(I) Electromagnetic waves and sound waves can have the same frequency. (a) What is the wavelength of a 1.00-kHz electromagnetic wave? (b) What is the wavelength of a 1.00-kHz sound wave? (The speed of sound in air is 341 m/s.) (c) Can you hear a 1.00-kHz electromagnetic wave?

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding the wavelength of two waves (an electromagnetic wave and a sound wave) with the same frequency of 1.00 kHz. Additionally, we need to determine if a 1.00-kHz electromagnetic wave is audible to humans.
2. Step 2: Recall the formula for the relationship between the speed of a wave, its frequency, and its wavelength: \( v = \lambda f \), where \( v \) is the speed of the wave, \( \lambda \) is the wavelength, and \( f \) is the frequency. Rearrange this formula to solve for wavelength: \( \lambda = \frac{v}{f} \).
3. Step 3: For part (a), calculate the wavelength of the electromagnetic wave. The speed of electromagnetic waves in a vacuum (and approximately in air) is \( c = 3.00 \times 10^8 \; \text{m/s} \). Using \( f = 1.00 \; \text{kHz} = 1.00 \times 10^3 \; \text{Hz} \), substitute into the formula: \( \lambda_{\text{EM}} = \frac{c}{f} = \frac{3.00 \times 10^8}{1.00 \times 10^3} \).
4. Step 4: For part (b), calculate the wavelength of the sound wave. The speed of sound in air is given as \( v = 341 \; \text{m/s} \). Using the same frequency \( f = 1.00 \; \text{kHz} = 1.00 \times 10^3 \; \text{Hz} \), substitute into the formula: \( \lambda_{\text{sound}} = \frac{v}{f} = \frac{341}{1.00 \times 10^3} \).
5. Step 5: For part (c), consider whether a 1.00-kHz electromagnetic wave is audible. Electromagnetic waves are not sound waves; they are light waves. Humans can only hear sound waves within the frequency range of approximately 20 Hz to 20 kHz. Therefore, a 1.00-kHz electromagnetic wave is not audible because it is not a sound wave.

Key Concepts

Wavelength

Wavelength is the distance between successive peaks of a wave, typically measured in meters. It is inversely related to frequency, meaning that as frequency increases, wavelength decreases. The relationship is described by the equation λ = v/f, where λ is the wavelength, v is the wave speed, and f is the frequency. This concept is crucial for calculating the wavelengths of both electromagnetic and sound waves in the given question.

Wave Speed

Wave speed refers to the speed at which a wave propagates through a medium. For electromagnetic waves, the speed is approximately 3.00 x 10^8 m/s in a vacuum, while sound waves travel at different speeds depending on the medium; in air, it is about 341 m/s. Understanding wave speed is essential for determining the wavelength of waves at a given frequency, as seen in part (a) and (b) of the question.

Nature of Electromagnetic Waves vs. Sound Waves

Electromagnetic waves are oscillations of electric and magnetic fields that can travel through a vacuum, while sound waves are mechanical waves that require a medium (like air) to propagate. This distinction is important for part (c) of the question, as it addresses whether a human can hear a 1.00-kHz electromagnetic wave. Humans can only perceive sound waves, not electromagnetic waves, which is why the answer to part (c) is no.