`vecE` & `vecB` in an electromagnetic wave oscillate along the directionhaving unit vectors hatk &`(hati - hatj)`. Find unit vector along direction of propagation

Options

A. `(hati-hatj)/sqrt2`
B. `(hati+hatj)/sqrt2`
C. `(hatj + hatk)/sqrt2`
D. `(hatj - hatk)sqrt2`

Answer

To find the unit vector along the direction of propagation of an electromagnetic wave where the electric field vector \( \vec{E} \) and magnetic field vector \( \vec{B} \) oscillate along given directions, we can follow these steps:

### Step 1: Identify the vectors
Given:
- The electric field vector \( \vec{E} \) is along the direction of \( \hat{k} \).
- The magnetic field vector \( \vec{B} \) is along the direction of \( \hat{i} - \hat{j} \).

### Step 2: Write the vectors in component form
We can express the vectors as:
- \( \vec{E} = 0\hat{i} + 0\hat{j} + 1\hat{k} \) (which is simply \( \hat{k} \))
- \( \vec{B} = 1\hat{i} - 1\hat{j} + 0\hat{k} \)

### Step 3: Calculate the cross product \( \vec{E} \times \vec{B} \)
To find the direction of propagation, we need to calculate the cross product \( \vec{E} \times \vec{B} \).

Using the determinant form:
\[ \vec{E} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 0 & 1 \\ 1 & -1 & 0 \end{vmatrix} \]

### Step 4: Calculate the determinant
Calculating the determinant:
\[ \vec{E} \times \vec{B} = \hat{i} \begin{vmatrix} 0 & 1 \\ -1 & 0 \end{vmatrix} - \hat{j} \begin{vmatrix} 0 & 1 \\ 1 & 0 \end{vmatrix} + \hat{k} \begin{vmatrix} 0 & 0 \\ 1 & -1 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
- The first determinant: \( (0)(0) - (1)(-1) = 1 \)
- The second determinant: \( (0)(0) - (1)(1) = -1 \)
- The third determinant: \( (0)(-1) - (0)(1) = 0 \)

Putting it all together:
\[ \vec{E} \times \vec{B} = \hat{i}(1) - \hat{j}(-1) + \hat{k}(0) = \hat{i} + \hat{j} \]

### Step 5: Find the magnitude of the cross product
Now, we find the magnitude of \( \vec{E} \times \vec{B} \):
\[ |\vec{E} \times \vec{B}| = \sqrt{(1)^2 + (1)^2} = \sqrt{2} \]

### Step 6: Find the unit vector along the direction of propagation
The unit vector \( \hat{n} \) along the direction of propagation is given by:
\[ \hat{n} = \frac{\vec{E} \times \vec{B}}{|\vec{E} \times \vec{B}|} = \frac{\hat{i} + \hat{j}}{\sqrt{2}} \]

### Final Result
Thus, the unit vector along the direction of propagation is:
\[ \hat{n} = \frac{\hat{i} + \hat{j}}{\sqrt{2}} \]

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