The rms value of the electric field of the light from the sun is `720 N//C` The total energy density of the electromagnetic wave is

Options

A. `3.3xx10^(-3)"J m"^(-3)`
B. `4.58xx10^(-6)"J m"^(-3)`
C. `6.37xx10^(-9)"J m"^(-3)`
D. `81.35xx10^(-12)"J m"^(-3)`.

Explanation

To find the total energy density of the electromagnetic wave given the RMS value of the electric field, we can follow these steps:

### Step 1: Understand the formula for energy density
The total energy density \( U \) of an electromagnetic wave can be expressed as the sum of the energy density due to the electric field \( U_E \) and the energy density due to the magnetic field \( U_B \):
\[ U = U_E + U_B \]

### Step 2: Write the formula for energy density due to the electric field
The energy density due to 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, approximately \( 8.85 \times 10^{-12} \, \text{F/m} \)
- \( E \) is the RMS value of the electric field.

### Step 3: Write the formula for energy density due to the magnetic field
The energy density due to the magnetic field is given by:
\[ U_B = \frac{1}{2} \frac{B^2}{\mu_0} \]
However, for electromagnetic waves, the energy densities due to the electric and magnetic fields are equal:
\[ U_E = U_B \]
Thus, we can express the total energy density as:
\[ U = 2 U_E \]

### Step 4: Substitute the expression for \( U_E \) into the total energy density formula
Substituting \( U_E \) into the total energy density formula gives:
\[ U = 2 \left( \frac{1}{2} \epsilon_0 E^2 \right) = \epsilon_0 E^2 \]

### Step 5: Substitute the values
Now, we can substitute the values into the equation. Given that \( E = 720 \, \text{N/C} \):
\[ U = \epsilon_0 E^2 = (8.85 \times 10^{-12} \, \text{F/m}) \times (720 \, \text{N/C})^2 \]

### Step 6: Calculate the result
Calculating the above expression:
\[ U = 8.85 \times 10^{-12} \times 720^2 \]
\[ = 8.85 \times 10^{-12} \times 518400 \]
\[ \approx 4.587 \times 10^{-6} \, \text{J/m}^3 \]

### Final Answer
The total energy density of the electromagnetic wave is approximately:
\[ U \approx 4.587 \times 10^{-6} \, \text{J/m}^3 \]