Problem 6 - Master Resource Book in JEE Main Physics

Question

Monochromatic light of wavelength \(589 \mathrm{~nm}\) is incident from air on a water surface. What are the wavelength and speed of refracted light? Refractive index of water is \(1.33 .\) (NCERT] (a) \(4.20 \times 10^{-6}\) and \(4.0 \times 10^{7} \mathrm{~m} / \mathrm{s}\) (b) \(3.68 \times 10^{-9} \mathrm{~m}\) and \(3.02 \times 10^{7} \mathrm{~m} / \mathrm{s}\) (c) \(1.9 \times 10^{-10} \mathrm{~m}\) and \(3.2 \times 10^{8} \mathrm{~m} / \mathrm{s}\) (d) \(4.42 \times 10^{-7} \mathrm{~m}\) and \(2.25 \times 10^{6} \mathrm{~m} / \mathrm{s}\)

Step-by-Step Solution

Answer

The wavelength of refracted light is \(4.42 \times 10^{-7} \text{ m}\), speed calculation in option (d) should be \(2.25 \times 10^8 \text{ m/s}\).

Step 1: Understand the Refraction Concepts

When monochromatic light passes from one medium to another, its speed and wavelength change according to the refractive index, but its frequency remains unchanged. In this case, light travels from air into water.

Step 2: Define Given Values

- Wavelength in air, \( \lambda_0 = 589 \text{ nm} = 589 \times 10^{-9} \text{ m}\).- Refractive index of water, \( n = 1.33 \).

Step 3: Calculate Wavelength in Water

The wavelength of light in a medium is given by \( \lambda = \frac{\lambda_0}{n} \). Substitute the given values:\[ \lambda = \frac{589 \times 10^{-9} \text{ m}}{1.33} \approx 4.42 \times 10^{-7} \text{ m} \]

Step 4: Calculate Speed of Light in Water

The speed of light in a medium is given by \( v = \frac{c}{n} \), where \( c = 3.00 \times 10^8 \text{ m/s} \) is the speed of light in a vacuum. Substitute the given values:\[ v = \frac{3.00 \times 10^8 \text{ m/s}}{1.33} \approx 2.25 \times 10^8 \text{ m/s} \]

Step 5: Compare with Given Options

The calculated wavelength and speed are approximately \( 4.42 \times 10^{-7} \text{ m} \) and \( 2.25 \times 10^8 \text{ m/s} \), which match none of the options completely due to the speed calculation issue in the options. However, the wavelength calculation correlates best with option (d).

Key Concepts

Monochromatic Light

Monochromatic light refers to light that consists of one single wavelength. This means it has a uniform color, not a combination of different colors like white light. One common example of monochromatic light is the light emitted by a sodium vapor lamp, which typically gives out light with a wavelength of about 589 nm.
This wavelength remains consistent when the light travels through the same medium. However, when monochromatic light enters a different medium, its speed and wavelength change due to refraction. Despite these transformations, its frequency remains constant because frequency is an intrinsic property of the light, determined by its source.
Importantly, understanding the behavior of monochromatic light when transitioning between media helps us better grasp the fundamental principles of optics, such as refraction and how light behaves in different environments.

Wavelength in Different Media

The wavelength of light changes when it travels from one medium to another. This change occurs due to the refraction of light, which is the bending of light as it enters a medium with a different refractive index.
The formula to determine the new wavelength in a different medium is given by \( \lambda = \frac{\lambda_0}{n}\)where \(\lambda_0\) is the original wavelength in the first medium (e.g., air), and \(n\) is the refractive index of the second medium (e.g., water).
In the original exercise, the wavelength of monochromatic light in water was calculated using this formula, demonstrating how the light's wavelength becomes shorter in a medium with a higher refractive index. This concept is crucial for technologies like fiber optics, where light needs to travel effectively through various media without excessive dispersion.

Speed of Light in Water

Light slows down when it passes from a vacuum or air into a denser medium like water. This slowing down is described by the refractive index \(n\) of the medium. The refractive index indicates how much the speed of light is reduced relative to its speed in a vacuum.
The formula to calculate the speed of light in a medium is:\(v = \frac{c}{n}\)where \(c\) is the speed of light in a vacuum, approximately \(3.00 \times 10^8\) m/s. In water, with a refractive index of 1.33, the speed of light becomes approximately \(2.25 \times 10^8\) m/s.
This understanding is vital not only for theoretical studies but also for real-world applications, such as designing lenses and understanding how light behaves underwater, impacting everything from cameras to scientific equipment used for undersea exploration.