To solve the given problem, we will analyze the electric field of the electromagnetic wave and derive the necessary parameters step by step.
### Step 1: Identify the given electric field equation
The electric field of the electromagnetic wave is given by:
\[ E = 10 \cos(10^7 t + kx) \hat{j} \, \text{V/m} \]
### Step 2: Determine the amplitude of the wave
The amplitude \( E_0 \) of the wave is the coefficient in front of the cosine function.
From the equation:
\[ E_0 = 10 \, \text{V/m} \]
### Step 3: Identify the angular frequency \( \omega \)
The angular frequency \( \omega \) is the coefficient of \( t \) in the argument of the cosine function.
From the equation:
\[ \omega = 10^7 \, \text{rad/s} \]
### Step 4: Calculate the wavelength \( \lambda \)
The relationship between the speed of light \( c \), angular frequency \( \omega \), and wavelength \( \lambda \) is given by:
\[ c = \omega \lambda \]
Rearranging gives:
\[ \lambda = \frac{c}{\omega} \]
Using \( c \approx 3 \times 10^8 \, \text{m/s} \):
\[ \lambda = \frac{3 \times 10^8}{10^7} = 30 \, \text{m} \]
### Step 5: Calculate the wave number \( k \)
The wave number \( k \) is related to the wavelength by:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( \lambda \):
\[ k = \frac{2\pi}{30} \approx 0.209 \, \text{rad/m} \]
### Step 6: Determine the direction of propagation
The wave is represented in the form \( E = E_0 \cos(\omega t + kx) \hat{j} \). The positive sign before \( kx \) indicates that the wave is propagating in the negative x-direction. However, since the electric field is in the \( \hat{j} \) direction, it implies that the wave is propagating in the positive x-direction.
### Summary of Results
1. Amplitude \( E_0 = 10 \, \text{V/m} \)
2. Angular frequency \( \omega = 10^7 \, \text{rad/s} \)
3. Wavelength \( \lambda = 30 \, \text{m} \)
4. Wave number \( k = 0.209 \, \text{rad/m} \)
5. Direction of propagation: +x direction
### Conclusion
Based on the calculations, the correct statements are:
- The wave amplitude is \( 10 \, \text{V/m} \) (True).
- The wave is propagating along the +x direction (True).
- The wavelength \( \lambda \) is not \( 188.4 \, \text{m} \) but \( 30 \, \text{m} \) (False).
- The wave number \( k \) is not \( 0.33 \, \text{rad/m} \) but \( 0.209 \, \text{rad/m} \) (False).
