Equation (9.45) provides two coupled relationships between the components of the electric field vector $\mathbf{E}_0$ and the magnetic field vector $\mathbf{B}_0$ in the context of wave propagation. Equation (9.46) is a compact vector form summarizing the same information. Let us break this down step by step to understand the transition:

Equation (9.45):

$-k(\mathbf{E}_0)_y = \omega (\mathbf{B}_0)_x, \quad k(\mathbf{E}_0)_x = \omega (\mathbf{B}_0)_y$ These are the component-wise relationships for the electric and magnetic fields in terms of the wave number $k$, angular frequency $\omega$, and the respective components of $\mathbf{E}_0$ and $\mathbf{B}_0$.

Vector Cross Product Interpretation:

A wave propagating in the $z$-direction (as implied by the unit vector $\hat{z}$) often involves fields that satisfy a right-hand rule. The magnetic field $\mathbf{B}_0$ is proportional to the cross product of the propagation direction $\hat{z}$ and the electric field $\mathbf{E}_0$: $\mathbf{B}_0 = -\frac{k}{\omega} (\hat{z} \times \mathbf{E}_0).$

Derivation of Equation (9.46):

1. The wave propagation is along the $z$-axis, meaning that $\mathbf{k} = k \hat{z}$, where $k$ is the wave number.

2. The cross product $\hat{z} \times \mathbf{E}_0$ gives a vector perpendicular to both $\hat{z}$ and $\mathbf{E}_0$. For the components, this implies:

   • In the $x$-direction: $(\hat{z} \times \mathbf{E}_0)_x = (\mathbf{E}_0)_y$,
   • In the $y$-direction: $(\hat{z} \times \mathbf{E}_0)_y = -(\mathbf{E}_0)_x$.

3. Substituting these relations into the expressions from (9.45), you find:

   • $(\mathbf{B}_0)_x = -\frac{k}{\omega} (\mathbf{E}_0)_y$,
   • $(\mathbf{B}_0)_y = \frac{k}{\omega} (\mathbf{E}_0)_x$.

4. These match the components of the vector cross product $-\frac{k}{\omega} (\hat{z} \times \mathbf{E}_0)$.

Hence, equation (9.46) is the vector compact form that encapsulates the relationships provided in (9.45).

Let me know if you’d like a deeper explanation! Here are some questions to expand on this:

1. What is the physical significance of the cross product in this context?
2. How does the wave equation relate to these field components?
3. Why is the direction of $\hat{z} \times \mathbf{E}_0$ important for electromagnetic waves?
4. Can we derive the general relation between $\mathbf{E}_0$ and $\mathbf{B}_0$ using Maxwell's equations?
5. How do these equations ensure that electromagnetic waves are transverse?

Tip: Always keep track of the right-hand rule when dealing with cross products in electromagnetism. It ensures that the field directions are consistent!