In a plane electromagnetic wave, the directions of electric field and magnetic field are represented by k and `2hat(i) - 2hat(j)` , respectively. What is the unit vector along direction of propagation of the wave.

Options

A. `1/(sqrt(2)) (hat(i)+hat(j))`
B. `1/(sqrt(5))(2hat(i)+hat(j))`
C. `1/(sqrt(5)) (2hat(i)+2hat(j))`
D. `1/(sqrt(2)) (hat(j)+hat(k))`

Explanation

To find the unit vector along the direction of propagation of the electromagnetic wave, we follow these steps:

### Step 1: Identify the Electric and Magnetic Field Vectors
Given:
- The direction of the electric field \( \mathbf{E} \) is represented by the unit vector \( \hat{k} \) (which indicates the z-direction).
- The magnetic field \( \mathbf{B} \) is given as \( \mathbf{B} = 2\hat{i} - 2\hat{j} \).

### Step 2: Calculate the Cross Product \( \mathbf{E} \times \mathbf{B} \)
To find the direction of propagation, we need to calculate the cross product \( \mathbf{E} \times \mathbf{B} \).

1. Represent the vectors:
- \( \mathbf{E} = 0\hat{i} + 0\hat{j} + 1\hat{k} \)
- \( \mathbf{B} = 2\hat{i} - 2\hat{j} + 0\hat{k} \)

2. Set up the determinant for the cross product:
\[ \mathbf{E} \times \mathbf{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 0 & 1 \\ 2 & -2 & 0 \end{vmatrix} \]

3. Calculate the determinant:
\[ \mathbf{E} \times \mathbf{B} = \hat{i} \begin{vmatrix} 0 & 1 \\ -2 & 0 \end{vmatrix} - \hat{j} \begin{vmatrix} 0 & 1 \\ 2 & 0 \end{vmatrix} + \hat{k} \begin{vmatrix} 0 & 0 \\ 2 & -2 \end{vmatrix} \]

This simplifies to:
\[ \mathbf{E} \times \mathbf{B} = \hat{i}(0 - (-2)) - \hat{j}(0 - 2) + \hat{k}(0) = 2\hat{i} + 2\hat{j} \]

### Step 3: Find the Unit Vector
The direction of propagation is given by the vector \( \mathbf{E} \times \mathbf{B} = 2\hat{i} + 2\hat{j} \).

1. Calculate the magnitude of \( \mathbf{E} \times \mathbf{B} \):
\[ |\mathbf{E} \times \mathbf{B}| = \sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

2. The unit vector \( \hat{s} \) in the direction of propagation is:
\[ \hat{s} = \frac{\mathbf{E} \times \mathbf{B}}{|\mathbf{E} \times \mathbf{B}|} = \frac{2\hat{i} + 2\hat{j}}{2\sqrt{2}} = \frac{1}{\sqrt{2}}(\hat{i} + \hat{j}) \]

### Final Answer
The unit vector along the direction of propagation of the wave is:
\[ \hat{s} = \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} \]