfaces is made into a mirror.

Given Data:

• Refractive index of the prism, $n = \sqrt{2}$
• Refracting angle of the prism, $A = 30^\circ$

The Concept:

When light enters a prism, it refracts at the first surface and again at the second surface. When one of the surfaces is mirrored, the light will reflect off the mirror instead of refracting through that surface. The key is to determine the angle of incidence that allows the light to retrace its path after reflecting off the mirrored surface.

Step-by-Step Explanation:

1. Understanding Refraction and Reflection:

   • When light enters a prism, it bends towards the normal due to refraction. The relationship can be described using Snell's Law: $n_1 \sin i = n_2 \sin r$ where $n_1$ is the refractive index of air (which we can take as 1), $i$ is the angle of incidence, $n_2$ is the refractive index of the prism, and $r$ is the angle of refraction.

2. Finding the Angle of Refraction:

   • For our prism, when light enters from air (with $n_1 = 1$) into the prism (with $n_2 = \sqrt{2}$), and refracts at the angle $A = 30^\circ$ (the angle between the two sides of the prism), we can use the formula: $n_1 \sin i = n_2 \sin r$ Here, we need to find $r$ at an angle $i$ that will allow light to reflect perfectly off the mirrored surface.

3. Critical Angles and Conditions for Retracing:

   • After reflecting off the mirrored surface, for the light to retrace its path, the angle of incidence on the mirror must equal the angle of reflection, which is a consequence of the law of reflection: $\text{Angle of incidence} = \text{Angle of reflection}$

4. Calculating Specific Angles:

   • To find the specific angle of incidence $i$ that allows for this retracing, we consider the geometry of the prism. The total internal angle at the vertex of the prism is $A = 30^\circ$.
   • If the light strikes the mirrored surface at an angle $r$, it will be reflected back into the prism. The angle of incidence $i$ can be deduced from the geometry of the prism:
     • The light coming out from the prism must also maintain an incident angle such that it equals the angle of refraction $r$ after reflection.

5. Determining the Correct Angle:

   • For the light beam to retrace its path precisely, the angle of incidence $i$ at the first surface must be $45^\circ$. This is because:
     • When $i = 45^\circ$, using Snell's Law: $\sin r = \frac{1}{\sqrt{2}} \rightarrow r = 30^\circ$
     • Therefore, the angle of incidence on the mirrored surface becomes $30^\circ$, which allows the light to reflect back into the prism, retracing its path.

Conclusion:

The correct answer is D) 45° because at this angle, the light will refract through the first surface, reflect off the mirror at $30^\circ$ and return through the prism.

Why Other Options are Incorrect:

• A) 60°: This angle would lead to a refraction that does not correspond to the conditions necessary for retracing.
• B) 0°: This implies no incidence, meaning the light would not enter the prism.
• C) 30°: While this may seem plausible, it would not satisfy the conditions for retracing as the angles would not align correctly after reflection.

In summary, the calculation and geometrical understanding of light behavior in prisms lead us to conclude that 45° is indeed the angle of incidence that allows the beam of light to retrace its path after reflecting off the mirrored surface.