Problem 10 - Applied Physics

Question

What is the wavelength of an electromagnetic wave with frequency \(50.0 \mathrm{MHz}\) ?

Step-by-Step Solution

Answer

The wavelength is 6 meters.

Step 1: Understand the Relation between Frequency and Wavelength

The speed of light (\( c \)) is related to frequency (\( f \)) and wavelength (\( \lambda \)) by the equation: \[ c = f \cdot \lambda \] where \( c \approx 3 \times 10^8 \text{ m/s} \).

Step 2: Rearrange the Equation to Solve for Wavelength

To find the wavelength, rearrange the equation to solve for \( \lambda \): \[ \lambda = \frac{c}{f} \]

Step 3: Substitute the Known Values

Substitute the values for the speed of light and frequency into the equation: \[ \lambda = \frac{3 \times 10^8 \text{ m/s}}{50.0 \times 10^6 \text{ Hz}} \]

Step 4: Perform the Calculation

Calculate the wavelength: \[ \lambda = \frac{3 \times 10^8}{50.0 \times 10^6} \] \[ \lambda = 6 \text{ m} \]

Key Concepts

Wavelength Calculation

When you're tasked with finding the wavelength of an electromagnetic wave, first, it's crucial to understand the equation that connects wavelength \( (\lambda) \), frequency \( (f) \), and the speed of light \( (c) \). This relationship is given by the equation:\[c = f \cdot \lambda\]To find the wavelength, you need to isolate \( \lambda \) by rearranging the equation to:\[\lambda = \frac{c}{f}\]By substituting in the known values of the speed of light, approximately \( 3 \times 10^8 \text{ m/s} \), and the given frequency, for example \( 50.0 \times 10^6 \text{ Hz} \), you can calculate the wavelength:

• \( \lambda = \frac{3 \times 10^8}{50.0 \times 10^6} \)
• \( \lambda = 6 \text{ m} \)

So, the wavelength of the wave is 6 meters. This kind of calculation is fundamental in understanding the characteristics of electromagnetic waves.

Frequency and Wavelength Relationship

One of the key concepts in the study of electromagnetic waves is the relationship between frequency and wavelength. They share an inverse relationship, meaning when one increases, the other decreases.In other words:

• If the frequency \((f)\) of a wave increases, the wavelength \((\lambda)\) must decrease.
• If the frequency decreases, the wavelength increases.

This relationship is expressed by the equation \( c = f \cdot \lambda \).This helps us understand why waves with very high frequencies, like gamma rays or X-rays, have very small wavelengths, whereas waves with low frequencies, such as radio waves, have longer wavelengths. Understanding this relationship is vital in various applications, like communications and medical imaging, where different parts of the electromagnetic spectrum are used based on their frequency and wavelength characteristics.

Speed of Light Calculation

The speed of light is a universal constant that is an essential part of calculating the properties of electromagnetic waves. The accepted value of the speed of light in a vacuum is \( c \approx 3 \times 10^8 \text{ m/s} \).In electromagnetic wave equations, such as \( c = f \cdot \lambda \), the speed of light allows us to find either the frequency or the wavelength when the other is known.

• This calculation is crucial for understanding how waves travel through different media.
• It provides insights into the behavior of light and all other types of electromagnetic waves.

The consistent speed of light across the universe is foundational to physics and helps to unify various wave phenomena into a coherent framework. By understanding and utilizing this fundamental constant, scientists and engineers can make precise predictions about wave behavior under various circumstances.