The average energy-density of electromagnetic wave given by `E=(50N//C) sin (omegat-kx)` will be nearly:

Options

A. `10^(-8)j//m^(3)`
B. `10^(-7)j//m^(3)`
C. `10^(-6)j//m^(3)`
D. `10^(5)j//m^(3)`

Explanation

To find the average energy density of the electromagnetic wave given by the electric field \( E = (50 \, \text{N/C}) \sin(\omega t - kx) \), we can follow these steps:

### Step 1: Identify the amplitude of the electric field
The amplitude \( E_0 \) of the electric field is given as \( 50 \, \text{N/C} \).

### Step 2: Use the formula for average energy density
The average energy density \( U_{av} \) of an electromagnetic wave can be calculated using the formula:
\[ U_{av} = \frac{1}{2} \epsilon_0 E_0^2 \]
where \( \epsilon_0 \) is the permittivity of free space.

### Step 3: Substitute the values
We know:
- \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{N m}^2/\text{C}^2 \)
- \( E_0 = 50 \, \text{N/C} \)

Substituting these values into the formula:
\[ U_{av} = \frac{1}{2} \times (8.85 \times 10^{-12}) \times (50)^2 \]

### Step 4: Calculate \( E_0^2 \)
Calculating \( E_0^2 \):
\[ E_0^2 = 50^2 = 2500 \, \text{(N/C)}^2 \]

### Step 5: Calculate the average energy density
Now substituting \( E_0^2 \) back into the equation:
\[ U_{av} = \frac{1}{2} \times (8.85 \times 10^{-12}) \times 2500 \]
\[ U_{av} = \frac{1}{2} \times 22125 \times 10^{-12} \]
\[ U_{av} = 11062.5 \times 10^{-12} \, \text{J/m}^3 \]

### Step 6: Convert to scientific notation
To express this in scientific notation:
\[ U_{av} = 1.10625 \times 10^{-8} \, \text{J/m}^3 \]
Rounding this, we get:
\[ U_{av} \approx 1.1 \times 10^{-8} \, \text{J/m}^3 \]

### Conclusion
Thus, the average energy density of the electromagnetic wave is approximately \( 1.1 \times 10^{-8} \, \text{J/m}^3 \).