Magnetic field of an electromagnetic wave is `vecB=12 × 10^(-9) sin(kx - omega t)hatk (T)`. The equation of corresponding electric field should be

Options

A. Option1 `E = 36 sin(kx - omega t ) hatj N/C`
B. Option2 `E = 3.6 sin(kx - omega t ) hatj N/C`
C. Option3 `E = 3.6 sin(kx - omega t )(- hatj) N/C`
D. Option4 `E = 36 sin(kx - omega t )(- hatj) N/C`

Answer

To find the equation of the corresponding electric field for the given magnetic field of an electromagnetic wave, we can follow these steps:

### Step 1: Identify the given magnetic field
The magnetic field is given as:
\[ \vec{B} = 12 \times 10^{-9} \sin(kx - \omega t) \hat{k} \, \text{(T)} \]
Here, \(B_0 = 12 \times 10^{-9} \, \text{T}\) is the amplitude of the magnetic field, and it oscillates in the \(z\)-direction (indicated by \(\hat{k}\)).

### Step 2: Use the relationship between electric field and magnetic field
In electromagnetic waves, the magnitudes of the electric field \(E_0\) and magnetic field \(B_0\) are related by the speed of light \(c\):
\[ c = \frac{E_0}{B_0} \]
Where \(c \approx 3 \times 10^8 \, \text{m/s}\).

### Step 3: Calculate the magnitude of the electric field
Rearranging the formula gives us:
\[ E_0 = c \cdot B_0 \]
Substituting the values:
\[ E_0 = (3 \times 10^8 \, \text{m/s}) \cdot (12 \times 10^{-9} \, \text{T}) = 3.6 \times 10^{0} \, \text{N/C} = 3.6 \, \text{N/C} \]

### Step 4: Determine the direction of the electric field
In an electromagnetic wave, the electric field \(\vec{E}\), magnetic field \(\vec{B}\), and the direction of wave propagation are mutually perpendicular. The direction of propagation is given by the \(x\)-axis (since the argument of the sine function is \(kx - \omega t\)), and the magnetic field is in the \(z\)-direction (\(\hat{k}\)).

Using the right-hand rule (or the cross product relationship):
\[ \vec{E} \times \vec{B} \propto \text{Direction of wave propagation} \]
If \(\vec{B}\) is in the \(\hat{k}\) direction (z-axis), and the wave propagates in the \(\hat{i}\) direction (x-axis), then \(\vec{E}\) must be in the \(\hat{j}\) direction (y-axis).

### Step 5: Write the equation of the electric field
Thus, the electric field can be expressed as:
\[ \vec{E} = E_0 \sin(kx - \omega t) \hat{j} \]
Substituting the value of \(E_0\):
\[ \vec{E} = 3.6 \sin(kx - \omega t) \hat{j} \, \text{(N/C)} \]

### Final Answer
The equation of the corresponding electric field is:
\[ \vec{E} = 3.6 \sin(kx - \omega t) \hat{j} \, \text{(N/C)} \]