Problem 76 - 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 \(n\) is the refractive index of the medium with respect to air, select the possible values of \(n\) from the following (a) \(1.3\) (b) \(1.4\) (c) \(1.5\) (d) \(1.60\)

Step-by-Step Solution

Answer

The possible values of \(n\) that satisfy the condition are 1.5 and 1.6.

Step 1: Understand Total Internal Reflection

Total internal reflection occurs when a ray of light traveling in a denser medium strikes the boundary at an angle greater than the critical angle, causing it to reflect entirely back into the medium.

Step 2: Determine Critical Angle Formula

The critical angle \( \theta_c \) for total internal reflection occurs when the refracted angle is \(90^{\circ}\). The formula is given by \( \sin \theta_c = \frac{1}{n} \), where \(n\) is the refractive index of the denser medium with respect to air.

Step 3: Calculate the Maximum Critical Angle

Given the angle of incidence is \(45^{\circ}\), the critical angle must be less than or equal to this for total internal reflection to occur. Using \( \sin \theta_c = \frac{1}{n} \), ensure \( \sin 45^{\circ} \geq \frac{1}{n} \). Since \( \sin 45^{\circ} = \frac{1}{\sqrt{2}} \approx 0.707 \), the condition becomes \( n \geq 1.414 \).

Step 4: Verify Suitable Options

Compare the options provided to see which satisfy \( n \geq 1.414 \). This condition is satisfied by the options where the refractive index is 1.5 and 1.6.

Key Concepts

Refractive Index

The refractive index, often symbolized by the letter \( n \), is a measure of how much light bends as it enters a different medium. It's a dimensionless number that compares the speed of light in a vacuum to its speed in the medium.

• If the refractive index is higher, it means light travels slower in that medium compared to a vacuum.
• The refractive index depends on the optical density of the material; denser materials have higher refractive indices.
• It's typically more than one, considering light slows down upon entering any medium other than a vacuum.

Imagine light as a runner on a track, with different sections of the track coated with different surfaces. Where the surface is slippery (a lower refractive index), the runner can maintain higher speed. When the track is sticky (a higher refractive index), the runner slows down more. Thus, the refractive index gives us a good indicator of how much bending will occur when light crosses the boundary between two media.

Example

Water has a refractive index of about 1.33, meaning light travels slower in water than in air. When light moves from air into water, it bends towards the normal (an imaginary line perpendicular to the surface). This bending allows us to understand refraction and is crucial for lens design, fiber optics, and various optical applications.

Critical Angle

The critical angle is an important concept related to the refractive index. It is the angle of incidence above which total internal reflection occurs when light attempts to move from a denser to a less dense medium.
When light hits the boundary at an angle greater than the critical angle, rather than refracting into the next medium, it reflects entirely back into the original medium.

• This happens because at the critical angle, the refracted ray travels along the boundary, forming an angle of \(90^{\circ}\) with the normal.
• To find the critical angle \( \theta_c \) for a given medium, we use the equation \( \sin \theta_c = \frac{1}{n} \), where \( n \) is the refractive index of the medium.
• If \( n \) is known, the critical angle can be calculated easily using this formula.

Understanding the critical angle is essential in applications like designing binoculars and fiber-optic cables. It ensures the light stays within the intended path, enhancing the functionality of optical devices.

Angle of Incidence

The angle of incidence is the angle between the incoming ray (such as a ray of light) and the normal to the surface at the point of incidence. This concept is fundamental in understanding how light behaves at surfaces.

• The normal is an imaginary line that stands perpendicular to the surface.
• A simple rule governs the angle’s behavior: the angle of incidence equals the angle of reflection.
• If the angle of incidence is greater than the critical angle, total internal reflection can occur, which means light doesn't pass through to the other medium but rather, reflects back entirely.

In our everyday life, this explains why we often see reflections on the surface of water or glass. When designing optical systems, understanding how the angle of incidence influences reflection and refraction is vital to ensure they work as intended. For example, in the exercise, the light hitting the surface at an angle of \(45^{\circ}\) means we need to consider whether this is above the critical angle for total internal reflection to happen.