If `u_(E )` and `u_(B)` be the time average of the electric and magnetic field energy densities at a point due to electromagnetic wave, then

Options

A. `u_(E )=u_(B)`
B. `u_(E )=2u_(B)`
C. `2u_(E )=mu_(B)`
D. none of these

Answer

To solve the problem of finding the relationship between the time average electric field energy density \( u_E \) and the magnetic field energy density \( u_B \) due to electromagnetic waves, we will follow these steps:

### Step 1: Write the expressions for energy densities
The energy density of the electric field \( u_E \) is given by:
\[ u_E = \frac{1}{2} \epsilon_0 E_0^2 \]
where \( \epsilon_0 \) is the permittivity of free space and \( E_0 \) is the maximum electric field strength.

The energy density of the magnetic field \( u_B \) is given by:
\[ u_B = \frac{1}{2} \frac{B_0^2}{\mu_0} \]
where \( B_0 \) is the maximum magnetic field strength and \( \mu_0 \) is the permeability of free space.

### Step 2: Relate \( E_0 \) and \( B_0 \)
In electromagnetic waves, the relationship between the electric field and the magnetic field is given by:
\[ c = \frac{E_0}{B_0} \]
where \( c \) is the speed of light in vacuum. This can be rearranged to express \( E_0 \) in terms of \( B_0 \):
\[ E_0 = c B_0 \]

### Step 3: Substitute \( E_0 \) into the expression for \( u_E \)
Now, substitute \( E_0 = c B_0 \) into the expression for \( u_E \):
\[ u_E = \frac{1}{2} \epsilon_0 (c B_0)^2 \]
This simplifies to:
\[ u_E = \frac{1}{2} \epsilon_0 c^2 B_0^2 \]

### Step 4: Use the relationship between \( c \), \( \epsilon_0 \), and \( \mu_0 \)
We know that:
\[ c^2 = \frac{1}{\mu_0 \epsilon_0} \]
Substituting this into the expression for \( u_E \):
\[ u_E = \frac{1}{2} \epsilon_0 \left(\frac{1}{\mu_0 \epsilon_0}\right) B_0^2 \]
This simplifies to:
\[ u_E = \frac{1}{2} \frac{B_0^2}{\mu_0} \]

### Step 5: Compare \( u_E \) and \( u_B \)
Now we can compare \( u_E \) and \( u_B \):
\[ u_E = \frac{1}{2} \frac{B_0^2}{\mu_0} = u_B \]
Thus, we find that:
\[ u_E = u_B \]

### Conclusion
The time average electric field energy density \( u_E \) is equal to the time average magnetic field energy density \( u_B \):
\[ u_E = u_B \]

### Final Answer
The correct option is that \( u_E = u_B \).

---