Problem 4 - University Physics with Modern Physics

Question

Light with a frequency of \(5.80 \times 10^{14} \mathrm{Hz}\) travels in a block of glass that has an index of refraction of \(1.52 .\) What is the wavelength of the light (a) in vacuum and (b) in the glass?

Step-by-Step Solution

Answer

(a) 517 nm in vacuum; (b) 340 nm in glass.

Step 1: Wavelength in Vacuum

To find the wavelength of light in a vacuum, we use the formula for the speed of light:\[\lambda_0 = \frac{c}{f}\]where \(\lambda_0\) is the wavelength in a vacuum, \(c\) is the speed of light in a vacuum \((3.00 \times 10^8 \text{ m/s})\), and \(f\) is the frequency of the light. Substituting the known values:\[\lambda_0 = \frac{3.00 \times 10^8 \text{ m/s}}{5.80 \times 10^{14} \text{ Hz}} = 5.17 \times 10^{-7} \text{ m} = 517 \text{ nm}\]So, the wavelength in a vacuum is 517 nm.

Step 2: Calculate Wavelength in Glass

To find the wavelength of the light in the glass, we use the formula:\[\lambda_{n} = \frac{\lambda_0}{n}\]where \(\lambda_n\) is the wavelength in the medium, \(\lambda_0\) is the wavelength in a vacuum, and \(n\) is the index of refraction of the medium. Substituting the known values:\[\lambda_{n} = \frac{517 \text{ nm}}{1.52} = 340 \text{ nm}\]Thus, the wavelength of the light in the glass is 340 nm.

Key Concepts

Wavelength

Wavelength is a fundamental concept in wave optics. It represents the distance between two consecutive peaks (or troughs) of a wave. This distance helps determine the wave's properties, such as its color in the case of light. For light traveling in a vacuum, the wavelength can be determined using the formula:\[\lambda_0 = \frac{c}{f}\] where \(\lambda_0\) is the wavelength in a vacuum, \(c\) is the speed of light \((3.00 \times 10^8 \text{ m/s})\), and \(f\) is the frequency. When light speeds through different mediums, its wavelength changes, but the frequency remains constant. This change is due to the wave's new speed in the given medium.

Frequency

Frequency refers to the number of wave cycles that pass a particular point in one second. It is measured in hertz (Hz). In the context of light waves, frequency determines the color of the light. The equation linking frequency and wavelength in a vacuum is:\[f = \frac{c}{\lambda_0}\]where \(\lambda_0\) is the wavelength and \(c\) is the speed of light.

• High frequency means a shorter wavelength and more cycles per second.
• Low frequency results in a longer wavelength and fewer cycles per second.

Understanding frequency is key to comprehending how light behaves and interacts with different media.

Index of Refraction

The index of refraction, often represented by \(n\), describes how much light bends as it enters a new medium. This value is a ratio comparing the speed of light in a vacuum to its speed in the substance. It is calculated using the formula:\[n = \frac{c}{v}\]where \(v\) is the speed of light in the medium.For instance, in glass where the index is 1.52, light travels more slowly than in a vacuum. This slowdown results in a shorter wavelength but the same frequency. This principle is crucial in designing lenses and other optical devices.

Medium

A medium is any substance through which light can travel. Different media affect the way light propagates. In wave optics, the medium's properties (like its index of refraction) significantly affect the wavelength and speed of light.

• When moving from a vacuum to glass, the speed of light decreases.
• This change alters the wavelength while keeping the frequency constant.

Understanding different media allows us to appreciate phenomena like refraction and allows for practical applications in technology like fiber optics and cameras. Different media transform how we see and understand waves.