Sure! Let's break this down step by step and use the right-hand rule to explain it in the context of a plane wave propagating in the z-direction.

Plane Wave Propagation:

• The problem states that a plane wave is propagating along the z-direction. This means the wave is moving forward along the $z$-axis.

Orientation of Electric and Magnetic Fields:

• We are told that the electric field (denoted as $\mathbf{E}$) and the magnetic field (denoted as $\mathbf{B}$) are oriented along the x and y directions (or vice versa). Let's assume the electric field is along the x-direction and the magnetic field is along the y-direction for this explanation (though swapping them doesn't change the result).

Cross Product and Right-Hand Rule:

The key concept here is that the cross product of the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ gives the direction of wave propagation, which in this case is along the z-direction.

This is where the right-hand rule comes in. Here's how to apply it:

1. Point your right hand's index finger in the direction of the electric field ($\mathbf{E}$). In this case, let's assume the electric field $\mathbf{E}$ is along the x-axis (to the right in 3D space).

2. Point your right hand's middle finger in the direction of the magnetic field ($\mathbf{B}$). Here, we assume $\mathbf{B}$ is along the y-axis (upward).

3. Your thumb (perpendicular to both fingers) will now point in the direction of the wave's propagation, which is the z-axis (outward).

Thus, using the right-hand rule, when the electric field is along the x-axis, and the magnetic field is along the y-axis, the plane wave propagates in the positive z-direction.

Summary:

• Electric field (E) along the x-axis.
• Magnetic field (B) along the y-axis.
• Wave propagation (the direction the wave moves) along the z-axis, determined by $\mathbf{E} \times \mathbf{B}$.

If you reverse the directions of $\mathbf{E}$ and $\mathbf{B}$, the propagation direction would be reversed as well, but the right-hand rule will still hold.

Do you want further details or have any questions?

Here are 5 follow-up questions to explore this topic further:

1. What happens if the magnetic field is along the x-axis and the electric field along the y-axis?
2. How does the right-hand rule apply in cases of circularly polarized waves?
3. Can the wave propagate in other directions if the electric and magnetic fields are aligned differently?
4. What is the significance of the cross product $\mathbf{E} \times \mathbf{B}$ in Maxwell's equations?
5. How would the scenario change if the wave were propagating in a medium instead of free space?

Tip: When applying the right-hand rule, always make sure to align your fingers carefully with the correct field directions to avoid errors in determining the propagation direction.