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. `1.41 xx 10^(-8)T`
B. `2.83 xx 10^(-8)T`
C. `0.70 xx 10^(-8) T`
D. `4.23 xx 10^(-8) T`

Explanation

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

### Step 1: Understand the relationship between electric and magnetic fields in electromagnetic waves.
In electromagnetic waves, the relationship between the RMS value of the electric field (E_rms) and the peak value of the magnetic field (B_0) is given by the equation:

\[ E_{rms} = \frac{B_0 \cdot c}{\sqrt{2}} \]

where \(c\) is the speed of light in free space, approximately \(3 \times 10^8 \, \text{m/s}\).

### Step 2: Calculate the peak value of the electric field (E_0).
The peak value of the electric field (E_0) can be calculated from the RMS value using the relationship:

\[ E_0 = E_{rms} \cdot \sqrt{2} \]

Given that \(E_{rms} = 6 \, \text{V/m}\):

\[ E_0 = 6 \cdot \sqrt{2} = 6 \cdot 1.414 \approx 8.49 \, \text{V/m} \]

### Step 3: Use the relationship between E_0 and B_0.
The relationship between the peak electric field (E_0) and the peak magnetic field (B_0) in free space is given by:

\[ E_0 = B_0 \cdot c \]

Rearranging this gives:

\[ B_0 = \frac{E_0}{c} \]

### Step 4: Substitute the values to find B_0.
Now, substituting the values we have:

\[ B_0 = \frac{8.49 \, \text{V/m}}{3 \times 10^8 \, \text{m/s}} \approx 2.83 \times 10^{-8} \, \text{T} \]

### Final Result:
The peak value of the magnetic field is approximately:

\[ B_0 \approx 2.83 \times 10^{-8} \, \text{T} \]