Problem 6 - College Physics

Question

\(\bullet\) Most people perceive light having a wavelength between 630 \(\mathrm{nm}\) and 700 \(\mathrm{nm}\) as red and light with a wavelength between 400 \(\mathrm{nm}\) and 440 \(\mathrm{nm}\) as violet. Calculate the approximate frequency ranges for (a) violet light and (b) red light.

Step-by-Step Solution

Answer

Violet light: 6.82 to 7.5 THz; Red light: 4.29 to 4.76 THz.

Step 1: Understand the Relationship

To find the frequency of light, we use the equation relating the speed of light \( c \), wavelength \( \lambda \), and frequency \( f \): \( c = \lambda f \). The speed of light \( c \) is approximately \( 3 \times 10^8 \ \text{m/s} \).

Step 2: Convert Wavelength from nm to m

Since the speed equation requires meters, convert the given wavelengths from nanometers to meters: \( 1 \ \text{nm} = 1 \times 10^{-9} \ \text{m} \). For violet: 400 nm becomes \( 400 \times 10^{-9} \ \text{m} \) and 440 nm becomes \( 440 \times 10^{-9} \ \text{m} \). For red: 630 nm becomes \( 630 \times 10^{-9} \ \text{m} \) and 700 nm becomes \( 700 \times 10^{-9} \ \text{m} \).

Step 3: Calculate Frequency for Violet Light

Using the speed of light equation \( c = \lambda f \), solve for frequency \( f = \frac{c}{\lambda} \). For 400 nm: \( f = \frac{3 \times 10^8 \ \text{m/s}}{400 \times 10^{-9} \ \text{m}} = 7.5 \times 10^{14} \ \text{Hz} \). For 440 nm: \( f = \frac{3 \times 10^8 \ \text{m/s}}{440 \times 10^{-9} \ \text{m}} = 6.82 \times 10^{14} \ \text{Hz} \). The frequency range for violet light is approximately 6.82 to 7.5 THz.

Step 4: Calculate Frequency for Red Light

Using \( f = \frac{c}{\lambda} \), calculate frequency for 630 nm: \( f = \frac{3 \times 10^8 \ \text{m/s}}{630 \times 10^{-9} \ \text{m}} = 4.76 \times 10^{14} \ \text{Hz} \). For 700 nm: \( f = \frac{3 \times 10^8 \ \text{m/s}}{700 \times 10^{-9} \ \text{m}} = 4.29 \times 10^{14} \ \text{Hz} \). The frequency range for red light is approximately 4.29 to 4.76 THz.

Key Concepts

Wavelength

Wavelength is a fundamental concept when discussing the electromagnetic spectrum and light. It describes the distance between successive crests (the highest points) of a wave. In the context of light, the wavelength determines what color the light appears to our eyes. For example, red light has longer wavelengths compared to violet light.

• Wavelength is usually measured in meters ( ext{m}), but for light, we commonly use nanometers ( ext{nm}), where 1 ext{nm} equals 1 x 10^{-9} ext{m}.
• The visible light spectrum ranges approximately from 400 nm (violet) to 700 nm (red).

Understanding wavelength helps us decode the properties of light and predict its behavior, such as how it interacts with different materials and how it is perceived by the human eye.

Frequency

Frequency refers to the number of wave cycles that pass a point in one second, expressed in hertz (Hz). In the context of light, frequency helps us determine the energy of light waves.

• Higher frequency corresponds to greater energy. Thus, violet light, which has a higher frequency, is more energetic than red light.
• The formula to find frequency is \( f = \frac{c}{\lambda} \), where \( f \) is frequency, \( c \) is the speed of light, and \( \lambda \) is wavelength.

As frequency increases from red to violet, the energy of the light waves also increases, affecting their color and visibility.

Speed of Light

The speed of light is a constant that plays a crucial role in the study of physics and the understanding of the universe. It is \(3 \times 10^8 \text{ m/s}\) in a vacuum, and this constant speed is a key part of equations involving electromagnetic waves.

• The relationship between the speed of light, frequency, and wavelength is given by \( c = \lambda f \).
• This equation is essential for converting between wavelength and frequency, as done in the exercise to calculate the frequencies for red and violet light.

The speed of light is not only a fundamental constant in physics but also a bridge between wavelength and frequency, allowing us to convert and understand these two properties in relation to each other.

Visible Light

Visible light is a small portion of the electromagnetic spectrum that is detectable by the human eye. This range encompasses a variety of colors from violet to red, each with its unique wavelength and frequency.

• Violet light has the shortest wavelength, around 400 nm, and highest frequency, while red light has the longest wavelength, around 700 nm, and lowest frequency.
• Light within this range can be split into various colors using prisms or diffraction grating, creating a spectrum like a rainbow.

Understanding visible light helps us understand the broader electromagnetic spectrum and how energy travels in different forms throughout the universe. It also highlights the fascinating ways in which physics becomes a part of our everyday experiences, from the colors we perceive to the technology we use.