Problem 88 - Master Resource Book in JEE Main Physics

Question

Total internal reflection is the phenomenon of reflection of light into denser medium at the interface of denser medium with a rarer medium. Light must. travel from denser to rarer and angle incidence denser medium must be greater than critical angle (C) for the pair of media in contact. We can show that $$ \mu=\frac{1}{\sin C} $$ Critical angle for glass air-interface where refractive index \(\mu\) of glass is \(3 / 2\), is (a) \(41.8^{*}\) (b) \(60^{*}\) (c) \(30^{*}\) (d) \(44.3^{*}\)

Step-by-Step Solution

Answer

The critical angle is approximately \(41.8^\circ\).

Step 1: Identify Given Values and Formula

The refractive index \( \mu \) of the glass is given as \( \frac{3}{2} \). The relationship between the refractive index and the critical angle \( C \) is expressed as \( \mu = \frac{1}{\sin C} \). We need to find the critical angle for the glass-air interface.

Step 2: Rearrange Formula to Find Critical Angle

Rearrange the formula \( \mu = \frac{1}{\sin C} \) to solve for \( \sin C \): \( \sin C = \frac{1}{\mu} \). Substituting the given refractive index, we have \( \sin C = \frac{1}{\frac{3}{2}} = \frac{2}{3} \).

Step 3: Calculate Critical Angle

Calculate the critical angle \( C \) by taking the arcsine of \( \sin C \). Use a calculator to find \( C = \arcsin \left( \frac{2}{3} \right) \). This calculation results in \( C \approx 41.8^\circ \).

Key Concepts

Critical Angle

The critical angle is an important concept when you study light refraction. It is the angle of incidence equal to the angle at which light, traveling from a denser medium to a less dense one, is refracted at 90 degrees along the boundary. This means the refracted light travels along the surface, rather than entering the less dense medium. The significance of the critical angle is that any angle of incidence larger than this will lead to total internal reflection.

To calculate the critical angle, we use the equation \( \mu = \frac{1}{\sin C} \), where \( \mu \) is the refractive index of the denser medium and \( C \) is the critical angle. By rearranging this equation, we find \( \sin C = \frac{1}{\mu} \). This formula helps to find the critical angle if we know the refractive index. For instance, when the refractive index is given as \( \frac{3}{2} \), we can find the critical angle by calculating \( C = \arcsin\left(\frac{2}{3}\right) \), resulting in approximately \( 41.8^\circ \). This calculated angle tells us when light will begin completely reflecting within the medium rather than passing through.

Refractive Index

The refractive index is a measure of how much a medium can bend light rays. It dictates how much a light beam will change its speed and direction when it enters a new medium. Every material has its own unique refractive index; for instance, glass has a higher refractive index than air, which means light slows down more in glass than in air, affecting how much it bends.

The refractive index (μ) is typically given as a dimensionless number. The higher the refractive index, the slower light travels through that medium. The relationship between the refractive index and the critical angle can be described using the formula \( \mu = \frac{1}{\sin C} \). This equation is handy for determining the critical angle, given a known refractive index.

For example, if the refractive index of glass is \( \frac{3}{2} \) (or 1.5), it implies light travels 1.5 times slower in glass than in a vacuum. Plugging this value into the equation \( \sin C = \frac{1}{\mu} \), we can compute the critical angle, which in the case of a glass-air interface, is \( 41.8^\circ \). This value highlights the optical behavior of glass compared to air.

Glass-Air Interface

The glass-air interface is a common example that helps students understand concepts like total internal reflection and critical angle. It represents the boundary between a denser medium (glass) and a less dense medium (air). When light travels from glass to air, it speeds up, bending. If the angle of incidence is greater than the critical angle, the light is not refracted into the air but instead reflects back into the glass, a phenomenon known as total internal reflection.

The behavior of light at the glass-air interface explains many optical devices' workings, such as optical fibers or glass prisms. Knowing the critical angle at this interface is essential for predicting when total internal reflection will occur. For glass with a refractive index of \( \frac{3}{2} \), calculations give a critical angle of \( 41.8^\circ \). This means any incidence angle greater than \( 41.8^\circ \) will cause the light to be reflected back into the glass rather than escaping into the air.

Understanding this interface also sheds light on other practical applications, like designing lenses and improving visibility through media with different optical densities, enhancing our control over light propagation.