If the magnetic field in a plane electromagnetic wave is given by ` vec(B) = 6 xx 10^(-8) sin (1.6 xx 10^(3) x + 48 xx 10^(10)t) hatj T` ,. What will be expression for electric filed ?

Options

A. `vecE=2sin(1.6xx10^(3)x+48xx10^(10)t)hativ//m`
B. `vecE=18 sin (1.6xx10^(3)x+48xx10^(10)t)hativ//m`
C. `vecB=6sin(1.6xx10^(3)x+48xx10^(10)t)hativ//m`
D. `vecE=6xx10^(-8)sin(1.6xx10^(3)x+48xx10^(10)t)hativ//m`

Explanation

To find the expression for the electric field \( \vec{E} \) corresponding to the given magnetic field \( \vec{B} \) in a plane electromagnetic wave, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given magnetic field**:
The magnetic field is given as:
\[ \vec{B} = 6 \times 10^{-8} \sin(1.6 \times 10^{3} x + 48 \times 10^{10} t) \hat{j} \, \text{T} \]

2. **Determine the amplitude of the magnetic field**:
The amplitude \( B_0 \) of the magnetic field is:
\[ B_0 = 6 \times 10^{-8} \, \text{T} \]

3. **Use the relationship between electric field and magnetic field**:
In electromagnetic waves, the relationship between the electric field \( \vec{E} \) and the magnetic field \( \vec{B} \) is given by:
\[ E_0 = c B_0 \]
where \( c \) is the speed of light in vacuum, approximately \( 3 \times 10^{8} \, \text{m/s} \).

4. **Calculate the amplitude of the electric field**:
Substitute the values into the equation:
\[ E_0 = (3 \times 10^{8} \, \text{m/s}) \times (6 \times 10^{-8} \, \text{T}) = 18 \, \text{N/C} \]

5. **Determine the wave vector and angular frequency**:
From the given magnetic field expression, we can identify:
- Wave number \( k = 1.6 \times 10^{3} \, \text{m}^{-1} \)
- Angular frequency \( \omega = 48 \times 10^{10} \, \text{s}^{-1} \)

6. **Write the expression for the electric field**:
The electric field \( \vec{E} \) will have the same wave vector and angular frequency as the magnetic field, and it will be perpendicular to \( \vec{B} \). Thus, the expression for the electric field is:
\[ \vec{E} = E_0 \sin(kx + \omega t) \hat{i} \]
Substituting the values:
\[ \vec{E} = 18 \sin(1.6 \times 10^{3} x + 48 \times 10^{10} t) \hat{i} \, \text{N/C} \]

### Final Expression:
Thus, the expression for the electric field is:
\[ \vec{E} = 18 \sin(1.6 \times 10^{3} x + 48 \times 10^{10} t) \hat{i} \, \text{N/C} \]