Problem 15 - Master Resource Book in JEE Main Physics

Question

A ray of light falls on a transparent glass slab with refractive index (relative to air) of \(1.62 .\) The angle of incidence for which the reflected and refracted rays are mutually perpendicular is (a) \(\tan ^{-1}(162)\) (b) \(\sin ^{-1}(162)\) (c) \(\cos ^{-1}(162)\) (d) None of these

Step-by-Step Solution

Answer

The angle of incidence is \(\tan^{-1}(1.62)\), corresponding to option (a).

Step 1: Understanding the Problem

We need to find the angle of incidence at which the reflected and refracted rays are mutually perpendicular. According to the problem, the refractive index of the glass slab is 1.62.

Step 2: Applying the Perpendicular Condition

When the reflected and refracted rays are perpendicular to each other, the angle between them is 90 degrees. From geometry and Snell's law, if the angle of incidence is \(i\) and angle of refraction is \(r\), then \(i + r = 90^\circ\).

Step 3: Using Snell's Law

Snell's law states \(n_1 \sin i = n_2 \sin r\). With relative refractive indices, where air has an index of 1 and glass 1.62, we have:\[\sin i = 1.62 \sin r\]

Step 4: Substituting the Perpendicular Condition

From \(i + r = 90^\circ\), we know \(r = 90^\circ - i\). Substitute this into Snell's Law:\[\sin i = 1.62 \sin (90^\circ - i)\]Using the trigonometric identity \(\sin(90^\circ - x) = \cos x\), this becomes:\[\sin i = 1.62 \cos i\]

Step 5: Solving for the Angle of Incidence

Rearrange \(\sin i = 1.62 \cos i\) to get:\[\tan i = 1.62\]Hence, the angle of incidence \(i\) is:\[ i = \tan^{-1}(1.62) \]

Step 6: Identifying the Correct Option

The correct angle of incidence for which the reflected and refracted rays are mutually perpendicular corresponds to option (a): \(\tan^{-1}(162)\).

Key Concepts

Refractive Index

The refractive index, often denoted as 'n', is a crucial concept in understanding how light travels through different media. It is a measure of how much the speed of light is reduced inside a medium compared to the speed of light in a vacuum, which is always considered 1.0.
For a medium like glass, the refractive index will be higher than that for air, indicating light travels slower inside the glass.

In our problem, the refractive index of the glass slab is given by 1.62 relative to air. This means when light enters the glass from air, its speed is about 1.62 times slower. The refractive index influences the bending, or refraction, of the light ray at the interface between two media.

• The higher the refractive index, the more bending occurs.
• This index aids in calculating the angles of incidence and refraction using Snell's Law.

Understanding how refractive index works helps in predicting how light will behave as it moves between substances of varying optical densities.

Angle of Incidence

The angle of incidence is the angle at which an incoming ray of light strikes the surface of a new medium. It is measured from the normal, which is an imaginary line perpendicular to the surface at the point of contact.
In optics, understanding this angle is vital as it determines how much the light will bend upon entering the new medium.

In the given problem, we seek the specific angle of incidence where reflected and refracted rays become perpendicular. This scenario is unique because it allows us to apply both Snell’s Law and certain geometrical relationships between angles. We started with knowing:

• The light approaches the glass slab, forming a specific angle of incidence 'i'.
• When the angle of incidence is such that the reflected and refracted rays are 90 degrees apart, we use Snell's Law to relate these angles.

Overall, the angle of incidence is a fundamental angle necessary for predicting light's behavior as it crosses into new environments.

Angle of Refraction

The angle of refraction is the angle between the refracted ray and the normal line at the point where the light enters a new medium. It represents the change in direction that occurs as light moves between substances with different refractive indices. This angle is intrinsic to understanding how a light wave transitions from one medium to another.

In our context, determining the angle of refraction relies heavily on both the angle of incidence and the refractive index of the glass slab, as formulated by Snell's Law:

• Snell’s law, expressed as \( n_1 \sin i = n_2 \sin r \), relates these key angles and media indices.
• For the problem, we know \( r = 90^\circ - i \), a vital condition under the perpendicular requirement.

This condition lets us derive the relationship \( \tan i = 1.62 \), ultimately aiding in solving for the precise angle of incidence needed. The angle of refraction, hence, is deeply tied to the setup and outcome of bending light through differing materials.