Question

For four different transparent medium \(\mathrm{n}_{41} \times \mathrm{n}_{12} \times \mathrm{n}_{21}=\) (A) \(\left(1 / \mathrm{n}_{41}\right)\) (B) \(\mathrm{n}_{41}\) (C) \(\mathrm{n}_{14}\) (D) \(\left(1 / \mathrm{n}_{14}\right)\)

Short answer

The correct answer is (A) \( \left(1 / \mathrm{n}_{41}\right) \).

Step-by-step solution

Understand the properties of refractive indices

Refractive index (n) is a dimensionless ratio that describes how light propagates through a medium. It is defined as the ratio of the speed of light in vacuum (c) to the speed of light in the medium (v): \( n = \frac{c}{v} \). When light travels from one medium to another, it changes its speed according to the refractive index of the second medium.

Determine the relationship between the given refractive indices

Let's examine the given refractive indices: n41, n12, and n21. We can rewrite these as \( n_{41} = \frac{n_4}{n_1} \), \( n_{12} = \frac{n_1}{n_2} \), and \( n_{21} = \frac{n_2}{n_1} \), where n1, n2, and n4 are the refractive indices of medium 1, medium 2, and medium 4 respectively.

Compute the product n41 × n12 × n21

Now, let's compute the product of the given refractive indices: \( n_{41} \times n_{12} \times n_{21} \). \[ n_{41} \times n_{12} \times n_{21} = \frac{n_4}{n_1} \times \frac{n_1}{n_2} \times \frac{n_2}{n_1} = \frac{n_4}{1} = n_{41}^{-1} \]

Compare the computed value with the given options

Comparing the computed value with the given options, we see that it matches option (A): \( n_{41} \times n_{12} \times n_{21} = \left(1 / \mathrm{n}_{41}\right) \) So, the correct answer is (A) \( \left(1 / \mathrm{n}_{41}\right) \).

Key concepts

Light Propagation

Light propagation is the process by which light travels through different media. When light encounters a new medium, its speed changes due to the medium's refractive index. This change in speed causes light to bend at the interface between two media—a phenomenon known as refraction. The refractive index of a medium () is crucial for understanding this behavior.

• The speed of light is fastest in a vacuum, with a speed denoted by c.
• In other media, light slows down, and the ratio of c to the speed of light in the medium (v) is the refractive index: n = \( \frac{c}{v} \).

The higher the refractive index, the slower light moves in the medium. Understanding light propagation is fundamental for exploring topics like fiber optics and lens design, which rely on precise control and manipulation of light paths.

Optics

Optics is the branch of physics that studies light and its interactions with matter. It encompasses various phenomena, such as reflection, refraction, diffraction, and interference. These concepts are essential in designing optical instruments like telescopes, glasses, and microscopes. In the context of the exercise, we focus on refraction, a key aspect of optics, which occurs due to a change in light propagation speed in different materials.
Refraction leads to the bending of light rays at the boundary between different transparent media. This bending depends on the refractive indices of the involved media.

• Snell's Law describes the relationship between angles and refractive indices when light passes through different media: \(\ n_1 \sin\theta_1 = n_2 \sin\theta_2 \).
• Optics helps in understanding how lenses focus light and form images by utilizing refraction and reflection principles.

Thus, a deeper grasp of optics allows for better comprehension of how light can be directed and manipulated for various technological applications.

Transparent Medium

A transparent medium is a material through which light can travel with little attenuation. Such mediums include air, glass, and water, which allow light to pass while being minimally absorbed. The concept of transparency is integral to fields like optics and photonics, as it determines how well and what type of optical phenomena occur within the material.

• In transparency, the alignment and structure of molecules often allow light waves to pass without significant scattering or absorption.
• Refraction, reflection, and transmission are all influenced by the medium’s transparency, altering how we perceive light passing through it.

The refractive index indicates computationally how transparent a material is, with lower indices often associated with higher transparency.