`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)/root2`
C. `(hatj + hatk)/root2`
D. `(hatj - hatk)root2`

Answer

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

### Step-by-Step Solution:

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

2. **Express the Vectors**:
- We can express the electric field vector as:
\[ \vec{E} = E \hat{k} \]
- And the magnetic field vector as:
\[ \vec{B} = B (\hat{i} - \hat{j}) = B \hat{i} - B \hat{j} \]

3. **Use the Cross Product to Find the Direction of Propagation**:
- The direction of propagation of the electromagnetic wave is given by the cross product \( \vec{E} \times \vec{B} \).
- We will calculate \( \vec{E} \times \vec{B} \):
\[ \vec{E} \times \vec{B} = (E \hat{k}) \times (B \hat{i} - B \hat{j}) \]

4. **Calculate the Cross Product**:
- Using the distributive property of the cross product:
\[ \vec{E} \times \vec{B} = E B (\hat{k} \times \hat{i}) - E B (\hat{k} \times \hat{j}) \]
- We know the following cross products:
- \( \hat{k} \times \hat{i} = \hat{j} \)
- \( \hat{k} \times \hat{j} = -\hat{i} \)
- Therefore:
\[ \vec{E} \times \vec{B} = E B \hat{j} + E B \hat{i} = E B (\hat{i} + \hat{j}) \]

5. **Find the Magnitude of the Resulting Vector**:
- The magnitude of \( \vec{E} \times \vec{B} \) is:
\[ |\vec{E} \times \vec{B}| = E B \sqrt{1^2 + 1^2} = E B \sqrt{2} \]

6. **Determine the Unit Vector**:
- The unit vector along the direction of propagation is given by:
\[ \hat{n} = \frac{\vec{E} \times \vec{B}}{|\vec{E} \times \vec{B}|} = \frac{E B (\hat{i} + \hat{j})}{E B \sqrt{2}} = \frac{\hat{i} + \hat{j}}{\sqrt{2}} \]

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