A plane electromagnectic wave propagating along x-direction can have the following pairs of `E` and `B`.

Options

A. `E_(x') B_(y)`
B. `E_(y)'B_(z)`
C. `b_(X')E_(y)`
D. `E_(z') B_(y)`

Explanation

To determine the pairs of electric field (E) and magnetic field (B) vectors for a plane electromagnetic wave propagating along the x-direction, we can use the right-hand rule and the relationship between E, B, and the direction of wave propagation. Here’s a step-by-step solution:

### Step 1: Understand the Relationship
The relationship between the electric field (E), magnetic field (B), and the direction of wave propagation (C) is given by the equation:
\[ C = E \times B \]
This means that the direction of the wave (C) is determined by the cross product of the electric field vector (E) and the magnetic field vector (B).

### Step 2: Apply the Right-Hand Rule
To find the direction of the wave, we can use the right-hand rule:
- Point your thumb in the direction of the electric field vector (E).
- Point your index finger in the direction of the magnetic field vector (B).
- Your middle finger will then point in the direction of wave propagation (C).

### Step 3: Analyze Each Pair
Now, we will analyze each given pair of E and B to see if the wave propagates in the x-direction.

1. **Pair A: \( E_x \) and \( B_y \)**
- Thumb (E) in the x-direction, Index finger (B) in the y-direction.
- Result: Middle finger points out of the plane (z-direction). **Not valid.**

2. **Pair B: \( E_y \) and \( B_z \)**
- Thumb (E) in the y-direction, Index finger (B) in the z-direction.
- Result: Middle finger points in the x-direction. **Valid.**

3. **Pair C: \( B_x \) and \( E_y \)**
- Thumb (B) in the x-direction, Index finger (E) in the y-direction.
- Result: Middle finger points out of the plane (z-direction). **Not valid.**

4. **Pair D: \( E_z \) and \( B_y \)**
- Thumb (E) in the z-direction, Index finger (B) in the y-direction.
- Result: Middle finger points in the negative x-direction. **Valid.**

### Conclusion
The valid pairs of \( E \) and \( B \) that allow the electromagnetic wave to propagate along the x-direction are:
- Pair B: \( E_y \) and \( B_z \)
- Pair D: \( E_z \) and \( B_y \)