Question

What is the wavelength of the electromagnetic waves used for cell phone communications at \(848.97 \mathrm{MHz} ?\)

Short answer

Answer: The wavelength of the electromagnetic wave with a frequency of 848.97 MHz for cell phone communications is approximately 0.3534 km.

Step-by-step solution

Convert frequency to Hz

First, we need to convert the given frequency from MHz to Hz. We know that: 1 MHz = \(10^6\) Hz So, 848.97 MHz = \(848.97 \times 10^6 \mathrm{Hz}\)

Apply the formula for the wavelength of an electromagnetic wave

The formula for the wavelength (\(\lambda\)) of an electromagnetic wave is given by: \(\lambda = \frac{c}{f}\) Where: \(c\) is the speed of light, which is approximately \(3 \times 10^8 \mathrm{\frac{m}{s}}\) \(f\) is the frequency of the wave in Hz

Calculate the wavelength

Using the formula, we can now find the wavelength of the electromagnetic wave: \(\lambda = \frac{c}{f} = \frac{3 \times 10^8 \mathrm{\frac{m}{s}}}{848.97 \times 10^6 \mathrm{Hz}}\)

Simplify and solve for the wavelength

Divide the numbers to find the wavelength: \(\lambda = \frac{3 \times 10^8 \mathrm{\frac{m}{s}}}{848.97 \times 10^6 \mathrm{Hz}} = 0.0003534 \mathrm{m}\)

Convert the wavelength to a more appropriate unit

The wavelength is quite small, so we can convert it to a more appropriate unit by multiplying it by \(10^3\) to get the value in kilometers: \(\lambda = 0.0003534 \mathrm{m} \times 10^3 = 0.3534 \mathrm{km}\) Therefore, the wavelength of the electromagnetic waves used for cell phone communications at \(848.97 \mathrm{MHz}\) is approximately \(0.3534 \mathrm{km}\).

Key concepts

Frequency to Wavelength Conversion

Understanding the relationship between frequency and wavelength is crucial when studying electromagnetic waves. To convert frequency to wavelength, one must use the fundamental formula: \[\begin{equation}\lambda = \frac{c}{f}\end{equation}\]where \(\lambda\) represents the wavelength, \(c\) is the speed of light in a vacuum, and \(f\) is the frequency of the wave.

In the context of our cell phone communications exercise, the given frequency was in megahertz (MHz), a commonly used unit in telecommunications. To use the formula, frequency must be converted into hertz (Hz), the standard unit for frequency. The conversion is straightforward since 1 MHz equals \(10^6\) Hz. For instance, \(848.97 \text{ MHz}\) is simply \(848.97 \times 10^6\) Hz. By plugging this frequency and the known speed of light into the formula, the wavelength can be determined in meters. It's helpful to convert the result to a more comprehensible unit such as kilometers or centimeters, depending on the size of the wavelength.

Speed of Light in Physics

The speed of light in a vacuum, denoted by \(c\text{,}\) is a fundamental constant in physics with a value of approximately \(3 \times 10^8\) meters per second. It's not just a speed limit for light but for any electromagnetic radiation, as well as information and gravitational effects.

Within the realm of our calculation involving cell phone frequencies, the speed of light provides the missing piece needed to calculate the wavelength of the electromagnetic wave used in communications. When applying the speed of light to the formula for wavelength, one can extract the size of the waves involved in carrying signals to and from our mobile devices.

Given its monumental role in physics, the speed of light is used to explore the cosmos, measure distances in space, and synchronize time across the globe. Its constancy across all frames of reference is a cornerstone of Einstein's theory of relativity.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, ranging from very low-frequency radio waves to very high-frequency gamma rays. Each category in the spectrum—including radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays—has a different range of wavelengths and frequencies.

Radio waves, used for cell phone communications, fall at the lower-frequency end of the spectrum with longer wavelengths. Conversely, gamma rays reside at the high-frequency end with much shorter wavelengths.

Understanding the electromagnetic spectrum is essential for applications like telecommunications, where the frequency determines the behavior of the waves, such as their ability to penetrate materials or travel long distances. Each region of the spectrum is utilized differently, for example, radio and microwaves for communication, infrared for heat sensors, and X-rays for medical imaging, showcasing the spectrum's versatility in technology and science.