Problem 89 - Master Resource Book in JEE Main Physics

Question

The cross-section of a glass prism has the form of an isoceles triangle. One of the refracting faces is silvered. A ray of light falls normally on the other refracting face. After being reflected twice, it emerges through the base of the prism perpendicular to it. The angles of the prism are (a) \(54^{*}, 54^{*}, 72^{*}\) (b) \(72^{*}, 72^{*}, 36^{*}\) (c) \(45^{\circ}, 45^{\prime \prime}, 90^{\circ}\) (d) \(57^{\prime}, 57^{\circ}, 76^{\circ}\)

Step-by-Step Solution

Answer

The correct angles of the prism are 72°, 72°, and 36° (option b).

Step 1: Identify Entry and Exit Conditions for the Ray

The ray of light hits one of the refracting faces normally, so it enters without deviation. It is then mentioned that it emerges through the base of the prism perpendicularly. This sets a condition for the angles of the prism since the ray goes through reflection twice.

Step 2: Understand Double Reflection in Prism

Since one face of the prism is silvered and the ray reflects there, it implies that the interior angles at which the light enters and exits must be complementary. Reflecting twice means that each reflection follows the law of reflection where the angle of incidence equals the angle of reflection.

Step 3: Analyze Angle Possibilities

For the ray to emerge perpendicular, the angle at the base of the prism must allow this. Since it's an isosceles triangle, consider what angle sets allow for a complementarity in the settings. The light striking normally suggests specific geometric constraints that generally hold only in particular angle configurations.

Step 4: Use Given Options to Determine Correct Angle

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Key Concepts

Reflection

Reflection is a fundamental concept in optics that refers to the bouncing back of light waves when they encounter a surface. When a ray of light hits a surface, such as a mirror or a silvered face of a prism, it is reflected back at the same angle it arrived with. This is known as the law of reflection, which states that the angle of incidence is equal to the angle of reflection.

In practical terms, if a light ray approaches a reflective surface at a 30-degree angle relative to a normal line (an imaginary line perpendicular to the surface), it will reflect away at the same 30-degree angle on the opposite side of the normal. Understanding this principle is crucial when analyzing the paths of light rays in prisms, especially when they reflect multiple times as described in this exercise.

Key points of reflection include:

• The angle of incidence equals the angle of reflection.
• Reflection occurs with different angles depending on the surface orientation.
• Reflective surfaces include mirrors and specially coated prism faces.

Isosceles prism

An isosceles prism is a type of triangular optical prism where two of its angles are equal. The interesting part about an isosceles prism is its symmetrical properties, which influence how light travels through it.

In this exercise, we deal with an isosceles prism where such symmetry is key for the path of the ray. Since the ray enters and exits at specific angles, these angles align with the isosceles nature of the prism. When analyzing the behavior of light in such prisms, symmetry helps in predicting how rays will reflect and refract.

Important characteristics of isosceles prisms include:

• Two equal angles that direct the symmetric passage of rays.
• Enhanced precision in directing multiple reflections along intended paths.
• Commonly used in optics for producing specific reflective and refractive effects.

Understanding these symmetry features aids in solving geometrical optics problems involving prisms.

Ray of light

A ray of light is a model used to represent a direction of light's travel. Light propagates in straight lines, and a ray represents this path ideally having no width. It is often depicted with lines and arrows, indicating the direction of light flow.

In the context of prism optics, the concept of a ray is powerful. It helps predict how light will interact with optical elements like lenses and prisms. For instance, when a ray enters a prism normally, it travels through without being bent. After entering, as it encounters angles and reflective surfaces within the prism, its path changes according to reflection and refraction laws.

Key insights about rays of light include:

• They travel in straight lines until interacting with surfaces.
• Refraction and reflection alter their path within optics materials.
• Diagrams of rays help visualize and solve path problems in exercises.

By understanding rays of light, students can accurately predict and visualize how optical systems will affect light's journey.