In an electromanetic wave, the average energy density associated with electric field is

Options

A. `CV^(2)//2`
B. `Q^(2)//2C`
C. `epsi_(0)^(2)//2E_(0)`
D. `epsi_(0)E^(2)//2`

Answer

To find the average energy density associated with the electric field in an electromagnetic wave, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Energy Density**:
The energy density (u) in an electromagnetic wave is the energy stored per unit volume. It is composed of contributions from both the electric field (E) and the magnetic field (B).

2. **Average Energy Density Formulas**:
The average energy density associated with the electric field (u_E) is given by:
\[ u_E = \frac{1}{2} \epsilon_0 E^2 \]
where \( \epsilon_0 \) is the permittivity of free space and \( E \) is the electric field strength.

3. **Average Energy Density of Magnetic Field**:
Similarly, the average energy density associated with the magnetic field (u_B) is given by:
\[ u_B = \frac{1}{2} \frac{B^2}{\mu_0} \]
where \( \mu_0 \) is the permeability of free space and \( B \) is the magnetic field strength.

4. **Relationship Between E and B**:
In an electromagnetic wave, the electric field (E) and magnetic field (B) are related by the equation:
\[ B = \frac{E}{c} \]
where \( c \) is the speed of light in vacuum, given by \( c = \frac{1}{\sqrt{\epsilon_0 \mu_0}} \).

5. **Substituting B in Terms of E**:
To express the magnetic energy density in terms of the electric field, we substitute \( B \) into the equation for \( u_B \):
\[ u_B = \frac{1}{2} \frac{(E/c)^2}{\mu_0} = \frac{1}{2} \frac{E^2}{\mu_0 c^2} \]
Since \( \mu_0 c^2 = \frac{1}{\epsilon_0} \), we can rewrite \( u_B \) as:
\[ u_B = \frac{1}{2} \epsilon_0 E^2 \]

6. **Total Average Energy Density**:
The total average energy density (u_total) in an electromagnetic wave is the sum of the energy densities from the electric and magnetic fields:
\[ u_{total} = u_E + u_B = \frac{1}{2} \epsilon_0 E^2 + \frac{1}{2} \epsilon_0 E^2 = \epsilon_0 E^2 \]

7. **Final Expression for Average Energy Density Associated with Electric Field**:
Since the question specifically asks for the average energy density associated with the electric field, we can conclude:
\[ u_E = \frac{1}{2} \epsilon_0 E^2 \]

### Conclusion:
The average energy density associated with the electric field in an electromagnetic wave is given by:
\[ u_E = \frac{1}{2} \epsilon_0 E^2 \]