Problem 52 - Master Resource Book in JEE Main Physics

Question

Light takes \(t_{1}\) second to travel a distance \(x\) in vacuum and the same light takes \(t_{2}\) second to travel \(10 x \mathrm{~cm}\) in a medium. Critical angle for corresponding medium will be (a) \(\sin ^{-1}\left(\frac{10 t_{2}}{t_{1}}\right)\) (b) \(\sin ^{-1}\left(\frac{t_{2}}{10 t_{1}}\right)\) (c) \(\sin ^{-1}\left(\frac{10 t_{1}}{t_{2}}\right)\) (d) \(\sin ^{-1}\left(\frac{t_{1}}{10 t_{2}}\right)\)

Step-by-Step Solution

Answer

(c) \(\sin^{-1}\left(\frac{10t_1}{t_2}\right)\) is the critical angle.

Step 1: Calculate Speed of Light in Vacuum

Given that light takes \(t_1\) seconds to travel a distance \(x\) in vacuum, the speed of light in vacuum \(c\) can be expressed as \[c = \frac{x}{t_1}.\]

Step 2: Calculate Speed of Light in Medium

Given that the same light takes \(t_2\) seconds to travel \(10x\) cm in the medium, the speed of light in the medium \(v\) is computed as \[v = \frac{10x}{t_2}.\]

Step 3: Determine Refractive Index of the Medium

The refractive index \(n\) of the medium is the ratio of the speed of light in vacuum to the speed in the medium: \[n = \frac{c}{v} = \frac{\frac{x}{t_1}}{\frac{10x}{t_2}} = \frac{t_2}{10t_1}.\]

Step 4: Determine the Critical Angle

The critical angle \(\theta_c\) is given by \(\theta_c = \sin^{-1}\left(\frac{1}{n}\right)\). Substituting the expression for \(n\): \[\theta_c = \sin^{-1}\left(\frac{1}{\frac{t_2}{10t_1}}\right) = \sin^{-1}\left(\frac{10t_1}{t_2}\right).\]

Step 5: Choose the Correct Option

From the given options, the correct expression for the critical angle is \(\sin^{-1}\left(\frac{10t_1}{t_2}\right)\), which corresponds to option (c).

Key Concepts

Refractive Index

The refractive index, often denoted by the symbol \(n\), is a measure of how much a ray of light bends when it enters a medium from a vacuum. It is a crucial concept in understanding how light propagates through different materials. Given by the ratio of the speed of light in vacuum \(c\) to the speed of light in the medium \(v\), the refractive index is calculated as:

• \( n = \frac{c}{v} \)

A higher refractive index indicates that light travels slower in the medium. Consequently, light bends more sharply upon entering the medium. This bending of light is what allows lenses to focus light, and for optical fibers to guide light over distances. Furthermore, the refractive index can be used to find critical angles, which are vital in applications like fiber optics where total internal reflection is used to trap light inside a medium.

Speed of Light in Vacuum

The speed of light in vacuum is one of the fundamental constants in physics. Represented commonly as \(c\), it is approximately \(3 \times 10^8\) meters per second. This speed is the ultimate speed limit of the universe. No information or matter can travel faster than this, according to current physical theories. In vacuum, light travels without encountering resistance, allowing it to maintain this constant velocity. Understanding this speed is essential, as it provides a baseline for comparing how light behaves in other media. When light transitions from vacuum into another medium like water or glass, it slows down, which leads to the phenomenon of refraction. Because of this crucial role, the speed of light in a vacuum is used as a reference point for calculating other optical properties such as the refractive index.

Speed of Light in Medium

When light travels through a medium other than vacuum, its speed decreases. This happens because light interacts with the particles within the medium, which causes it to slow down. The speed of light in a specific medium is denoted as \(v\) and varies depending on the medium's density and composition.For example:

• In water, the speed of light is about \(2.25 \times 10^8\) meters per second.
• In glass, it might drop to around \(2 \times 10^8\) meters per second.

This reduction in speed is responsible for the bending or refraction of light as it enters the surface of a new medium at an angle. Knowing this speed allows us to compute the refractive index, which tells us how much the light will bend. Hence, understanding how light slows in different media is central to the fields of optics and photonics, impacting technologies like lenses, microscopes, and even corrective eyewear.