If the magnetic field in plane electromagnetic was is given by ` vecB = 3 xx 10 ^( -8 ) sin ( 1.6 xx 10 ^( 3 ) x + 48 xx 10 ^( 10 ) t ) hatj T`, then what will be expression for electric field ?

Options

A. ` vec E = ( 3 xx 10 ^( - 8 ) sin ( 1.6 xx 1 0 ^(3) x + 48 xx 10 ^( 10) t ) hati V//m) `
B. ` vec E = ( 9 sin (1.6 xx 10 ^( - 3) x + 48 xx 10 ^( 10 ) t ) hatk V//m) `
C. ` vecE = ( 3xx 10 ^( - 8 ) sin ( 1.6 xx 10 ^( 3 ) x + 48 xx 10 ^( 10 ) t ) hatj V//m) `
D. `vecE = ( 60 sin (1.6 xx 10 ^( 3 ) x + 48 xx 10 ^( 10 ) t ) hatk V //m) `

Answer

To find the expression for the electric field (E) corresponding to the given magnetic field (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 by:
\[ \vec{B} = 3 \times 10^{-8} \sin(1.6 \times 10^{3} x + 48 \times 10^{10} t) \hat{j} \, \text{T} \]

2. **Identify Parameters:**
From the equation, we can identify:
- Amplitude of the magnetic field, \( B_0 = 3 \times 10^{-8} \, \text{T} \)
- Wave number \( k = 1.6 \times 10^{3} \, \text{m}^{-1} \)
- Angular frequency \( \omega = 48 \times 10^{10} \, \text{s}^{-1} \)

3. **Use the Relationship Between Electric and Magnetic Fields:**
In an electromagnetic wave, the relationship between the electric field (E) and the magnetic field (B) is given by:
\[ E = cB \]
where \( c \) is the speed of light in vacuum, approximately \( 3 \times 10^{8} \, \text{m/s} \).

4. **Calculate the Electric Field Amplitude:**
Substituting the values:
\[ E_0 = cB_0 = (3 \times 10^{8} \, \text{m/s})(3 \times 10^{-8} \, \text{T}) = 9 \, \text{V/m} \]

5. **Determine the Direction of the Electric Field:**
The magnetic field is in the \( \hat{j} \) direction (y-axis), and the direction of propagation is along the x-axis. According to the right-hand rule, if the magnetic field is along \( \hat{j} \) and the wave propagates along the x-axis, the electric field will be along the \( \hat{k} \) direction (z-axis).

6. **Write the Expression for the Electric Field:**
The electric field can be expressed as:
\[ \vec{E} = E_0 \sin(kx + \omega t) \hat{k} \]
Substituting the values of \( E_0 \), \( k \), and \( \omega \):
\[ \vec{E} = 9 \sin(1.6 \times 10^{3} x + 48 \times 10^{10} t) \hat{k} \, \text{V/m} \]

### Final Expression:
Thus, the expression for the electric field is:
\[ \vec{E} = 9 \sin(1.6 \times 10^{3} x + 48 \times 10^{10} t) \hat{k} \, \text{V/m} \]