Problem 36 - Arihant AIEEE Physics

Question

There is a glass prism of refractive index \(\mu\) and angle of prism is \(A\). A ray of light enter the side \(A B\) face of the prism at an angle of incidence \(i\). The value of angle of incidence \(i\) so, that no ray emerges from the face \(A C\) of the prism, is: (a) \(\sin ^{-1}\left[\sqrt{\mu^{2}-1} \sin A-\cos A\right]\) (b) \(\sin ^{-1}\left[\sqrt{\mu^{2}+1} \sin A-\cos A\right]\) (c) \(\sin ^{-1}\left[\sqrt{\mu^{2}+1} \sin A+\cos A\right]\) (d) none of the above

Step-by-Step Solution

Answer

The correct answer is (a).

Step 1: Understanding the Situation

A ray of light is entering through a glass prism and we need to determine the condition so that it doesn't emerge through the other face of the prism. This typically involves a situation where total internal reflection occurs, which depends on the critical angle.

Step 2: Total Internal Reflection Condition

Total internal reflection occurs when the angle of incidence inside the prism is greater than the critical angle. The critical angle, \( \theta_c \), can be calculated using the formula: \( \theta_c = \sin^{-1}\left(\frac{1}{\mu}\right) \).

Step 3: Relating Angles Inside the Prism

Using Snell's Law, relate the angle of incidence \( i \) to the angle of refraction \( r \) when light enters the prism: \( \mu \sin r = \sin i \). As the angle \( r \) becomes the angle of incidence at the face \( AC \), for total internal reflection \( r > \theta_c \).

Step 4: Applying Geometry of the Prism

Given the prism's angle \( A \), apply the geometry of prism: \( r = A - \theta_c \). Therefore, for total internal reflection to occur, \( r = A - \sin^{-1}\left(\frac{1}{\mu}\right) \).

Step 5: Determine the Expression for Incidence Angle

By substituting the value of \( r \) in Snell's Law and rearranging, we get the expression \( \sin i = \mu \sin(A - \sin^{-1}(\frac{1}{\mu})) \). This simplifies to \( \sin i = \sqrt{\mu^2 - 1} \sin A - \cos A \).

Step 6: Identify the Correct Option

The derived expression for the angle of incidence \( i \) when no ray emerges matches option (a).

Key Concepts

Total Internal Reflection

Total internal reflection is a fascinating phenomenon that occurs when a light ray traveling inside a medium (like glass) hits the boundary with another medium (like air) at an angle large enough that it does not cross into the second medium, but instead reflects entirely back into the first medium. This angle is critical, because once it is exceeded, the light does not refract out of the material anymore.
Total internal reflection happens when:

• The light is traveling from a medium with a higher refractive index to one with a lower refractive index.
• The angle of incidence is greater than the critical angle for the medium boundary.

In the case of a prism, if light meets these conditions, it won't emerge through a face on the opposite side. This is what makes total internal reflection essential in applications like fiber optics, where keeping the light contained is crucial for signal transmission.

Critical Angle

The critical angle is integral to understanding total internal reflection. It is defined as the minimum angle of incidence at which total internal reflection occurs. At the critical angle, the refracted ray travels along the boundary between the two media, rather than entering the second medium.
The formula for calculating the critical angle \( \theta_c \) is quite straightforward:

• \( \theta_c = \sin^{-1}\left(\frac{1}{\mu}\right) \)

where \( \mu \) is the refractive index of the medium from which the light is coming. If the angle of incidence exceeds \( \theta_c \), the light will reflect entirely inside the medium instead of refracting out. In practical terms, this means that light in a prism won't escape out of a face if the incidence angle is large enough, and understanding this angle is crucial for optical technologies.

Snell's Law

Snell's Law is the backbone of studying refraction and total internal reflection. It provides the relationship between the angles of incidence and refraction when a light ray passes between two media with different refractive indices. It's mathematically expressed as:

• \( \mu_1 \sin i = \mu_2 \sin r \)

where \( \mu_1 \) and \( \mu_2 \) are the refractive indices of the first and second mediums, respectively; \( i \) is the angle of incidence; and \( r \) is the angle of refraction.
Using Snell's Law, we can predict whether the light will refract into the second medium or reflect entirely if total internal reflection occurs. In the exercise, Snell's Law helps relate various angles of incidence and refraction within the prism environment to determine conditions for total internal reflection.

Angle of Incidence

The angle of incidence is the angle between the incoming ray of light and the perpendicular (normal) to the surface it strikes. In optics, identifying this angle is essential because it influences whether light will reflect or refract as it meets different media.
For a prism, the angle of incidence at the entry face determines how the light will interact inside the prism. If the incidence angle is larger than the critical angle, the light will undergo total internal reflection. For our specific exercise, determining the correct angle of incidence ensures that the light does not emerge from the opposite face, which is a unique characteristic of prisms used in optical boundaries.
Understanding how to manipulate and measure the angle of incidence is essential for precise optical applications such as lenses, cameras, and optical networking.