A plane electromagnetic wave travels in free space along X-direction. If the value of `vecB` (in tesla) at a particular point in space and time is `1.2 xx 10^(-8) hatk`. The value of `vecE` (in `Vm^(-1)`) at that point is

Options

A. `1.2hatj`
B. `3.6hatk`
C. `1.2hatk`
D. `3.6hatj`

Explanation

To find the value of the electric field vector \( \vec{E} \) given the magnetic field vector \( \vec{B} \) in a plane electromagnetic wave traveling in free space, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- The magnetic field \( \vec{B} \) is given as \( 1.2 \times 10^{-8} \hat{k} \) T (Tesla).
- The speed of light in free space \( c \) is approximately \( 3 \times 10^8 \) m/s.

2. **Use the Relationship Between Electric Field and Magnetic Field**:
- The relationship between the electric field \( \vec{E} \) and the magnetic field \( \vec{B} \) in an electromagnetic wave is given by the equation:
\[ c = \frac{E}{B} \]
- Rearranging this equation to find \( E \):
\[ E = B \cdot c \]

3. **Substitute the Known Values**:
- Substitute the value of \( B \) and \( c \) into the equation:
\[ E = (1.2 \times 10^{-8} \, \text{T}) \cdot (3 \times 10^8 \, \text{m/s}) \]

4. **Calculate the Magnitude of the Electric Field**:
- Performing the multiplication:
\[ E = 1.2 \times 3 \times 10^{-8} \times 10^8 \]
\[ E = 3.6 \, \text{V/m} \]

5. **Determine the Direction of the Electric Field**:
- The magnetic field \( \vec{B} \) is in the \( \hat{k} \) direction (Z-direction).
- The wave is traveling in the \( \hat{i} \) direction (X-direction).
- According to the right-hand rule, if \( \vec{B} \) is in the Z-direction and the wave is propagating in the X-direction, then the electric field \( \vec{E} \) must be in the Y-direction (which is represented by \( \hat{j} \)).

6. **Final Expression for the Electric Field**:
- Therefore, the electric field vector can be expressed as:
\[ \vec{E} = 3.6 \, \hat{j} \, \text{V/m} \]

### Final Answer:
The value of \( \vec{E} \) at that point is \( 3.6 \, \hat{j} \, \text{V/m} \).