Problem 11 - Applied Physics

Question

Find the frequency of an electromagnetic wave if its wavelength is \(85.5 \mathrm{~m}\).

Step-by-Step Solution

Answer

The frequency is approximately \(3.51 \times 10^6\) Hz.

Step 1: Understanding the Wave Equation

The frequency of an electromagnetic wave can be found using the wave equation, which states that the speed of a wave is equal to the product of its frequency and wavelength. This is represented as: \[ c = \lambda \times f \]where \( c \) is the speed of light in a vacuum (approximately \(3 \times 10^8\) m/s), \( \lambda \) is the wavelength, and \( f \) is the frequency.

Step 2: Rearranging the Wave Equation

Since we are looking for the frequency \( f \), rearrange the wave equation to solve for \( f \): \[ f = \frac{c}{\lambda} \]This tells us that the frequency is equal to the speed of light divided by the wavelength.

Step 3: Plugging in the Values

Now, substitute the given wavelength and the constant speed of light into the equation: \[ f = \frac{3 \times 10^8 \text{ m/s}}{85.5 \text{ m}} \]

Step 4: Calculating the Frequency

Perform the division to find the frequency: \[ f \approx 3.51 \times 10^6 \text{ Hz} \]

Key Concepts

Wave Equation

The wave equation is a fundamental concept in physics that connects the speed, frequency, and wavelength of waves. This relationship is crucial for understanding how electromagnetic waves, like light and radio waves, travel through space. The general form of the wave equation is \( c = \lambda \times f \), where:

• \( c \) is the speed of the wave;
• \( \lambda \) (lambda) is the wavelength;
• \( f \) is the frequency.

For electromagnetic waves in a vacuum, the speed \( c \) is a constant approximately equal to \(3 \times 10^8\) meters per second. This equation helps us understand that as the wavelength increases, the frequency decreases, provided the speed remains constant. By rearranging the formula, we can solve for any of the three variables, revealing the interconnected nature of wave properties.

Frequency Calculation

Calculating the frequency of an electromagnetic wave involves rearranging the wave equation to solve for frequency. In its rearranged form: \[ f = \frac{c}{\lambda} \]This equation expresses that frequency \( f \) can be calculated by dividing the speed of light \( c \) by the wavelength \( \lambda \). This relationship shows the inverse proportion between frequency and wavelength: as one increases, the other decreases. To calculate the frequency in the original exercise, we used the known speed of light and the given wavelength:\[ f = \frac{3 \times 10^8 \text{ m/s}}{85.5 \text{ m}} \] After performing the division, the resulting frequency was approximately \(3.51 \times 10^6\) Hertz (Hz), which indicates the number of wave cycles passing a point per second.

Wavelength

The concept of wavelength refers to the distance between successive peaks of a wave. In the context of electromagnetic waves, the wavelength helps determine the energy and nature of the wave.Understanding wavelength is important because:

• The type (or color, in the case of visible light) of electromagnetic radiation is defined by its wavelength.
• Wavelength and frequency are inversely related via the wave equation. A shorter wavelength means a higher frequency, and vice versa.

In our original exercise, the wavelength was provided as \(85.5\) meters. By applying the wave equation, it helps us calculate the frequency given this specific wavelength. This example illustrates how crucial the concept of wavelength is in analyzing and identifying various electromagnetic waves.