Question

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

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

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.

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 \(5\times10^6\,\text{m}\).

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.