Problem 49 - Master Resource Book in JEE Main Physics

Question

A ray of light travelling in a transparent medium falls on a surface separating the medium from air, at an angle of incidence of \(45^{\circ}\). The ray undergoes total internal reflection. If, \(\mu\) is refractive index of the medium w.r.t. air, select the possible values of \(\mu\) from the following(a) \(1.5\) (b) \(1.6\) [c) \(1.2\) (d) \(1.3\)

Step-by-Step Solution

Answer

Possible values of \(\mu\) are 1.5 and 1.6.

Step 1: Understand Total Internal Reflection Conditions

Total internal reflection occurs when a ray of light traveling from a medium with a higher refractive index \( \mu \) to a medium with a lower refractive index (such as air, where the refractive index is 1) hits the boundary at an angle greater than the critical angle \( \theta_c \). The critical angle \( \theta_c \) is given by the formula \( \sin(\theta_c) = \frac{1}{\mu} \).

Step 2: Determine Critical Angle for Given Refractive Index

To determine the critical angle \( \theta_c \) for each provided refractive index \( \mu \), we use \( \sin(\theta_c) = \frac{1}{\mu} \).- For \( \mu = 1.5 \): \( \sin(\theta_c) = \frac{1}{1.5} = 0.666\ldots \rightarrow \theta_c = \sin^{-1}(0.666\ldots) \approx 41.8^\circ \).- For \( \mu = 1.6 \): \( \sin(\theta_c) = \frac{1}{1.6} = 0.625 \rightarrow \theta_c = \sin^{-1}(0.625) \approx 38.7^\circ \).- For \( \mu = 1.2 \): \( \sin(\theta_c) = \frac{1}{1.2} = 0.833\ldots \rightarrow \theta_c = \sin^{-1}(0.833\ldots) \approx 56.4^\circ \).- For \( \mu = 1.3 \): \( \sin(\theta_c) = \frac{1}{1.3} = 0.769\ldots \rightarrow \theta_c = \sin^{-1}(0.769\ldots) \approx 50.3^\circ \).

Step 3: Compare Angle of Incidence with Critical Angle

The critical angle must be smaller than the angle of incidence for total internal reflection to occur. The given angle of incidence is \(45^\circ\).- For \( \mu = 1.5 \), \( \theta_c = 41.8^\circ \), total internal reflection occurs because \(45^\circ > 41.8^\circ\).- For \( \mu = 1.6 \), \( \theta_c = 38.7^\circ \), total internal reflection occurs because \(45^\circ > 38.7^\circ\).- For \( \mu = 1.2 \), \( \theta_c = 56.4^\circ \), no total internal reflection occurs because \(45^\circ < 56.4^\circ\).- For \( \mu = 1.3 \), \( \theta_c = 50.3^\circ \), no total internal reflection occurs because \(45^\circ < 50.3^\circ\).

Key Concepts

Refractive Index

The refractive index, often denoted as \(\mu\), is a crucial property of a medium that determines how much it can bend light. When light enters a medium at an angle, it changes speed, causing it to bend either towards or away from the normal line. This change in direction is quantified by the refractive index, which compares the speed of light in a vacuum to that in the medium. The formula for refractive index is given by \(\mu = \frac{c}{v}\), where \(c\) is the speed of light in a vacuum and \(v\) is the speed in the medium.

Understanding the refractive index helps us explain phenomena like refraction and total internal reflection, where light behaves differently depending on the medium's optical density. A higher refractive index indicates that light travels slower in the medium, and typically the medium is denser. This is essential for solving problems involving total internal reflection, as we'll see when discussing the critical angle.

Critical Angle

The critical angle \(\theta_c\) is a specific angle of incidence at which light traveling from a denser medium to a less dense medium (such as air) is refracted along the boundary, making the angle of refraction equal to 90 degrees. Total internal reflection occurs when the angle of incidence exceeds this critical angle. Mathematically, the critical angle can be calculated with the formula \(\sin(\theta_c) = \frac{1}{\mu}\), where \(\mu\) is the refractive index of the denser medium relative to the less dense medium.

• If the angle of incidence is greater than the critical angle, light will not pass into the less dense medium, and instead, it will be reflected back completely into the denser medium.
• If the angle of incidence is less than the critical angle, the light will refract and pass into the less dense medium.
• If the angle of incidence is exactly equal to the critical angle, the light travels along the boundary between the two mediums.

Find the critical angle for specific materials to determine when total internal reflection might occur. This concept is key, particularly in optics, where precise control over light paths is essential.

Angle of Incidence

The angle of incidence plays an essential role in determining the behavior of light as it interacts with different surfaces and mediums. It is the angle between the incoming light ray and the normal line (a line perpendicular to the surface at the point of contact).

• When light travels from a denser medium to a less dense one, the angle of incidence can decide if the light will undergo refraction or total internal reflection.
• For total internal reflection to occur, the angle of incidence must be greater than the critical angle.
• If the angle of incidence is less than the critical angle, part of the light will refract, passing through to the other medium.

In the context of optical systems and fiber optics, controlling the angle of incidence allows engineers to manage how light propagates through systems. By manipulating this angle, it's possible to maintain light within a medium using total internal reflection, optimizing signal transmission with minimal loss.