Total energy of density of electromagnetic waves in vacuum is given by the relation

Options

A. `1/2.(E^(2))/(epsi_(0)) + (B^(2))/(2mu_(0))`
B. `1/(2)epsi_(0)E^(2) + 1/2mu_(0)B^(2)`
C. `(E^(2) + B^(2))/(c)`
D. `(1)/(2) epsi_(0)E^(2) + (B^(2))/(2mu_(0))`

Explanation

To find the total energy density of electromagnetic waves in a vacuum, we can break down the problem into several steps.

### Step-by-Step Solution:

1. **Understanding Energy Density**:
The total energy density (u) of electromagnetic waves in a vacuum is the sum of the energy densities of the electric field (u_E) and the magnetic field (u_B).

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

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

4. **Relating Electric and Magnetic Fields**:
In electromagnetic waves, 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.

5. **Substituting B in Terms of E**:
We can express \( B^2 \) in terms of \( E^2 \):
\[ B^2 = \left(\frac{E}{c}\right)^2 = \frac{E^2}{c^2} \]

6. **Total Energy Density**:
Now, substituting \( B^2 \) into the equation for \( u_B \):
\[ u_B = \frac{1}{2} \frac{E^2/c^2}{\mu_0} = \frac{E^2}{2\mu_0 c^2} \]
Since \( c^2 = \frac{1}{\epsilon_0 \mu_0} \), we can substitute this into the equation:
\[ u_B = \frac{E^2}{2\mu_0 \cdot \frac{1}{\epsilon_0 \mu_0}} = \frac{E^2 \epsilon_0}{2} \]

7. **Combining the Energy Densities**:
Now we can combine \( u_E \) and \( u_B \):
\[ u = u_E + u_B = \frac{1}{2} \epsilon_0 E^2 + \frac{1}{2} \epsilon_0 E^2 = \epsilon_0 E^2 \]

8. **Final Expression**:
Thus, the total energy density of electromagnetic waves in vacuum is:
\[ u = \epsilon_0 E^2 \]

### Final Answer:
The total energy density of electromagnetic waves in vacuum is given by:
\[ u = \epsilon_0 E^2 \]