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. `vec(E) = 2 sin (1.6 xx 10^(3) x 48xx 10^(10) t) hati v//m`
B. `vec(E) = 18 sin (1.6 xx 10^(3) x+ 48xx 10^(10) t) hatk v//m`
C. `vec(E) = 6 sin (1.6 xx 10^(3) x+ 48xx 10^(10) t) hati v//m`
D. `vec(E) = 6xx 10^(-18)sin (1.6 xx 10^(3) x+ 48xx 10^(10) t) hati v//m`

Answer

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 1: Identify the given magnetic field
The magnetic field is given by:
\[ \vec{B} = 6 \times 10^{-8} \sin(1.6 \times 10^{3} x + 4.48 \times 10^{10} t) \hat{j} \, \text{T} \]

### Step 2: Use the relationship between electric field and magnetic field
In an electromagnetic wave, the magnitudes of the electric field \( E \) and magnetic field \( B \) are related by the equation:
\[ \frac{E}{B} = c \]
where \( c \) is the speed of light in vacuum, approximately \( 3 \times 10^8 \, \text{m/s} \).

### Step 3: Calculate the magnitude of the electric field
From the equation above, we can express the electric field \( E \) as:
\[ E = c \cdot B \]
Substituting the values:
\[ E = (3 \times 10^8) \cdot (6 \times 10^{-8}) \]

### Step 4: Simplify the expression
Calculating the above expression:
\[ E = 3 \times 6 = 18 \, \text{V/m} \]

### Step 5: Determine the direction of the electric field
Since the magnetic field \( \vec{B} \) is in the \( \hat{j} \) direction and the wave is propagating in the \( \hat{i} \) direction (along the x-axis), the electric field \( \vec{E} \) will be in the \( \hat{k} \) direction (along the z-axis) due to the right-hand rule.

### Step 6: Write the final expression for the electric field
Thus, the electric field can be expressed as:
\[ \vec{E} = 18 \sin(1.6 \times 10^{3} x + 4.48 \times 10^{10} t) \hat{k} \, \text{V/m} \]

### Final Answer
\[ \vec{E} = 18 \times 10^{-8} \sin(1.6 \times 10^{3} x + 4.48 \times 10^{10} t) \hat{k} \, \text{V/m} \]

---