Problem 31 - Master Resource Book in JEE Main Physics

Question

At what angle should a ray of light be incident on the face of a prism of refracting angle \(60^{\circ}\) so that it just suffers total internal reflection at the other face? The refractive index of the material of the prism is \(1.524\). [NCERT] (a) \(16^{\circ}\) (b) \(29^{\circ}\) (c) \(45^{\circ}\) (d) \(58^{\circ}\)

Step-by-Step Solution

Answer

The angle of incidence is approximately \( 29^{\circ} \).

Step 1: Identify Critical Angle Criteria

For total internal reflection (TIR) to occur, the angle of incidence must be greater than the critical angle on the second face of the prism. The critical angle \( C \) is determined using the formula \( \sin C = \frac{1}{\text{refractive index}} \). Since the refractive index \( n = 1.524 \), we have \( \sin C = \frac{1}{1.524} \approx 0.6562 \). Thus, \( C \approx \arcsin(0.6562) \approx 41.1^{\circ} \).

Step 2: Analyze Prism Geometry

The given prism has a refracting angle \( A = 60^{\circ} \). We need to ensure the angle of incidence at the second face is equal to or greater than the critical angle \( C = 41.1^{\circ} \).

Step 3: Use Snell's Law at First Face

Let \( i_1 \) be the angle of incidence at the first face and \( r_1 \) be the angle of refraction into the prism, so \( \sin i_1 = n \sin r_1 \). The angle \( r_2 \) at the second face is \( r_2 = A - r_1 \). For TIR, we require \( r_2 \geq C \).

Step 4: Calculate Maximum Refracted Angle at Second Face

From the relationship \( r_2 = A - r_1 \), ensure \( A - r_1 = 41.1^{\circ} \). Therefore, \( r_1 = A - C = 60^{\circ} - 41.1^{\circ} = 18.9^{\circ} \).

Step 5: Solve for Angle of Incidence

Using Snell's Law at the first surface \( \sin i_1 = n \sin r_1 \), substitute \( r_1 = 18.9^{\circ} \).\[ \sin i_1 = 1.524 \times \sin(18.9^{\circ}) \approx 1.524 \times 0.3249 \approx 0.4947 \]. Solving gives \( i_1 = \arcsin(0.4947) \approx 29.5^{\circ} \), approximately \( 29^{\circ} \).

Step 6: Conclusion

The ray of light should be incident at an angle of \( 29^{\circ} \) for it to just suffer total internal reflection at the second face of the prism.

Key Concepts

Critical Angle

The critical angle is an important concept when discussing the behavior of light as it travels between different mediums. It is the angle of incidence at which light is refracted along the boundary, rather than passing through to the next medium. Light strikes the surface in a manner that causes it to reflect back into the original medium, a phenomenon known as Total Internal Reflection (TIR).

• The critical angle is specific to the pair of media and depends on their refractive indices.
• The mathematical expression to calculate the critical angle, \( C \), is given by \( \sin C = \frac{1}{n} \), where \( n \) is the refractive index of the denser medium relative to the less dense medium.

In our exercise, the prism's refractive index is 1.524. Thus, the critical angle, \( C \), is derived as \( \arcsin(0.6562) \approx 41.1^{\circ} \). For total internal reflection to occur, the light must hit the interface at an angle greater than this critical angle.

Refractive Index

The refractive index, denoted as \( n \), is crucial in determining how much light is bent, or refracted, when entering a new medium. It quantifies the speed reduction of light as it passes through different substances.

• A higher refractive index indicates that light travels more slowly through that medium.
• The refractive index is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium.

For example, in the provided exercise, the prism has a refractive index of 1.524. This indicates that light travels slower in the prism than in air, affecting how light rays bend when they enter or exit the prism. The refractive index influences both the critical angle and how much the light ray bends inside the prism.

Snell's Law

Snell's Law is a fundamental principle that describes the relationship between the angles of incidence and refraction when a light wave passes from one medium into another. It is a key equation in optics and can be written as:

• \( n_1 \sin i = n_2 \sin r \)

Where:

• \( n_1 \) and \( n_2 \) are the refractive indices of the initial and second medium.
• \( i \) is the angle of incidence.
• \( r \) is the angle of refraction.

In the exercise, Snell's Law was used to determine the angle of incidence required to achieve total internal reflection in a prism. At the first face of the prism, Snell's Law helps us calculate that the angle of incidence \( i_1 \) should be approximately \( 29^{\circ} \) to ensure that light suffers total internal reflection upon reaching the second face, given the critical angle and refractive index values.