Problem 36 - Master Resource Book in JEE Main Physics

Question

A ray of light from a denser medium strikes a rarer medium at angle of incidence \(\angle\) The reflected and refracted rays make an angle of \(90^{\circ}\) with each other. The angles of reflection and refraction are \(r\) and \(r^{\prime}\) respectively. The critical angle is (a) \(\sin ^{-1}\left(\tan r^{*}\right)\) (b) \(\sin ^{-1}(\tan r)\) (c) \(\tan ^{-1}\left(\tan t^{\prime}\right)\) (d) \(\tan ^{-1}(\tan i]\)

Step-by-Step Solution

Answer

The critical angle is (b) \(\sin^{-1}(\tan r)\).

Step 1: Understand the scenario

A ray of light is passing from a denser medium to a rarer medium, which means it can both reflect and refract at the boundary. The reflected and refracted rays make an angle of 90 degrees with each other.

Step 2: Recall relevant concepts

The reflection angle (\(r\)) is equal to the angle of incidence for a flat boundary as per the law of reflection. The refraction angle (\(r'\)) is related to the angle of incidence by Snell's Law: \(n_1 \sin i = n_2 \sin r'\).

Step 3: Apply the 90-degree angle condition

Since the reflected ray makes a 90-degree angle with the refracted ray, \(r + r' = 90^{\circ}\). This implies \(r' = 90^{\circ} - r\).

Step 4: Use Snell's Law

Snell's Law provides the relation \(n_1 \sin i = n_2 \sin r'\). Substituting \(r' = 90^{\circ} - r\) (hence \(\sin r' = \cos r\)), we have: \(n_1 \sin i = n_2 \cos r\).

Step 5: Consider the critical angle condition

At the critical angle, \(r' = 90^{\circ}\), and thus \(\sin r' = 1\). We can solve the equation from step 4: \(\sin c = \frac{n_2}{n_1}\), where \(c\) is the critical angle.

Step 6: Use trigonometric identity

Given \(\sin r' = \cos r\), then \(\cos r = \frac{n_1}{n_2} \sin i\). For the critical angle, \(\tan r = \frac{\sin i}{\cos i}\), implying \(\sin c = \tan r\). The critical angle is \(c = \sin^{-1}(\tan r)\), matching option (b).

Key Concepts

Critical Angle

The critical angle is a key concept when light travels from a medium with higher refractive index (denser) to one with a lower refractive index (rarer). This angle is the point at which light, instead of refracting into the second medium, is entirely reflected back into the first medium, a phenomenon known as total internal reflection.

Let's explore how this relates to Snell's Law. When the angle of incidence exceeds this critical angle, the refractive ray no longer exists, and reflection is complete. To determine the critical angle ( \(c \)), we use the formula: \[ \sin c = \frac{n_2}{n_1} \]where \(n_1\) and \(n_2\) are the refractive indices of the denser and rarer media, respectively. This relation arises because at the critical angle, the angle of refraction is \(90^{\circ}\), causing \(\sin(90^{\circ}) = 1\).

Understanding this concept helps in applications like fiber optics, where light is kept entirely within the optical fibers due to total internal reflection, enabling efficient data transmission.

Refraction

Refraction is the bending of light as it passes from one medium into another, changing speed and direction. This behavior is governed by Snell's Law, which states that the angle of incidence and refraction are related by the refractive indices of the two media.Snell's Law is expressed as:\[ n_1 \sin i = n_2 \sin r' \]where:

• \(n_1\) and \(n_2\) are the refractive indices of the first and second media, respectively.
• \(i\) is the angle of incidence (the angle at which light strikes the boundary).
• \(r'\) is the angle of refraction (the angle at which light travels in the second medium).

When light travels from a denser to a rarer medium, it bends away from the normal, while it bends towards the normal when transitioning from a rarer to a denser medium.

The knowledge of refraction is crucial in designing lenses for glasses, cameras, and even in creating optical illusions. Understanding how light bends can also help explain natural phenomena, like why a straw looks bent when placed in a glass of water.

Law of Reflection

The law of reflection is a fundamental principle that describes how light reflects off surfaces. According to this law, the angle of incidence is always equal to the angle of reflection.Here's how it works:

• The angle of incidence \( i \) is the angle between the incoming ray and a line perpendicular to the surface at the point of incidence, known as the normal line.
• The angle of reflection \( r \) is the angle between the reflected ray and the normal line.

The relation can be expressed as:\[ i = r \]This principle holds true for various surfaces and is crucial for understanding how mirrors work. It explains how an image forms on a plain mirror as light reflects at the same angle it hit the mirror.

This law is not only vital in optics but also finds applications in fields such as radar technology and even in arts, to understand how light interacts with different surfaces.