Problem 11 - Arihant AIEEE Physics

Question

In a medium of refractive index \(n_{1}\), a monochromatic light of wavelength \(\lambda_{1}\) is travelling. When it enters in a denser medium of refractive index \(n_{2}\), the wavelength of the light in the second medium is : (a) \(\lambda_{1}\left(\frac{n_{1}}{n_{2}}\right)\) (b) \(\lambda_{1}\left(\frac{n_{2}}{n_{1}}\right)\) (c) \(\frac{\lambda_{1}\left(n_{2}-n_{1}\right)}{n_{2}}\) (d) \(\frac{\lambda_{1}\left(n_{2} \cdots n_{1}\right)}{n_{1}}\)

Step-by-Step Solution

Answer

The wavelength in the second medium is \( \lambda_{1} \left( \frac{n_{1}}{n_{2}} \right) \), corresponding to option (a).

Step 1: Understand the Problem

We need to find the wavelength of light when it enters a denser medium, given two different refractive indices and an initial wavelength. We will use the relationship between the refractive indices and wavelengths in different media.

Step 2: Recall the Formula for Wavelength in Medium

When light enters a medium with a different refractive index, its speed and wavelength change, but its frequency remains constant. The formula relating the wavelength in different media is given by \( \frac{\lambda_{2}}{\lambda_{1}} = \frac{n_{1}}{n_{2}} \), where \( \lambda_{2} \) is the wavelength in the second medium.

Step 3: Solve for the Wavelength in the Second Medium

Using the formula from Step 2, we solve for \( \lambda_{2} \), the wavelength in the second medium: \[ \lambda_{2} = \lambda_{1} \left( \frac{n_{1}}{n_{2}} \right). \] This shows how the wavelength changes due to the different refractive indices.

Step 4: Identify the Correct Option

Comparing the solved expression \( \lambda_{1} \left( \frac{n_{1}}{n_{2}} \right) \) with the provided options, we find that option (a) matches our expression.

Key Concepts

Wavelength in Media

When light crosses from one medium to another, its speed and wavelength can change, though its frequency remains the same. This is a crucial idea in optics. The wavelength of light in a medium is influenced by the medium's refractive index.
The refractive index () is a measure of how much the speed of light is reduced inside a medium compared to its speed in a vacuum. The formula that governs the change in wavelength as light moves from one medium to another is:

• \[ \frac{\lambda_{2}}{\lambda_{1}} = \frac{n_{1}}{n_{2}} \]

Here, \(\lambda_{1}\) and \(\lambda_{2}\) represent the wavelengths of the light in the first and second medium, respectively, and \(n_{1}\) and \(n_{2}\) are their respective refractive indices.
Therefore, the wavelength of the light becomes compressed or elongated depending on whether the medium is denser or less dense, respectively. Understanding this relationship is key to solving many problems in optics.

Monochromatic Light

Monochromatic light refers to light that has a single wavelength or a very narrow band of wavelengths. It is often associated with a single color of visible light because each color corresponds to a particular wavelength range.
When considering monochromatic light, it simplifies many optical calculations since changes in wavelength due to different media are predictable using the properties of the initial wavelength. This type of light is commonly used in experiments and applications where precise and consistent behavior of light is required, such as lasers.
Key characteristics of monochromatic light include:

• Uniform color or frequency.
• Predictable changes across different media while maintaining constant frequency.
• Essential in experiments involving interference, diffraction, and refraction.

Monochromatic light is critical to understanding how light behaves in different materials and under varying conditions. It remains unabated by the refractive index changes of mediums, thereby serving as a reliable source in various scientific and industrial applications.

Snell's Law

Snell's Law is the fundamental principle that describes how light refracts when moving from one medium to another. It is pivotal for understanding the path that light takes under such conditions.
The law states that the ratio of the sines of the angles of incidence and refraction is equivalent to the reciprocal of the ratio of the refractive indices of the two media:

• \[ \frac{\sin \theta_{1}}{\sin \theta_{2}} = \frac{n_{2}}{n_{1}} \]

Where \(\theta_{1}\) and \(\theta_{2}\) are the angles of incidence and refraction, respectively. This relationship shows how the direction of light changes due to differences in optical density between materials.
In addition to changing direction, light also changes speed and wavelength upon entering a new medium. Snell's Law helps predict these attributes by quantifying the exact degree of bending and the change in wave properties, playing a major role in the design of lenses, prisms, and optical fibers.