Question

Mono chromatic light of wavelength \(399 \mathrm{~nm}\) is incident from air on a water \((\mathrm{n}=1.33)\) Surface. The wavelength of refracted light is \(\mathrm{nm}\) (A) 300 (B) \(\overline{600}\) (C) 333 (D) 443

Short answer

The wavelength of refracted light in water is 300 nm. The correct option is (A) 300.

Step-by-step solution

Write down the given information

The given information is: - Wavelength of incident light in air, \(\lambda_{air} = 399 \ nm\) - Refractive index of water, \(n_{water} = 1.33\)

Write the formula relating the wavelength in two different media

The formula relating the wavelength in two different media is given by: \(n_{1}\lambda_{1}=n_{2}\lambda_{2}\), where \(n_1\) and \(n_2\) are the refractive indices of media 1 and 2 respectively, and \(\lambda_1\) and \(\lambda_2\) are the wavelengths in media 1 and 2 respectively.

Plug in the given values and solve for the wavelength in water

In our case, we have: - \(n_{1} = 1\) (refractive index of air) - \(\lambda_{1} = 399 \ nm\) (wavelength of incident light in air) - \(n_{2} = 1.33\) (refractive index of water) We need to find \(\lambda_{2}\), the wavelength of refracted light in water. Using the formula: \(n_{1}\lambda_{1}=n_{2}\lambda_{2}\), \(1 \times 399 = 1.33 \times \lambda_{2} \) Now, solve for \(\lambda_{2}\): \(\lambda_{2}\) = \(\frac{399}{1.33}\) = \(300 nm\) So, the wavelength of refracted light in water is 300 nm. The correct option is (A) 300.

Key concepts

Refractive Index

The refractive index is a dimensionless number that describes how light propagates through a medium. It is represented by the symbol \( n \), and it essentially measures how much the speed of light is reduced inside the medium. The refractive index can be calculated using the formula:\[ n = \frac{c}{v} \]where \( c \) is the speed of light in a vacuum, and \( v \) is the speed of light in the medium.
This is why a medium with a higher refractive index slows down light more compared to a medium with a lower refractive index. In the exercise, water has a refractive index \( n = 1.33 \), meaning light travels slower in water compared to air.

• A higher refractive index indicates slower light speed in that medium.
• Common refractive indices: Air is close to 1, water is about 1.33, and glass varies but often around 1.5.

Create an image in your mind of light 'bending' as it enters a water surface, due to its slowing down. This bending and slowing down are roots of understanding how light behaves in different substances.

Monochromatic Light

Monochromatic light is light that has one single wavelength or color. This type of light is often considered for theoretical exercises due to its simplicity since it doesn’t split into different colors like white light does. The wavelength of monochromatic light is constant in a given medium, only changing when the light enters a new medium with a different refractive index. In this exercise, the light entering the water has a wavelength of 399 nanometers when in air.
When moving to water, its speed changes due to the water's refractive index, thus altering its wavelength.

• Monochromatic effectively means 'one color'.
• Lasers are often examples of monochromatic light in practical applications.
• The change in wavelength is calculated using the relationship between wavelength and refractive indices of different media.

Understanding monochromatic light is crucial in physics and optics, offering simple yet powerful insights into the nature of light behavior.

Snell's Law

Snell's Law describes the relationship between the angle of incidence and the angle of refraction of waves passing through a boundary between two different isotropic media. It is defined by the formula:\[n_1 \sin \theta_1 = n_2 \sin \theta_2 \]where \( n_1 \) and \( n_2 \) are the refractive indices of the two media, and \( \theta_1 \) and \( \theta_2 \) are the angles formed by the light ray with the normal in each respective medium.
This law is crucial for understanding how light 'bends' as it moves between substances like air and water. Even though in the exercise the emphasis is on wavelength change, Snell's Law often accompanies refraction discussions.

• Snell's Law helps in calculating how sharply a light ray bends as it enters a new medium.
• It's vital for designing lenses and understanding natural phenomena like rainbows.
• It provides a quantitative way to relate path changes to speed changes.

The bending or refraction is due to the change in speed of light, precisely explained through Snell's Law. So next time you see light rays bending in a glass of water, think of Snell and his laws at work!