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. `6.37 xx 10^(-9) J m^(-3)`
B. `81.35 xx 10^(-12) J m^(-3)`
C. `3.5 xx 10^(-3) J m^(-3)`
D. `4.58 xx 10^(-6) 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 relationship between RMS value and peak electric field
The RMS (Root Mean Square) value of the electric field \( E_{\text{rms}} \) is related to the peak electric field \( E_0 \) by the formula:
\[ E_{\text{rms}} = \frac{E_0}{\sqrt{2}} \]
Given that \( E_{\text{rms}} = 720 \, \text{N/C} \), we can rearrange this to find \( E_0 \):
\[ E_0 = E_{\text{rms}} \times \sqrt{2} = 720 \times \sqrt{2} \]

### Step 2: Calculate the peak electric field \( E_0 \)
Calculating \( E_0 \):
\[ E_0 = 720 \times \sqrt{2} \approx 720 \times 1.414 \approx 1015.84 \, \text{N/C} \]

### Step 3: Use the formula for energy density
The average energy density \( u \) of an electromagnetic wave is given by:
\[ u = \frac{1}{2} \epsilon_0 E_0^2 \]
where \( \epsilon_0 \) (the permittivity of free space) is approximately \( 8.85 \times 10^{-12} \, \text{F/m} \).

### Step 4: Substitute \( E_0 \) into the energy density formula
Substituting \( E_0 \) into the formula:
\[ u = \frac{1}{2} \times (8.85 \times 10^{-12}) \times (1015.84)^2 \]

### Step 5: Calculate \( E_0^2 \)
Calculating \( E_0^2 \):
\[ E_0^2 \approx (1015.84)^2 \approx 1031870.65 \, \text{(N/C)}^2 \]

### Step 6: Calculate the energy density \( u \)
Now substituting back into the energy density formula:
\[ u = \frac{1}{2} \times (8.85 \times 10^{-12}) \times (1031870.65) \approx \frac{1}{2} \times 9.136 \times 10^{-6} \approx 4.568 \times 10^{-6} \, \text{J/m}^3 \]

### Step 7: Final result
Rounding the value, we find:
\[ u \approx 4.58 \times 10^{-6} \, \text{J/m}^3 \]

### Conclusion
The total energy density of the electromagnetic wave is approximately:
\[ \boxed{4.58 \times 10^{-6} \, \text{J/m}^3} \]