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Chapter 23: Problem 64

A ray of light travelling in glass \(\left(\mu=\frac{3}{2}\right)\) is incident on a horizontal glass air surface at the critical angle \(\theta_{c}\) If thin layer of water \(\left(\mu=\frac{4}{3}\right)\) is now poured on the glass air surface, the angle at which the ray emerges into air at the water-air surface is (a) \(60^{\circ}\) (b) \(45^{\circ}\) (c) \(\underline{90^{\circ}}\) (d) \(180^{*}\)

Short Answer

Expert verified
The ray emerges at a \( 90^{\circ} \) angle into the air.

Step by step solution

01

Understanding Critical Angle

The critical angle \( \theta_c \) is the angle of incidence in the denser medium (glass) for which the angle of refraction in the less dense medium (air) is \(90^{\circ}\). From Snell's law, \( n_1 \sin \theta_c = n_2 \sin 90^{\circ} \) where \( n_1 = \frac{3}{2} \) and \( n_2 = 1 \). This gives \( \sin \theta_c = \frac{1}{n_1} = \frac{2}{3} \).
02

Adding Water Layer

When water is added, the light first refracts from the glass into the water. Use Snell鈥檚 Law again: \( n_{glass} \sin \theta_c = n_{water} \sin \theta_w \). Substitute \( n_{glass} = \frac{3}{2} \), \( \sin \theta_c = \frac{2}{3} \), and \( n_{water} = \frac{4}{3} \). This gives \( \sin \theta_w = 1 \).
03

Light Path in Water

Since \( \sin \theta_w = 1 \), it follows that \( \theta_w = 90^{\circ} \). This means that the refracted ray in water travels along the water-air interface and then emerges into the air without deviating from its path.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Critical Angle
The critical angle is a concept in refraction that occurs when light travels from a denser medium to a less dense medium, such as from glass to air. At this angle, the light ray is refracted along the boundary and does not pass into the less dense medium.
To find the critical angle, we rely on Snell's Law, which is expressed as:
  • \( n_1 \sin \theta_c = n_2 \sin 90^{\circ} \)
Here, \( n_1 \) is the refractive index of the denser medium (in our case, glass), and \( n_2 \) is the refractive index of the less dense medium (air). When the angle of refraction is \(90^{\circ}\), the path of the refracted light lies along the boundary. Calculating the critical angle ensures we know the exact point at which total internal reflection occurs.
Refraction
Refraction is the bending of light as it passes through a medium with a different refractive index. When light travels from one medium to another, like from glass to air or water, its speed and direction change.
This behavior is described by Snell's Law:
  • \( n_1 \sin \theta_1 = n_2 \sin \theta_2 \),
where \( \theta_1 \) is the angle of incidence and \( \theta_2 \) is the angle of refraction. The refractive indices, \( n_1 \) and \( n_2 \), depend on the nature of the materials. If the first medium is denser than the second, the light bends away from the normal.
In the original exercise, when a layer of water is added, the path of light changes once more. Now the light has to pass from the glass into the water before emerging into the air, showcasing both the complexity and the predictability of refraction.
Optics in Physics
Optics is a branch of physics that deals with the behavior and properties of light. It encompasses the study of reflection, refraction, and the various phenomena associated with light's interactions with matter.
Key concepts in optics include:
  • Reflection: The bouncing back of light when it hits a surface.
  • Refraction: The bending of light as it moves between media with different refractive indices.
  • Dispersion: The splitting of light into its constituent colors, often seen through a prism.
Understanding optics allows us to develop technologies like lenses, glasses, and other optical instruments. In the exercise you encountered, the laws governing refraction and the critical angle help explain how light behaves as it transfers from one medium to another. Optics not only helps us explain these phenomena but also enables countless applications ranging from vision correction to sophisticated scientific instruments.

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Most popular questions from this chapter

A plane mirror is approaching you at \(10 \mathrm{cms}^{-1}\). Your image shall approach you with a speed of (a) \(+10 \mathrm{cms}^{-1}\) (b) \(-10 \mathrm{cms}^{-1}\) (c) \(+20 \mathrm{cms}^{-1}\) (d) \(-20 \mathrm{cms}^{-1}\)

A car is moving with at a constant speed of \(60 \mathrm{kmh}^{-1}\) on a straight road. Looking at the rear view mirror, the driver finds that the car following him is at distance of \(100 \mathrm{~m}\) and is approaching with a speed of \(5 \mathrm{~km} \mathrm{~h}^{-1}\). In order to keep track of the car in the rear, the driver begins to glance alternatively at the rear and side mirror of his car after every \(2 \mathrm{~s}\) till the other car overtakes. If the two cars were maintaining their speeds, which of the following statement (s) is/are correct? (a) The speed of the car in the rear is \(65 \mathrm{~km} \mathrm{~h}^{-1}\) (b) In the side mirror the car in the rear would appear to approach with a speed of \(5 \mathrm{~km} \mathrm{~h}^{-1}\) to the driver of the leading car (c) In the rear view minor, the speed of the approaching car would appeat to decrease as the distance between the. cars decreases (d) In the side mirror, the speed of the approaching car would appear to increase as the distance between the cars decreases

A concave lens of focal length \(20 \mathrm{~cm}\) produces an image half in size of the real object. The distance of the real object is (a) \(20 \mathrm{~cm}\) (b) \(30 \mathrm{~cm}\) (c) \(10 \mathrm{~cm}\) (d) \(60 \mathrm{~cm}\)

A ray light passes through an equilateral prism such that the angle of incidence and the angle of emergence are both equal to \(3 / 4\) th of the angle of prism. The angle of minimum deviation is (a) \(15^{\circ}\) (b) \(30^{\circ}\) (c) \(45^{\circ}\) (d) \(60^{\circ}\)

At what angle should a ray of light be incident on the face of a prism of refracting angle \(60^{\circ}\) so that it just suffers total internal reflection at the other face? The refractive index of the material of the prism is \(1.524\). [NCERT] (a) \(16^{\circ}\) (b) \(29^{\circ}\) (c) \(45^{\circ}\) (d) \(58^{\circ}\)
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