Problem 55 - Physics

Question

A certain type of laser emits light of known frequency. The light, however, occurs as a series of short pulses, each lasting for a time \(t_{0}\). (a) How is the wavelength of the light related to its frequency? (b) How is the length (in meters) of each pulse related to the time \(t_{0}\) ?

Step-by-Step Solution

Answer

(a) \( \lambda = \frac{c}{f} \), (b) \( L = c \cdot t_0 \)

Step 1: Understanding Wavelength and Frequency

The relationship between the wavelength \( \lambda \) and the frequency \( f \) of light is given by the equation \( c = \lambda f \), where \( c \) is the speed of light in a vacuum, approximately \( 3 \times 10^8 \) meters per second.

Step 2: Calculating Wavelength

To find the wavelength \( \lambda \), rearrange the equation: \( \lambda = \frac{c}{f} \). This shows that the wavelength is inversely proportional to the frequency, meaning as frequency increases, wavelength decreases.

Step 3: Pulse Length Understanding

The length of each pulse in meters depends on the time duration \( t_0 \) of the pulse and the speed of light. The pulse length \( L \) (in meters) is related to the time duration by the equation \( L = c \cdot t_0 \).

Step 4: Calculating Pulse Length

Substitute the known speed of light into the equation: \( L = (3 \times 10^8 \text{ m/s}) \cdot t_0 \). This equation shows that the pulse length is directly proportional to the time duration.

Key Concepts

Wavelength and Frequency Relationship

Light exhibits a fascinating relationship between its wavelength and frequency, two fundamental characteristics of any wave. These two properties are linked by the speed of light, denoted by the symbol \(c\). In this particular case, the speed of light in a vacuum is approximately 3 x 10^8 meters per second. This constant serves as the bridge connecting wavelength \(\lambda\) and frequency \(f\). We can express this relationship with the formula: \[ c = \lambda \cdot f\]From this equation, it's clear that frequency and wavelength are inversely related:

• As frequency increases, the wavelength decreases.
• Conversely, if the frequency decreases, the wavelength increases.

To find the wavelength given a certain frequency, you can rearrange the formula to: \[ \lambda = \frac{c}{f} \]This helps us understand that knowing the frequency allows us to easily calculate the wavelength. This inverse relationship is key when working with electromagnetic waves, such as light, allowing us to deduce one property when the other is known. Remembering this relationship can help when analyzing wave behavior in various scientific and practical applications.

Speed of Light

The speed of light is a fundamental constant in physics, symbolized by \(c\). In a vacuum, the speed of light is approximately 3 x 10^8 meters per second. This value is crucial not only for understanding optical phenomena but also in many areas of physics and engineering. It essentially sets the speed limit for how fast information can travel in the universe.
The speed of light plays a pivotal role in the equation that relates wavelength and frequency: \[ c = \lambda \cdot f\]Light travels incredibly fast, which means its wavelength and frequency carry a significant impact. Furthermore, this speed is used to measure astronomical distances and calculate time in relativistic physics. The constancy of light's speed is a foundation of Einstein's theory of relativity, influencing concepts like time dilation and the curvature of space-time.
Understanding this constant allows us to grasp not only basic optic principles but also complex theories about the universe itself.

Pulse Duration

Pulse duration refers to the length of time a brief pulse of light lasts. In the context of laser light emissions, the duration and properties of these pulses become quite significant. This characteristic can impact how light interacts with materials and how it is used in technological applications.
The length of each light pulse can be calculated based on the speed of light and its time duration, \(t_0\). The formula for determining the pulse length \(L\) in meters is:\[ L = c \cdot t_0\]Here's how it works:

• Since \(c\) is a constant (3 x 10^8 meters per second), multiplying it by the time duration \(t_0\) in seconds gives the total distance covered by the pulse.
• This formula shows that the longer the pulse lasts, the greater distance it travels.

Understanding pulse duration is essential, especially in applications like optical communications, where precision and clarity are crucial. By knowing the pulse duration and speed of the light, one can determine how far the pulse will travel, which is vital for effectively designing systems like laser-guided devices and fiber-optic communications.