Question

A light ray is incident perpendicular to one face of \(90^{\circ}\) prism and is totally internally reflected at the glass-air interface. If the angle of reflection is \(45^{\circ} .\) We conclude that the refractive index \(\ldots\) (A) \(\mu>\sqrt{2}\) (B) \(\mu>(1 / \sqrt{2})\) (C) \(\mu<\sqrt{2}\) (D) \(\mu<(1 / \sqrt{2})\)

Short answer

(A) \(\mu>\sqrt{2}\)

Step-by-step solution

Draw the Prism

Draw a 90° prism with a light ray incident perpendicularly to one face.

Identify the Perpendicular Angle

Since the light ray is incident perpendicularly to the prism's surface, the angle of incidence is 0°.

Use Snell's Law on the Incident Ray

Apply Snell's Law to the incident ray, which relates the angle of incidence (\(i\)) and the angle of refraction (\(r\)) with the refractive index \(\mu\): \(\mu = \frac{\sin i}{\sin r}\)

Input the Known Values and Solve for r

We are given the angle of incidence, \(i = 0°\). The refractive index, \(\mu\), is unknown. Thus, we can rewrite Snell's Law as: \(\mu = \frac{\sin 0^\circ}{\sin r} \Rightarrow \mu\sin r = 0\) Since \(\mu \ne 0\), then \(\sin r = 0\) which implies that the angle of refraction, \(r = 0^\circ\).

Apply Amici's Law

Amici's Law states that for any prism, the angles of incidence (\(a\)) and deviation (\(d\)) satisfy the following condition: \(a - r + d = 90^\circ\)

Find d using the Refraction Angle

In our case, the angle of refraction, \(r = 0^\circ\). Thus, the angle of deviation, \(d\), can be found by substituting the values of \(a\) and \(r\) in Amici's Law: \(90^\circ - 0^\circ + d = 90^\circ \Rightarrow d = 90^\circ\)

Apply the Critical Angle Formula

The critical angle, \(C\), is the angle at which a light ray is refracted at 90° relative to the normal. It is calculated as: \(C = \arcsin(\frac{1}{\mu})\)

Apply Total Internal Reflection Condition

For Total Internal Reflection to occur, the angle of incidence (\(d\)) must be greater than or equal to the critical angle (\(C\)): $d \ge C \Rightarrow 90^\circ \ge \arcsin(\frac{1}{\mu})\)

Identify the Condition for Refractive Index

Using the information from Step 8, we can find the condition for the refractive index \(\mu\): \(\arcsin\left(\frac{1}{\mu}\right) \le 45^\circ \Rightarrow \frac{1}{\mu} \le \sin 45^\circ\) Since \(\sin 45^\circ = \frac{1}{\sqrt{2}}\), the condition for the refractive index is: \(\mu \ge \sqrt{2}\) Thus, the correct answer is: (A) \(\mu>\sqrt{2}\)

Key concepts

Critical Angle

The critical angle is a pivotal concept in understanding how light behaves at the boundary between two different media, like glass and air. When light travels from a medium with a higher refractive index to a lower one, it bends away from the normal. At some specific angle, known as the critical angle, the refracted ray slides along the boundary. Beyond this angle, total internal reflection occurs, and the light doesn't exit the medium at all. The formula to calculate the critical angle, often denoted as \( C \), is:\[C = \arcsin\left(\frac{1}{\mu}\right)\]where \( \mu \) is the refractive index of the initial medium relative to the second medium.

• When light hits at the critical angle, it refracts at 90° along the boundary.
• If the angle of incidence surpasses the critical angle, total internal reflection keeps the light inside the first medium.

Thus, understanding the critical angle is crucial to predicting when a ray is trapped by total internal reflection.

Snell's Law

Snell's Law is key to understanding how light bends as it passes between two different media. This foundational principle relates the angle of incidence and the angle of refraction to the refractive indices of both materials. The law is given by the formula:\[\mu_1 \sin i = \mu_2 \sin r\]where:

• \( \mu_1 \) and \( \mu_2 \) are the refractive indices of the two media.
• \( i \) is the angle of incidence.
• \( r \) is the angle of refraction.

For example, if light enters a prism perpendicular to its face, Snell's Law predicts the behavior of the light within the prism. Notably, if the incident angle were zero degrees (as in the exercise), the sin of that angle is zero, showing no refraction. Snell's Law is essential in optics, explaining effects like lens focusing and the spread of rainbows.

Refractive Index

The refractive index \( \mu \) is a dimensionless number that describes how fast light travels through a material. It is defined as the speed of light in a vacuum divided by the speed of light in the medium. The relationship is:\[\mu = \frac{c}{v}\]where:

• \( c \) is the speed of light in a vacuum.
• \( v \) is the speed of light in the medium.

A higher refractive index indicates that light travels more slowly in that medium. In the context of the exercise, knowing the refractive index helps determine how much a ray bends when entering or exiting the prism. Total internal reflection relies on this principle because only light traveling from a medium with a higher refractive index to one with a lower refractive index can experience it. For example, light within a glass prism being internally reflected meets these criteria, highlighting \( \mu \)'s importance.

Prism

A prism is a geometric and optical element made of a transparent material with flat, polished surfaces that refract light. Prisms are renowned for dispersing light into its component colors, demonstrating the spectrum. However, in the context of the exercise, we focus on how prisms can direct and contain light through total internal reflection.

• Prisms often have a triangular shape and are characterized by their angles, like the 90° angle in the exercise.
• Light entering perpendicular to one face travels without deviation until it encounters another surface at an adequate angle for total internal reflection.

This behavior allows prisms to trap light and is pivotal in tools like periscopes and binoculars. The refractive properties, including the material's refractive index, play a role in where and how light exits after reflection. Understanding the geometry and refractive characteristics of a prism clarifies why light behaves as it does within the prism's confines.