The average value of electric energy density in an electromagnetic wave is (`E_(0)` is peak value):

Options

A. `(1)/(2)epsilon_(0)E^(2)`
B. `(E^(2))/(2epsilon_(0))`
C. `epsilon_(0)E^(2)`
D. `(1)/(4)epsilon_(0)E^(2)`

Answer

To find the average value of electric energy density in an electromagnetic wave, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Electric Energy Density**:
The electric energy density \( u \) in an electromagnetic wave is given by the formula:
\[ u = \frac{1}{2} \epsilon_0 E_{\text{rms}}^2 \]
where \( \epsilon_0 \) is the permittivity of free space and \( E_{\text{rms}} \) is the root mean square (rms) value of the electric field.

2. **Relating Peak Value to RMS Value**:
The rms value of the electric field \( E_{\text{rms}} \) can be expressed in terms of the peak value \( E_0 \):
\[ E_{\text{rms}} = \frac{E_0}{\sqrt{2}} \]

3. **Substituting RMS Value into Energy Density Formula**:
Substitute the expression for \( E_{\text{rms}} \) into the energy density formula:
\[ u = \frac{1}{2} \epsilon_0 \left( \frac{E_0}{\sqrt{2}} \right)^2 \]

4. **Calculating the Square**:
Calculate the square of \( E_{\text{rms}} \):
\[ \left( \frac{E_0}{\sqrt{2}} \right)^2 = \frac{E_0^2}{2} \]

5. **Final Expression for Energy Density**:
Substitute this back into the energy density formula:
\[ u = \frac{1}{2} \epsilon_0 \cdot \frac{E_0^2}{2} \]
Simplifying this gives:
\[ u = \frac{1}{4} \epsilon_0 E_0^2 \]

6. **Conclusion**:
Thus, the average value of electric energy density in an electromagnetic wave is:
\[ u = \frac{1}{4} \epsilon_0 E_0^2 \]

### Final Answer:
The average value of electric energy density in an electromagnetic wave is \( \frac{1}{4} \epsilon_0 E_0^2 \).

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