The electric and the magnetic field, associated with a electromagnetic, wave, propagating along the `+z-"axis"`, can be represented by

Options

A. (a) `[E = E_(0)hatk, B = B_(0)hati]`
B. (b) `[E = E_(0)hatj, B = B_(0) hatj]`
C. (c) `[E= E_(0)hatj, B = B_(0)hatk]`
D. (d) `[E = E_(0) hati, B = B_(0)hatj]`

Answer

To solve the problem of determining the representation of the electric and magnetic fields associated with an electromagnetic wave propagating along the +z-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Propagation Direction**:
- The question states that the electromagnetic wave is propagating along the +z-axis. This means the direction of propagation is represented by the unit vector \( \hat{k} \).

2. **Using the Right-Hand Rule**:
- The direction of propagation of an electromagnetic wave is given by the cross product of the electric field vector \( \vec{E} \) and the magnetic field vector \( \vec{B} \). Mathematically, this is represented as:
\[ \vec{k} = \vec{E} \times \vec{B} \]
- For the wave to propagate in the +z direction, the cross product \( \vec{E} \times \vec{B} \) must equal \( \hat{k} \).

3. **Choosing the Electric and Magnetic Field Directions**:
- To achieve a propagation in the +z direction, we can choose \( \vec{E} \) to be in the x-direction (along \( \hat{i} \)) and \( \vec{B} \) to be in the y-direction (along \( \hat{j} \)).
- According to the right-hand rule:
\[ \hat{i} \times \hat{j} = \hat{k} \]
- This confirms that if \( \vec{E} \) is along \( \hat{i} \) and \( \vec{B} \) is along \( \hat{j} \), the wave will propagate in the +z direction.

4. **Identifying the Correct Option**:
- Based on the above analysis, the correct representation of the electric and magnetic fields can be expressed as:
\[ \vec{E} = E_0 \hat{i} \quad \text{and} \quad \vec{B} = B_0 \hat{j} \]
- Therefore, the correct option from the given choices is the one that represents \( \vec{E} \) along the x-axis and \( \vec{B} \) along the y-axis.

### Final Answer:
The electric field \( \vec{E} \) is along the x-axis (i.e., \( \hat{i} \)), and the magnetic field \( \vec{B} \) is along the y-axis (i.e., \( \hat{j} \)). Thus, the correct representation is option D.