Which of the following pairs of space and time -varying E and B fields generate a plane electromagnetic wave traveling along z -direction ?

Options

A. `E_(x) B_(y)`
B. `E_(y),B_(x)`
C. `E_(x),B_(z)`
D. `E_(z),B_(x)`

Answer

To determine which pairs of space and time-varying electric (E) and magnetic (B) fields generate a plane electromagnetic wave traveling along the z-direction, we need to ensure that the E and B fields are perpendicular to each other and also perpendicular to the direction of wave propagation.

### Step-by-Step Solution:

1. **Understanding the Direction of Propagation**:
- The wave is traveling along the z-direction. This means that the wave vector (direction of propagation) is in the z-direction.

2. **Identifying the Required Orientation of E and B Fields**:
- For a plane electromagnetic wave, the electric field (E) and magnetic field (B) must be perpendicular to each other and also perpendicular to the direction of wave propagation (z-direction).
- Therefore, if the wave is traveling in the z-direction, E can be in the x-direction or y-direction, and B must be in the remaining direction.

3. **Analyzing the Given Options**:
- **Option 1**: E is in the x-direction, B is in the y-direction.
- Here, E (x) and B (y) are perpendicular to each other, and both are perpendicular to the z-direction. This option is valid.

- **Option 2**: E is in the y-direction, B is in the x-direction.
- Similar to the first option, E (y) and B (x) are perpendicular to each other, and both are perpendicular to the z-direction. This option is also valid.

- **Option 3**: E is in the x-direction, B is in the z-direction.
- Here, E (x) is perpendicular to the direction of propagation (z), but B (z) is parallel to the direction of propagation. This violates the requirement that E and B must be perpendicular. This option is invalid.

- **Option 4**: E is in the z-direction, B is in the x-direction.
- In this case, E (z) is parallel to the direction of propagation (z), which again violates the requirement that E and B must be perpendicular. This option is also invalid.

4. **Conclusion**:
- The valid pairs that generate a plane electromagnetic wave traveling along the z-direction are:
- Option 1: E in x-direction, B in y-direction.
- Option 2: E in y-direction, B in x-direction.

### Final Answer:
The pairs that generate a plane electromagnetic wave traveling along the z-direction are:
- Option 1: E in x, B in y
- Option 2: E in y, B in x