Question

The currents in household wiring and power lines alternate at a frequency of \(60.0 \mathrm{Hz}\). (a) What is the wavelength of the EM waves emitted by the wiring? (b) Compare this wavelength with Earth's radius. (c) In what part of the EM spectrum are these waves?

Short answer

(a) The wavelength is \( 5 \times 10^6 \text{ m} \). (b) It is slightly smaller than Earth's radius. (c) These waves are in the ELF range of the EM spectrum.

Step-by-step solution

Understanding the Relationship between Frequency and Wavelength

The speed of electromagnetic waves in a vacuum, which is the speed of light, is approximately \( c = 3 \times 10^8 \text{ m/s} \). The relationship between speed \( c \), frequency \( f \), and wavelength \( \lambda \) of a wave is given by the equation: \[ c = f \times \lambda \]Given that the frequency \( f = 60.0 \text{ Hz} \), we can rearrange the equation to find the wavelength:\[ \lambda = \frac{c}{f} \]

Calculating the Wavelength

Using the equation from the previous step, we substitute the given values:\[ \lambda = \frac{3 \times 10^8 \text{ m/s}}{60.0 \text{ Hz}} \]\[ \lambda = 5 \times 10^6 \text{ m} \]Thus, the wavelength of the EM waves is \( 5 \times 10^6 \text{ meters} \).

Comparing the Wavelength to Earth's Radius

Earth's average radius is approximately \( 6.371 \times 10^6 \text{ m} \). We compare this with the wavelength calculated:\( 5 \times 10^6 \text{ m} \) versus \( 6.371 \times 10^6 \text{ m} \).The wavelength of the EM waves is slightly smaller than Earth's radius.

Identifying the Spectrum Part

The calculated wavelength \( 5 \times 10^6 \text{ m} \) lies in the extremely low frequency (ELF) range within the electromagnetic spectrum. Radio waves, particularly those with low frequencies, can have very long wavelengths, such as this one.

Key concepts

Wavelength Calculation

In physics, understanding how to calculate the wavelength of electromagnetic (EM) waves is crucial. The wavelength (4") is determined by dividing the speed of light (
3 98 ext{ m/s}") by the frequency (6 ext{ Hz}") of the wave. This calculation helps us find out how long the wave is from one peak to the next. Wavelength calculation is an important step in analyzing EM waves. By knowing the frequency, such as the typical household 60 Hz, we can use the formula:\[ \lambda = \frac{c}{f} \] Wavelength helps us understand how different EM waves interact with the environment and affect us.Using this equation, for a household frequency, the wavelength is \( 5 \times 10^6 \text{ meters} \), illustrating the vast stretch these waves encompass.

Frequency and Wavelength Relationship

The frequency and wavelength of electromagnetic waves are interconnected through the speed of light. Frequency refers to how often the wave oscillates, while wavelength is the distance between consecutive peaks of the wave. Given the speed of light (
3 98 ext{ m/s}") is constant in a vacuum, changes in one will inversely affect the other.The equation connecting frequency (6 ext{ Hz}") and wavelength (d") is:\[ c = f \times \lambda \]If frequency increases, wavelength decreases, and vice versa. This relationship helps us determine characteristics of electromagnetic waves, from radio to gamma rays, across diverse applications.Understanding this relationship is key in fields such as telecommunications and astronomy.

Electromagnetic Spectrum

The electromagnetic spectrum is the range of all types of EM radiation. EM waves vary from very long radio waves to extremely short gamma rays. The spectrum includes multiple regions such as radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. Each type of wave has unique properties:

• Radio waves typically have the longest wavelengths.
• Visible light has wavelengths that the human eye can detect.
• Gamma rays possess very high energy and short wavelengths.

The placement of a wave within the spectrum is determined by its frequency and wavelength. This allows us to categorize and study different waves for numerous scientific and practical uses.

Extremely Low Frequency (ELF)

The extremely low frequency (ELF) range lies at one end of the electromagnetic spectrum. ELF waves have frequencies from 3 to 30 Hz and correspondingly long wavelengths, stretching even thousands of kilometers. These waves are produced by natural phenomena like the planet's atmosphere and man-made sources such as power lines and submarine communications. Despite their long wavelengths and low frequency, ELF waves can penetrate deep into water and the Earth's surface, making them valuable for certain kinds of communication.

Speed of Light

The speed of light is a fundamental constant of nature, approximately \( 3 \times 10^8 \text{ meters per second}\) in a vacuum. It's crucial in connecting frequency and wavelength of electromagnetic waves. Since light speed is constant, it governs how variations in frequency and wavelength interact. For example:

• If frequency increases, wavelength must decrease to maintain the light's speed.
• These principles are applied in technologies involving optics, wireless communications, and more.

Understanding the speed of light helps correctly navigate the relationship between wavelength and frequency.