Question

A light ray of wavelength \(700 . \mathrm{nm}\) traveling in air \(\left(n_{1}=1.00\right)\) is incident on a boundary with a liquid \(\left(n_{2}=1.63\right)\). a) What is the frequency of the refracted ray? b) What is the speed of the refracted ray? c) What is the wavelength of the refracted ray?

Short answer

Answer: (a) The frequency of the refracted ray is approximately 4.29 x 10^{14} Hz. (b) The speed of the refracted ray in the liquid is approximately 1.84 x 10^{8} m/s. (c) The wavelength of the refracted ray in the liquid is approximately 429 nm.

Step-by-step solution

Identify the given variables and constants.

We are given the following information: - Wavelength of the light ray in air, λ1 = 700 nm = 700 x 10^{-9} m - Refractive index of air, n1 = 1.00 - Refractive index of the liquid, n2 = 1.63 We know that the speed of light in a vacuum, c = 3 x 10^8 m/s.

Find the frequency of the light ray.

We will first find the frequency of the light ray. To do this, we first need to find the speed of the light ray in air. We can use the formula for refractive index: n = c / v Rearranging the formula to solve for v: v = c / n Now, substitute the values for air: v1 = (3 x 10^8 m/s) / 1.00 v1 = 3 x 10^8 m/s Next, we will use the wave equation v = λ × f: v1 = λ1 × f Rearranging the formula to solve for f: f = v1 / λ1 Now, substitute the values: f = (3 x 10^8 m/s) / (700 x 10^{-9} m) f ≈ 4.29 x 10^{14} Hz The frequency of the light ray is approximately 4.29 x 10^{14} Hz.

Find the speed of the refracted ray in the liquid.

Once again, we will use the formula for refractive index: n = c / v Rearranging the formula to solve for v: v = c / n Now, substitute the values for the liquid: v2 = (3 x 10^8 m/s) / 1.63 v2 ≈ 1.84 x 10^8 m/s The speed of the refracted ray in the liquid is approximately 1.84 x 10^{8} m/s.

Find the wavelength of the refracted ray in the liquid.

Since the frequency remains constant when the light ray enters the liquid, we can use the wave equation again to find the wavelength of the refracted ray: v2 = λ2 × f Rearranging the formula to solve for λ2: λ2 = v2 / f Now, substitute the values: λ2 = (1.84 x 10^8 m/s) / (4.29 x 10^{14} Hz) λ2 ≈ 4.29 x 10^{-7} m Converting back to nanometers: λ2 = 429 nm The wavelength of the refracted ray in the liquid is approximately 429 nm. To conclude: a) The frequency of the refracted ray is 4.29 x 10^{14} Hz. b) The speed of the refracted ray in the liquid is 1.84 x 10^{8} m/s. c) The wavelength of the refracted ray in the liquid is 429 nm.

Key concepts

Wavelength

The wavelength of light is the distance between consecutive peaks of a light wave. It is usually denoted as \( \lambda \) and measured in meters (m), with subunits such as nanometers (nm) used for visible light. The wavelength determines the color of light in the visible spectrum, with longer wavelengths corresponding to red and shorter ones to violet. In the context of refraction, when light passes from one medium to another, its velocity changes while its frequency remains constant. Consequently, the wavelength changes in proportion to the change in speed due to the new medium.

For students to better understand wavelength changes during refraction, it's useful to envision a light wave as similar to a series of water waves entering a shallow area from the deep: the waves slow down and their spacing (or wavelength) decreases, but the timing between the waves passing remains the same.

Frequency

In physics, frequency refers to the number of oscillations a wave completes per second, often described in Hertz (Hz). It reflects how often a wave crests pass through a stationary point. Frequency plays a crucial role in electromagnetic waves like light. When light crosses the boundary between two media with different refractive indices, such as air to water, one might imagine its frequency could change.

However, frequency remains constant during this transition, analogous to the number of water waves hitting the shore remaining consistent even as the waves slow down entering shallower water. This is a fundamental property that students should grasp, which provides the basis for understanding other phenomena such as the Doppler effect and the conservation of energy during wave transmission.

Speed of Light

The speed of light, colloquially denoted as \( c \) and approximately equal to \(3 \times 10^8 \) meters per second (m/s), is a fundamental constant of nature. It represents the fastest speed at which energy, matter, and information can travel. In a vacuum, light's speed is constant, though it can be reduced when passing through materials such as glass or water — each material is characterized by its refractive index.

This concept is key to understanding how light propagates through different media. For instance, when light enters a denser medium from a rarer medium, like air to glass, its speed decreases proportionally to the medium's refractive index. Recognizing that light slows down but doesn't change its intrinsic oscillation rate (frequency) helps students to explore how light behaves under various conditions.

Refractive Index

The refractive index, typically represented by \( n \), is a dimensionless number that expresses how much light slows down in a particular medium compared to a vacuum. It is calculated as the ratio of the speed of light in a vacuum \( c \) to the speed of light \( v \) in the medium. Substances with higher refractive indices are denser and slow down light to a greater extent.

Most importantly in the context of our problem, the refractive index determines how much light bends, or refracts, when entering the medium. Higher refractive indices result in greater bending. Students should realize that refractive index is a powerful tool for understanding the behavior of light in different mediums and is used in the design of lenses and optical devices. Making a connection between the refractive index and everyday experiences, such as the bending of a straw in a glass of water, helps in grasping this abstract concept and applying it to various problems in optics.