A certain 60.0-hertz ac powerline radiates an electromagnetic wave having a maximum electric field strength of 13.0 kilovolts per meter. What is the wavelength of this very-low-frequency electromagnetic wave? What is the magnitude of the strongest magnetic field present in this wave?
We’re told in this statement that the powerline has a frequency of 60.0 hertz; we’ll call this value 𝑓. We’re also told that the wave it radiates has a maximum electric field strength of 13.0 kilovolts per meter; we’ll call this 𝐸 sub max. We want to know the wavelength of this wave, which we’ll call 𝜆, and we want to know the magnitude of the strongest magnetic field present in the wave, which we’ll call 𝐵 sub max.
To begin our solution, we can recall that, in general, the speed of a wave 𝑣 is equal to the frequency of that wave times its wavelength. When we apply this relationship to our scenario, since we have an electromagnetic wave, our wave speed is 𝑐, the speed of light, which is equal to 𝑓 times 𝜆. When we rearrange to solve for 𝜆, the wavelength, we see that it’s equal to 𝑐 over 𝑓.
We’ve been told 𝑓 in the problem statement, and we’ll assume that 𝑐 is exactly 3.00 times 10 to the eighth meters per second. We can then plug in for 𝑐 and 𝑓. And when we calculate this fraction, we find that 𝜆 is equal to 5.00 times 10 to the sixth meters. That’s the wavelength of this electromagnetic wave.
Next, we move on to solving for the maximum magnetic field strength of this wave. And to do that, we can recall a relationship between electric and magnetic fields. Given an electric field 𝐸, the magnitude of the corresponding magnetic field 𝐵 is equal to 𝐸 divided by the speed of light 𝑐. So in our case, 𝐵 sub max equals 𝐸 sub max divided by 𝑐.
Plugging in for these values and making sure to write 𝐸 sub max in units of volts per meter, when we calculate this fraction, we find that 𝐵 sub max is equal to 4.33 times 10 to the negative fifth tesla. That’s the maximum magnetic field magnitude produced by this electromagnetic wave.