In an electromagnetic wave in free space the root mean square value of the electric field is `E_(rms)=6 V//m`. The peak value of the magnetic field is

Options

A. `1.41xx10^(-8)T`
B. `2.83xx10^(-8)T`
C. `0.70xx10^(-8)T`
D. `4.23xx10^(-8)T`

Answer

To find the peak value of the magnetic field (B₀) in an electromagnetic wave given the root mean square (RMS) value of the electric field (Eₘₛ), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values**:
- The root mean square value of the electric field, \( E_{\text{rms}} = 6 \, \text{V/m} \).

2. **Use the relationship between RMS values of electric and magnetic fields**:
- For electromagnetic waves in free space, the relationship between the RMS values of the electric field and the magnetic field is given by:
\[ \frac{E_{\text{rms}}}{B_{\text{rms}}} = c \]
where \( c \) is the speed of light in vacuum, approximately \( 3 \times 10^8 \, \text{m/s} \).

3. **Rearranging the formula to find \( B_{\text{rms}} \)**:
- We can rearrange the equation to find the RMS value of the magnetic field:
\[ B_{\text{rms}} = \frac{E_{\text{rms}}}{c} \]

4. **Substituting the known values**:
- Substitute \( E_{\text{rms}} = 6 \, \text{V/m} \) and \( c = 3 \times 10^8 \, \text{m/s} \) into the equation:
\[ B_{\text{rms}} = \frac{6 \, \text{V/m}}{3 \times 10^8 \, \text{m/s}} = 2 \times 10^{-8} \, \text{T} \]

5. **Finding the peak value of the magnetic field \( B_0 \)**:
- The peak value of the magnetic field \( B_0 \) is related to the RMS value by:
\[ B_0 = B_{\text{rms}} \times \sqrt{2} \]

6. **Calculating \( B_0 \)**:
- Substitute \( B_{\text{rms}} = 2 \times 10^{-8} \, \text{T} \):
\[ B_0 = 2 \times 10^{-8} \, \text{T} \times \sqrt{2} \approx 2.83 \times 10^{-8} \, \text{T} \]

### Final Answer:
The peak value of the magnetic field is approximately \( B_0 \approx 2.83 \times 10^{-8} \, \text{T} \).

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