The average intensity of electromagnetic wave is (where symbols have their usual meanings)

Options

A. `epsilon_0 × E^2_RMS × c`
B. `(B^2_RMS/ mu_0) × c`
C. `(1/2) × (epsilon_0) × E^2_RMS × c`
D. Both (A) & (B)

Answer

To find the average intensity of an electromagnetic wave, we can use two formulas that relate the average intensity (I_avg) to the electric field (E_rms) and magnetic field (B_rms) components of the wave.

### Step-by-Step Solution:

1. **Understanding the Formulas**:
The average intensity of an electromagnetic wave can be expressed using two different formulas:
\[ I_{\text{avg}} = \epsilon_0 E_{\text{rms}}^2 c \]
and
\[ I_{\text{avg}} = \frac{B_{\text{rms}}^2}{\mu_0} c \]
where:
- \(I_{\text{avg}}\) is the average intensity (in watts per meter squared, W/m²).
- \(\epsilon_0\) is the permittivity of free space (\(8.85 \times 10^{-12} \, \text{F/m}\)).
- \(E_{\text{rms}}\) is the root mean square value of the electric field (in newtons per coulomb, N/C).
- \(c\) is the speed of light (\(3 \times 10^8 \, \text{m/s}\)).
- \(B_{\text{rms}}\) is the root mean square value of the magnetic field (in tesla, T).
- \(\mu_0\) is the permeability of free space (\(1.26 \times 10^{-6} \, \text{N/A}^2\)).

2. **Using the First Formula**:
Let's calculate the average intensity using the first formula:
\[ I_{\text{avg}} = \epsilon_0 E_{\text{rms}}^2 c \]
Substitute the known values:
\[ I_{\text{avg}} = (8.85 \times 10^{-12}) E_{\text{rms}}^2 (3 \times 10^8) \]

3. **Using the Second Formula**:
Now, let's calculate the average intensity using the second formula:
\[ I_{\text{avg}} = \frac{B_{\text{rms}}^2}{\mu_0} c \]
Substitute the known values:
\[ I_{\text{avg}} = \frac{B_{\text{rms}}^2}{1.26 \times 10^{-6}} (3 \times 10^8) \]

4. **Equating the Two Formulas**:
Since both expressions represent the average intensity of the same electromagnetic wave, we can equate them:
\[ (8.85 \times 10^{-12}) E_{\text{rms}}^2 (3 \times 10^8) = \frac{B_{\text{rms}}^2}{1.26 \times 10^{-6}} (3 \times 10^8) \]

5. **Solving for the Relationship**:
From the above equation, we can derive a relationship between \(E_{\text{rms}}\) and \(B_{\text{rms}}\):
\[ E_{\text{rms}}^2 = \frac{B_{\text{rms}}^2}{\mu_0 \epsilon_0} \]
This shows how the electric and magnetic fields are related in an electromagnetic wave.

6. **Final Answer**:
The average intensity of an electromagnetic wave can be expressed in terms of either the electric field or the magnetic field using the formulas derived above.