91Ó°ÊÓ

The frequency of an electromagnetic wave in vacuum is directly related to its angular wave number. The wave number, denoted as \( k \), is the spatial frequency of the wave and is calculated by dividing \( 2 \pi \) by the wavelength \( \lambda \). In this problem, the wave number \( k \) is given as \( 4.00 \text{ m}^{-1} \). By using the relationship \[ k = \frac{2 \pi f}{c} \], where \( c \) is the speed of light in a vacuum (\( 3.00 \times 10^8 \text{ m/s} \)), we can solve for the frequency \( f \).
Rearrange the equation to get:
\[ f = \frac{k c}{2 \pi} \]
Substituting the given values:
\[ f = \frac{4.00 \times 3.00 \times 10^8}{2 \pi} \approx 1.91 \times 10^8 \text{ Hz} \]
This frequency represents how many oscillations the wave completes per second.

The root mean square (RMS) value of the electric field component of an electromagnetic wave is an important measure that reflects the field's effective value. First, we need to determine the peak electric field \( E_0 \) using the given peak magnetic field \( B_0 \). The given value is \( 85.8 \text{ nT} = 85.8 \times 10^{-9} \text{ T} \). The relationship between the peak electric and magnetic fields in vacuum is \[ E_0 = c B_0 \]
Substituting the known values:
\[ E_0 = 3.00 \times 10^8 \times 85.8 \times 10^{-9} = 25.7 \text{ V/m} \]
To find the RMS value, we use the formula:
\[ E_{\text{rms}} = \frac{E_0}{\sqrt{2}} \]
Substituting \( E_0 \):
\[ E_{\text{rms}} = \frac{25.7}{\sqrt{2}} \approx 18.2 \text{ V/m} \]
This RMS value is crucial for calculating other properties, such as the intensity of the light.

The intensity of an electromagnetic wave represents the energy per unit area per unit time that the wave carries. It can be calculated using the peak electric field \( E_0 \). The formula for the intensity \( I \) of light is:
\[ I = \frac{1}{2} c \epsilon_0 E_0^2 \]
where \( \epsilon_0 \) is the permittivity of free space \( 8.85 \times 10^{-12} \text{ F/m} \).
Using the given values:
\[ I = \frac{1}{2} \times 3.00 \times 10^8 \times 8.85 \times 10^{-12} \times (25.7)^2 \]
Compute the value to find:
\[ I \approx 8.74 \text{ W/m}^2 \]
This intensity value tells us the power of the wave spread over a specific area, which is fundamental in understanding the energy dynamics of electromagnetic waves.