Question

The velocity of light in glass whose refractive index with respect to air is \(1.5\) is \(2 \times 10^{8} \mathrm{~m} / \mathrm{s}\). In a certain liquid, the velocity of light is found to be \(2.5 \times 10^{8} \mathrm{~m} / \mathrm{s}\). What is the refractive index of the liquid with respect to air? (A) \(1.44\) (B) \(0.80\) (C) \(1.20\) (D) \(0.64\)

Short answer

The refractive index of the liquid with respect to air is \(1.20\).

Step-by-step solution

Recall the relation between refractive index, speed of light in vacuum and speed of light in a medium

The refractive index (n) of a medium is defined as the ratio of the speed of light (c) in vacuum to the speed of light (v) in that medium: \[n = \frac{c}{v}\]

Given values and required value

We are given the following values in the problem: Refractive index of glass with respect to air (ng) = 1.5 Speed of light in glass (vg) = \(2 \times 10^8 m/s\) Speed of light in liquid (vl) = \(2.5 \times 10^8 m/s\) We need to find the refractive index of the liquid with respect to air (nl).

Find the speed of light in vacuum

We can find the speed of light in vacuum (c) using the formula for the refractive index of the glass: \[n_g = \frac{c}{v_g}\] Rearranging the formula, we can find the value of c as: \[c = n_g \times v_g\] Substituting the given values, \[c = 1.5 \times (2 \times 10^8) m/s = 3 \times 10^8 m/s\]

Calculate the refractive index of the liquid with respect to air

Now that we have the value of the speed of light in vacuum, we can find the refractive index of the liquid with respect to air using the same formula for refractive index (n): \[n_l = \frac{c}{v_l}\] Substituting the given values, \[n_l = \frac{3 \times 10^8}{2.5 \times 10^8} = \frac{3}{2.5} = 1.20\]

Compare the calculated refractive index with the given options

We have found that the refractive index of the liquid with respect to air is 1.20. Comparing this value with the given options, we find that it matches with option (C). So, the correct answer to this problem is option (C) \(1.20\).

Key concepts

Speed of Light

Understanding the speed of light is crucial in grasping the concept of refractive index in optics. The speed of light, often denoted by the letter 'c', is a constant in a vacuum and is approximately 299,792,458 meters per second (usually rounded to 3 × 10⁸ m/s). This constant speed is what makes light one of the universe’s fundamental constants.
However, when light travels through different materials (like glass or water), it slows down due to interactions with the material's particles. The velocity of light in a medium (v) becomes slower than c.
This slower speed in a medium is crucial for determining properties like refractive index. Knowing c and comparing it to v in a particular material enables scientists and engineers to calculate and understand the light behavior in various environments.

Optics

Optics is the branch of physics that deals with the study of light and its interactions with various materials. A primary concept in optics is the refractive index, which tells us how much light slows down when it enters a different medium.
Using the refractive index (n), we can predict how light will travel through substances, which is essential for designing lenses, glasses, and other optical instruments.
In simple terms, the refractive index is a measure of how much slower light travels through a medium compared to vacuum. For example, in glass with a refractive index of 1.5, light travels 1.5 times slower than it does in vacuum. This reduction in speed is what causes light to bend or refract when changing mediums, leading to phenomena like rainbows or the apparent bending of a straw in water.

Medium

The term medium in optics refers to any material through which light can travel, such as air, water, or glass. Each medium affects the speed of light differently, which is why we observe variation in refractive indices.
When light enters a medium, its speed changes, a property characterized by the medium's refractive index. This index defines how much the light's path bends as it enters or exits the material.
Each medium has its unique refractive index that can be calculated using the formula \(n = \frac{c}{v}\), where c is the speed of light in vacuum, and v is the speed of light within the medium. Understanding the concept of medium helps in numerous applications, from creating better lenses and prisms to improving fiber optic communication systems.