Problem 4 - Applied Physics

Question

(a) What is the frequency of a light wave with wavelength \(5.00 \times 10^{-7} \mathrm{~m}\) and velocity \(3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}\) ? (b) Find the period of the wave.

Step-by-Step Solution

Answer

The frequency is \(6.00 \times 10^{14} \mathrm{~Hz}\) and the period is \(1.67 \times 10^{-15} \mathrm{~s}\).

Step 1: Understanding the Relationship

To find the frequency of a light wave, we use the formula for the speed of light: \[ c = \lambda \cdot f \]where \( c \) is the speed of light, \( \lambda \) is the wavelength, and \( f \) is the frequency of the wave. Rearrange the formula to solve for frequency \( f \).

Step 2: Solving for Frequency

Using the formula \( f = \frac{c}{\lambda} \), substitute the given values. The speed of light \( c = 3.00 \times 10^8 \mathrm{~m/s} \), and the wavelength \( \lambda = 5.00 \times 10^{-7} \mathrm{~m} \). Calculate the frequency:\[ f = \frac{3.00 \times 10^8 \mathrm{~m/s}}{5.00 \times 10^{-7} \mathrm{~m}} = 6.00 \times 10^{14} \mathrm{~Hz} \].

Step 3: Understanding Period and Frequency Relationship

The period \( T \) of a wave is the reciprocal of the frequency \( f \). This relationship is given by the formula: \[ T = \frac{1}{f} \].

Step 4: Solving for the Period

Now that we have the frequency \( f = 6.00 \times 10^{14} \mathrm{~Hz} \), use the formula to find the period:\[ T = \frac{1}{6.00 \times 10^{14} \mathrm{~Hz}} \approx 1.67 \times 10^{-15} \mathrm{~s} \].

Key Concepts

Wavelength

Wavelength is a fundamental concept when discussing light waves. Essentially, it is the distance between consecutive crests (or troughs) in a wave pattern. For light waves, wavelength is usually measured in meters and denoted by the Greek letter lambda \( \lambda \). Knowing the wavelength of a light wave is crucial, as it can determine the color of visible light, or the type of electromagnetic radiation if we are dealing with waves outside the visible spectrum.Key characteristics related to wavelength:

• The shorter the wavelength, the higher the frequency of the wave.
• Wavelength and frequency are inversely related. As one increases, the other decreases.

In typical light wave problems, you are given the wavelength to calculate either the frequency or speed. Wavelength is measured in meters in scientific contexts (e.g., the wavelength in the original problem is \( 5.00 \times 10^{-7} \mathrm{~m} \)).

Speed of Light

The speed of light is a constant and is one of the most crucial values in physics. It is usually denoted by \( c \) and has a value of approximately \( 3.00 \times 10^{8} \; \mathrm{m/s} \). This speed is crucial in understanding how light and electromagnetic waves propagate in space.Light travels at this speed in a vacuum, and it can slow down when moving through different mediums due to interactions with particles in those mediums.Significant points about the speed of light:

• It limits the speed at which information can be transferred through the universe.
• It's a constant that helps define relationships between other properties of light, such as wavelength and frequency.

In any calculation involving light waves, speed is a critical component and is always considered when working in physics and engineering fields.

Frequency Calculation

Frequency determines how often the crests of a wave pass a given point in a specified amount of time. In our light wave example, the frequency \( f \) is calculated using the formula: \[ f = \frac{c}{\lambda} \]where

• \( c \) is the speed of light \( (3.00 \times 10^{8} \; \mathrm{m/s}) \)
• \( \lambda \) is the wavelength \( (5.00 \times 10^{-7} \; \mathrm{m}) \)

Calculate this by substituting in the values, yielding a frequency of \( 6.00 \times 10^{14} \; \mathrm{Hz} \).Points to remember about frequency:

• Expressed in Hertz (Hz), which corresponds to cycles per second.
• Frequency relates to energy (higher frequency waves have higher energy).
• Inverse relationship with wavelength; more frequent waves have smaller wavelengths.

Understanding frequency is fundamental when exploring the wave nature of light, aiding in visualizing how light behaves and interacts with matter.

Period of Wave

The period of a wave is the time it takes for one complete wave cycle to pass a given point, measured in seconds. This attribute is closely linked with frequency, as they are inversely related. The formula describing this link is:\[ T = \frac{1}{f} \]Where \( T \) is the period and \( f \) is the frequency.Substituting a frequency of \( 6.00 \times 10^{14} \; \mathrm{Hz} \), we calculate\( T \approx 1.67 \times 10^{-15} \; \mathrm{s} \). Important considerations about wave period:

• A longer period means a slower occurrence of waves.
• The concept of period is essential when describing wave cycles over time.
• It acts as a reciprocal measure to the frequency.

The period of a wave is a vital piece of information, particularly when examining how waves propagate and interact in various environments.