Certainly! Let's analyze the given equation for the magnetic field of the plane-polarized electromagnetic wave and extract the relevant information to determine the wavelength and frequency of the wave.
The magnetic field is given by:
[tex]\[ B = 1.2 \times 10^{-6} \sin \left[2 \pi \left(\frac{z}{240} - \frac{t \times 10^7}{8}\right)\right] \][/tex]
Let's break this down step-by-step:
### Part (a) - Wavelength of the EM Wave
The general form for a wave traveling in the [tex]\( z \)[/tex]-direction is:
[tex]\[ \sin \left(2 \pi \left(\frac{z}{\lambda} - f t\right)\right) \][/tex]
By comparing this with our given equation:
[tex]\[ \sin \left[2 \pi \left(\frac{z}{240} - \frac{t \times 10^7}{8}\right)\right] \][/tex]
We can identify that:
[tex]\[ \frac{z}{\lambda} = \frac{z}{240} \][/tex]
This implies:
[tex]\[ \lambda = 240 \, \text{meters} \][/tex]
Therefore, the wavelength ([tex]\(\lambda\)[/tex]) of the EM wave is:
[tex]\[ \lambda = 240 \, \text{meters} \][/tex]
### Part (b) - Frequency of the EM Wave
Next, we'll find the frequency [tex]\( f \)[/tex]. Again, comparing the general form with the given equation, we have:
[tex]\[ f t = \frac{t \times 10^7}{8} \][/tex]
This implies:
[tex]\[ f = \frac{10^7}{8} \, \text{s}^{-1} \][/tex]
Therefore, the frequency ([tex]\(f\)[/tex]) of the EM wave is:
[tex]\[ f = \frac{10^7}{8} \, \text{Hz} \][/tex]
Simplifying:
[tex]\[ f = 1.25 \times 10^6 \, \text{Hz} \][/tex]
Or in standard form:
[tex]\[ f = 1,250,000 \, \text{Hz} \][/tex]
In summary:
- The wavelength ([tex]\(\lambda\)[/tex]) of the EM wave is 240 meters.
- The frequency ([tex]\(f\)[/tex]) of the EM wave is 1,250,000 Hz (or 1.25 MHz).
