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Chapter 22: Problem 25

In the United States, the ac household current oscillates at a frequency of \(60 \mathrm{Hz}\). In the time it takes for the current to make one oscillation, how far has the electromagnetic wave traveled from the current carrying wire? This distance is the wavelength of a \(60-\mathrm{Hz}\) EM wave. Compare this length with the distance from Boston to Los Angeles $(4200 \mathrm{km})$.

Short Answer

Expert verified
Answer: The wavelength of a 60 Hz electromagnetic wave is 5 x 10^6 m, which is greater than the distance between Boston and Los Angeles (4.2 x 10^6 m).

Step by step solution

01

Recall the formula for the speed of light in terms of frequency and wavelength

The speed of light (c) can be found using the formula \(c = f\lambda\), where f is the frequency and \(\lambda\) is the wavelength.
02

Use the given frequency to find the wavelength

We are given the frequency, \(f = 60 \,\text{Hz}\). The speed of light, \(c = 3 \times 10^8 \,\text{m/s}\). Plugging these values into the formula, we get: \(\lambda = \frac{c}{f}=\frac{3 \times 10^8 \,\text{m/s}}{60\,\text{Hz}}=5\times10^6\,\text{m}\) So, the wavelength of a \(60-\mathrm{Hz}\) EM wave is \(5\times10^6\,\text{m}\).
03

Compare the wavelength with the given distance

We're asked to compare this wavelength with the distance from Boston to Los Angeles, which is \(4200\,\text{km}\) or \(4.2\times10^6\,\text{m}\). The wavelength of the EM wave, \(5\times10^6\,\text{m}\), is greater than the distance between Boston and Los Angeles, \(4.2\times10^6\,\text{m}\). In conclusion, the wavelength of a \(60-\mathrm{Hz}\) EM wave is \(5\times10^6\,\text{m}\), which is greater than the distance from Boston to Los Angeles.

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Most popular questions from this chapter

The electric field in a microwave traveling through air has amplitude $0.60 \mathrm{mV} / \mathrm{m}\( and frequency \)30 \mathrm{GHz}$. Find the amplitude and frequency of the magnetic field.
The electric field in a radio wave traveling through vacuum has amplitude $2.5 \times 10^{-4} \mathrm{V} / \mathrm{m}\( and frequency \)1.47 \mathrm{MHz}$. (a) Find the amplitude and frequency of the magnetic field. (b) The wave is traveling in the \(+x\) -direction. At \(x=0\) and \(t=0,\) the electric field is \(1.5 \times 10^{-4} \mathrm{V} / \mathrm{m}\) in the \- y-direction. What are the magnitude and direction of the magnetic field at \(x=0\) and \(t=0 ?\).
In musical acoustics, a frequency ratio of 2: 1 is called an octave. Humans with extremely good hearing can hear sounds ranging from \(20 \mathrm{Hz}\) to \(20 \mathrm{kHz}\), which is approximately 10 octaves (since $2^{10}=1024 \approx 1000$ ). (a) Approximately how many octaves of visible light are humans able to perceive? (b) Approximately how many octaves wide is the microwave region?
An AM radio station broadcasts at \(570 \mathrm{kHz}\). (a) What is the wavelength of the radio wave in air? (b) If a radio is tuned to this station and the inductance in the tuning circuit is \(0.20 \mathrm{mH},\) what is the capacitance in the tuning circuit? (c) In the vicinity of the radio, the amplitude of the electric field is \(0.80 \mathrm{V} / \mathrm{m} .\) The radio uses a coil antenna of radius \(1.6 \mathrm{cm}\) with 50 turns. What is the maximum emf induced in the antenna, assuming it is oriented for best reception? Assume that the fields are sinusoidal functions of time.
The index of refraction of water is \(1.33 .\) (a) What is the speed of light in water? (b) What is the wavelength in water of a light wave with a vacuum wavelength of \(515 \mathrm{nm} ?\)
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