Question

Choose the correct statement. (A) The speed of light in the meta-material is \(v=\mathrm{c}|n|\) (B) The speed of light in the meta-material is \(v=\frac{c}{|n|}\) (C) The speed of light in the meta-material is \(v=c\). (D) The wavelength of the light in the meta-material \(\left(\lambda_{m}\right)\) is given by \(\lambda_{m}=\lambda_{\text {air }}|n|\), where \(\lambda_{\text {air }}\) is the wavelength of the light in air.

Short answer

The correct statement is (B) The speed of light in the meta-material is \(v=\frac{c}{|n|}\).

Step-by-step solution

Understanding the Speed of Light in Different Media

The speed of light in any medium is given by the equation v = c/n, where v is the speed of light in the medium, c is the speed of light in vacuum, and n is the refractive index of the medium. The refractive index is a dimensionless number which indicates how much the speed of light is reduced in the medium compared to its speed in a vacuum.

Analyzing the Options Given

(A) This option incorrectly uses the absolute value of n, which is not necessary since the refractive index is already a positive quantity that represents the relative speed reduction. (B) This option correctly presents the relationship between the speed of light in a medium and the refractive index. This is the standard equation for the speed of light in a material. (C) This suggests that the speed of light is the same in the material as in vacuum, which is not true for materials with a refractive index different from 1. (D) This option is addressing the wavelength, not the speed, of light in the material. The wavelength of light in a medium can be found by dividing the wavelength in air by the refractive index, λ_m = λ_air / n, not by multiplying it by the absolute value of n.

Selecting the Correct Option

Based on our analysis, the equation v = c/n correctly describes the speed of light in a medium with refractive index n. Therefore, we have to choose the option that accurately reflects this relationship without the incorrect use of absolute value.

Key concepts

Refractive Index

The refractive index, often denoted by the symbol 'n', is a crucial measure in optics as it describes how light propagates through different materials. Essentially, it is the ratio of the speed of light in a vacuum, which is a constant approximately equal to 299,792,458 meters per second, to the speed of light in the material. If we were to imagine the journey of a beam of light, as it moves from a vacuum into another medium, its velocity slows down due to the interaction with the medium's particles. This slowing down is precisely what the refractive index quantifies.

The greater the refractive index of a medium, the more it slows down the light passing through it, and the more the light path bends. This bending is often observed as light goes from air into water, causing objects to appear at a different position when looked at from above the surface. In mathematical terms, we express the speed of light in any medium using the equation: \( v = \frac{c}{n} \), where 'v' represents the speed of light within the material. Knowing the refractive index is also vital for designing lenses and optical devices, allowing us to predict how they will manipulate light to serve various functions like focusing or dispersion.

Meta-materials

Meta-materials are engineered materials with properties not found in naturally occurring substances. They are crafted to have a unique internal structure that can manipulate electromagnetic waves in unconventional ways. This manipulation is often achieved through the arrangement of multiple elements smaller than the wavelength of the light being affected. Imagine them as complex puzzles, where each piece is strategically placed to control how light traverses the overall structure.

These materials can bend light to such an extent that they can make objects appear invisible, a concept commonly known as 'cloaking'. This is possible due to their negative refractive index—a counterintuitive notion—as conventional materials have positive refractive indices. In the context of our problem, we need to understand that the equations governing light's behavior in standard materials might not directly apply to meta-materials, especially if negative refractive indices are involved. They challenge our traditional understanding of optics and open the door to innovative technologies in telecommunications, imaging, and beyond.

Wavelength of Light

The wavelength of light is the distance over which a wave's shape repeats. It is one of the fundamental properties of light that determines its color within the visible spectrum. However, the wavelength of light is not fixed; it changes depending on the medium through which the light travels. The equation \( \lambda_m = \frac{\lambda_{\text{air}}}{n} \) helps us calculate the wavelength of light in a medium, where \( \lambda_m \) is the wavelength in the medium, \( \lambda_{\text{air}} \) is the wavelength in air (or vacuum, which can be considered equivalent for most cases), and 'n' is the refractive index of the medium.

As light enters a medium like water or glass from air, it not only slows down due to the higher refractive index but also experiences a change in wavelength. The frequency, however, remains constant. This implies that the energy of a photon is also unaltered because it is directly proportional to frequency. Understanding the interplay between wavelength and refractive index is essential for applications such as fiber-optic communication and various types of spectroscopy, where precise control over light is needed.