Problem 86 - 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 water air interface is \(48.6^{\circ}\). What is the refractive index of water? (a) (b) \(3 / 2\) (c) \(4 / 3\) (d) \(3 / 4\)

Step-by-Step Solution

Answer

The refractive index of water is \(4/3\) (option c).

Step 1: Understanding the Formula

The formula given is \( \mu = \frac{1}{\sin C} \), where \( \mu \) is the refractive index of the denser medium (water, in this case), and \( C \) is the critical angle for total internal reflection.

Step 2: Identify Given Values

The problem provides the critical angle \( C \) for the water-air interface as \( 48.6^{\circ} \). This will be used in the formula to calculate the refractive index of water.

Step 3: Calculate \( \sin C \)

Calculate the sine of the critical angle: \( \sin 48.6^{\circ} \). Using a calculator, \( \sin 48.6^{\circ} \approx 0.7497 \).

Step 4: Apply the Formula

Substitute \( \sin C \) into the formula: \[ \mu = \frac{1}{\sin 48.6^{\circ}} = \frac{1}{0.7497} \].

Step 5: Calculate \( \mu \)

Perform the division to find \( \mu \): \( \mu \approx \frac{1}{0.7497} \approx 1.334 \).

Step 6: Match the Solution to Options

The calculated refractive index \( \mu \approx 1.334 \) matches option (c) \( \frac{4}{3} \), as \( 1.3333 \approx \frac{4}{3} \).

Key Concepts

Critical Angle

The critical angle is a specific angle of incidence above which light cannot pass through a medium but is instead completely reflected. This property is observed only when light travels from a denser medium to a rarer one, such as from water to air. For light to reflect internally, the angle of incidence must exceed this critical angle. The critical angle can be determined by the refractive index of the two media in contact. Some important points about critical angles include:

• It is different for every pair of media.
• Occurs only when light travels from denser to rarer medium.
• At the critical angle, the refracted ray skims along the interface.

For instance, in the water-air scenario, light shining from the water into the air must hit at an angle greater than 48.6° (the critical angle) for total internal reflection to occur.

Refractive Index

The refractive index is a measure of how much a substance can bend or refract light. It indicates the speed at which light travels through the medium in comparison to the speed of light in a vacuum. The higher the refractive index, the slower light travels in the medium and the more it bends.Key aspects of the refractive index are:

• A ratio of the speed of light in a vacuum to its speed in the medium.
• Represented by the symbol \( \mu \).
• Can be calculated using the formula \( \mu = \frac{1}{\sin C} \) when the critical angle \( C \) is known.

In our exercise, the refractive index of water, calculated with the given critical angle, is approximately 1.334. This means that light travels 1.334 times slower in water than in a vacuum.

Snell's Law

Snell's Law is fundamental in understanding how light refracts or bends when it passes through different mediums. It mathematically expresses the relationship between the angles of incidence and refraction and the indices of refraction of the two media. The law is given by:\[ n_1 \sin \theta_1 = n_2 \sin \theta_2 \]Where:

• \( n_1 \) and \( n_2 \) are the refractive indices of medium 1 and medium 2, respectively.
• \( \theta_1 \) is the angle of incidence.
• \( \theta_2 \) is the angle of refraction.

Using Snell's Law, if light travels from a medium with a high refractive index to a medium with a lower one, and if the angle of incidence is greater than the critical angle, total internal reflection occurs instead of refraction. This principle was crucial in solving the exercise and determining the refractive index of water when light moves from water to air.