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

Options

A. `E = 36 Ssin(kx - omega t ) hatj N/C`
B. `E = 3.6 Ssin(kx - omega t ) hatj N/C`
C. `E = 3.6 Ssin(kx - omega t )(- hatj) N/C`
D. `E = 36 Ssin(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:
\[ \vec{B} = 12 \times 10^{-9} \sin(kx - \omega t) \hat{k} \, \text{T} \]
This indicates that the magnetic field is oscillating in the z-direction (positive \(\hat{k}\) direction).

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

From the given magnetic field, we can identify:
\[ B_0 = 12 \times 10^{-9} \, \text{T} \]

Now, substituting \(B_0\) into the equation:
\[ E_0 = c \cdot B_0 = (3 \times 10^8) \cdot (12 \times 10^{-9}) = 3.6 \times 10^{-1} \, \text{N/C} = 36 \times 10^{-9} \, \text{N/C} \]

### Step 3: 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 (\(\vec{k}\)) are mutually perpendicular.

Given:
- The magnetic field \(\vec{B}\) is in the \(\hat{k}\) direction (z-direction).
- The wave is propagating in the positive x-direction (as indicated by \(kx\)).

Using the right-hand rule:
- If \(\vec{B}\) is in the \(\hat{k}\) direction, and the wave is propagating in the \(\hat{i}\) (x) direction, then the electric field \(\vec{E}\) must be in the \(\hat{j}\) (y) direction.

### Step 4: Write the equation of the electric field
The equation of the electric field can be written as:
\[ \vec{E} = E_0 \sin(kx - \omega t) \hat{j} \]
Substituting the value of \(E_0\):
\[ \vec{E} = 36 \times 10^{-9} \sin(kx - \omega t) \hat{j} \, \text{N/C} \]

### Final Answer
Thus, the equation of the corresponding electric field is:
\[ \vec{E} = 36 \times 10^{-9} \sin(kx - \omega t) \hat{j} \, \text{N/C} \]

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