The electric field in an electromagnetic wave is given by E = 56.5 sin `omega(t-x//c) NC^(-1)`. Find the intensity of the wave if it is propagating along x-axis in the free space. (Given `epsi_0=8.85 xx 10^(-12)C^2N^(-1)m^(-2))`

Options

A. `5.65Wm^(-2)`
B. `4.24Wm^(-2)`
C. `1.9xx10^(-7)Wm^(-2)`
D. `56.5Wm^(-2)`

Explanation

To find the intensity of the electromagnetic wave given the electric field \( E = 56.5 \sin(\omega(t - \frac{x}{c})) \, \text{N/C} \), we can follow these steps:

### Step 1: Identify the amplitude of the electric field
The electric field is given in the form \( E = E_0 \sin(\omega(t - \frac{x}{c})) \). Here, \( E_0 \) is the amplitude of the electric field. From the given equation, we have:
\[ E_0 = 56.5 \, \text{N/C} \]

### Step 2: Use the formula for intensity
The intensity \( I \) of an electromagnetic wave can be calculated using the formula:
\[ I = \frac{1}{2} \epsilon_0 E_0^2 c \]
where:
- \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \) (permittivity of free space),
- \( c = 3 \times 10^8 \, \text{m/s} \) (speed of light in vacuum).

### Step 3: Substitute the values into the intensity formula
Now, we can substitute the known values into the intensity formula:
\[ I = \frac{1}{2} (8.85 \times 10^{-12}) (56.5)^2 (3 \times 10^8) \]

### Step 4: Calculate \( E_0^2 \)
First, calculate \( E_0^2 \):
\[ E_0^2 = (56.5)^2 = 3196.25 \, \text{(N/C)}^2 \]

### Step 5: Substitute \( E_0^2 \) back into the intensity formula
Now substitute \( E_0^2 \) back into the intensity formula:
\[ I = \frac{1}{2} (8.85 \times 10^{-12}) (3196.25) (3 \times 10^8) \]

### Step 6: Perform the calculations
Calculating step-by-step:
1. Calculate \( (8.85 \times 10^{-12}) \times (3196.25) \):
\[ 8.85 \times 10^{-12} \times 3196.25 \approx 2.83 \times 10^{-8} \]
2. Now multiply by \( 3 \times 10^8 \):
\[ 2.83 \times 10^{-8} \times 3 \times 10^8 = 8.49 \]
3. Finally, divide by 2:
\[ I = \frac{8.49}{2} \approx 4.245 \, \text{W/m}^2 \]

### Final Result
Thus, the intensity of the wave is approximately:
\[ I \approx 4.245 \, \text{W/m}^2 \]

---