Question

A ray of light traveling in water is incident on its surface open to air. The angle of incidence is \(\theta\), which is less than the critical angle. Then there will be (A) only a reflected ray and no refracted ray (B) only a refracted ray and no reflected ray (C) a reflected ray and a refracted ray and the angle between them would be less than \(180^{\circ}-2 \theta\) (D) a reflected ray and a refracted ray and the angle between them would be greater than \(180^{\circ}-2 \theta\)

Short answer

There will be a reflected ray and a refracted ray, and the angle between them would be greater than \(180^\circ-2\theta\) (Option D).

Step-by-step solution

Understand the concept of Refraction and Reflection

When a ray of light travels from one medium to another, it generally undergoes both reflection and refraction if the angle of incidence is less than the critical angle. Refraction occurs as the light passes into a different medium and changes direction, adhering to Snell's law. Reflection happens according to the law of reflection, where the angle of incidence equals the angle of reflection.

Analyze the given conditions

The light is traveling from water to air and the angle of incidence is less than the critical angle. This means some of the light will be refracted into the air according to Snell's law, and some of it will be reflected back into the water, obeying the law of reflection.

Determine the behavior of reflected and refracted rays

Since the angle of incidence is less than the critical angle, we will indeed have both a reflected and a refracted ray. The angle of reflection in the water will be equal to the angle of incidence, and the refracted ray will deviate from the normal line.

Calculate the angle between the reflected and refracted rays

If the angle of the refracted ray with the normal is \(\theta_r\), then according to Snell's law, \(\sin\theta = n_{air}\sin\theta_r\), where \(n_{air} \) is the refractive index of air. The angle between the reflected and refracted rays inside the water is therefore \(180^\circ - (\theta + \theta_r)\).

Determine the angle relationship

Since \(\theta_r\) is different from \(\theta\) due to the change in medium, we can infer that the sum of the angles \(\theta + \theta_r\) will not be equal to \(2\theta\). Thus, the angle between the reflected and refracted ray will be greater than \(180^\circ-2\theta\).

Key concepts

Snell's Law

Snell's law is a fundamental concept in the study of optics, particularly when discussing the behavior of light as it passes from one medium to another. This law relates the angle of incidence to the angle of refraction, connected through the refractive indices of the two media. Mathematically, Snell's law is expressed as:
\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \]
where \(n_1\) is the refractive index of the first medium, \(\theta_1\) is the angle of incidence, \(n_2\) is the refractive index of the second medium, and \(\theta_2\) is the angle of refraction. When light travels from a medium with a higher refractive index to one with a lower refractive index, the angle of refraction is greater than the angle of incidence, causing the light to bend away from the normal. This bending alters the direction of the light wave and is why objects may look 'bent' or 'shifted' when viewed through a water surface or a lens.

• Snell's Law is crucial to understanding lens design, optical fibers, and even natural phenomena like mirages.
• The precise prediction of the refracted ray's angle requires knowledge of both media's refractive indices.
• This law is essential for solving problems involving the path of light between mediums, like the exercise described.

Critical Angle

The notion of the critical angle is significant in the context of light travelling from a denser to a rarer medium, such as water to air. The critical angle is defined as the smallest angle of incidence for which light gets refracted and emerges along the boundary. Any incidence angle larger than this will result in total internal reflection, without any refraction occurring.

Mathematically, the critical angle \( \theta_c \) can be found using the refractive indices of the two media:
\[ \theta_c = \sin^{-1} \left( \frac{n_2}{n_1} \right) \]
with \(n_1\) being the refractive index of the denser medium and \(n_2\) that of the rarer medium. Whenever light strikes a boundary at an angle less than the critical angle, both refracted and reflected rays are produced. This underpins the scenario presented in the original exercise where the light ray does not exceed the critical angle, resulting in a refracted ray into the air and a reflected ray within the water.

• The critical angle plays a crucial role in fiber optic technology, allowing light to be confined within the fiber, thus enabling data transmission over long distances.
• Understanding and calculating the critical angle is necessary when dealing with total internal reflection applications.

Law of Reflection

The law of reflection is a principle that applies to both specular (mirror-like) and diffuse (scattering) reflections of light. According to this law, the angle at which the light ray arrives, or the angle of incidence, is exactly equal to the angle at which the light ray leaves, or the angle of reflection, with respect to the normal to the surface.
This can be written as:
\[ \theta_{\text{incident}} = \theta_{\text{reflected}} \]
The incident ray, the reflected ray, and the normal line at the point of incidence all lie in the same plane. In the context of the exercise given, when light from the water strikes the water-air boundary with an angle of incidence less than the critical angle, it adheres to both Snell's law and the law of reflection. This results in part of the light being reflected within the water at an angle equal to its angle of incidence, and part refracted into the air, with its direction changed according to Snell's law.

• The law of reflection is used in the design of optical devices such as mirrors, periscopes, and telescopes to direct light predictably.
• A surface's ability to reflect light perfectly without scattering is known as its reflectivity, and is central to how we see and use various objects.