f E→ and K→ represent electric field and propagation vectors of the EM waves in vacuum, then magnetic field vector is given by: ( ω - angular frequency) :

To derive the magnetic field vector $\mathbf{B}$ for electromagnetic (EM) waves in vacuum using the electric field $\mathbf{E}$ and the propagation vector $\mathbf{k}$, we can use the relationships defined by Maxwell's equations.

Step 1: Understanding the Relationship

In a vacuum, the electromagnetic wave relations are as follows:

1. The electric field $\mathbf{E}$, magnetic field $\mathbf{B}$, and propagation vector $\mathbf{k}$ are mutually perpendicular.
2. The speed of light $c$ relates these fields and frequency via the following equations.

Step 2: Relevant Concepts

The relationships are given by: $\mathbf{B} = \frac{1}{c} \mathbf{k} \times \mathbf{E}$ Using the angular frequency $\omega$, the speed of light $c$ can be expressed as: $c = \frac{\omega}{k}$ where $k$ is the wave number.

Step 3: Deriving Magnetic Field Vector

Given that $\mathbf{E}$ and $\mathbf{k}$ are defined, we can express the magnetic field vector $\mathbf{B}$ in terms of these variables. Since $c = \frac{\omega}{k}$, we can rewrite:

$\mathbf{B} = \frac{1}{c} \mathbf{k} \times \mathbf{E} = \frac{k}{\omega} \mathbf{k} \times \mathbf{E}$

Noting the right-hand rule for cross products, the direction of $\mathbf{B}$ is determined by the right-hand rule and is perpendicular to both $\mathbf{E}$ and $\mathbf{k}$.

Step 4: Final Representation

Thus, the magnetic field vector $\mathbf{B}$ can be expressed in terms of the electric field vector $\mathbf{E}$ and the propagation vector $\mathbf{k}$ as follows: $\mathbf{B} = \frac{1}{c} \mathbf{k} \times \mathbf{E}$

Final Answer

In conclusion, the magnetic field vector $\mathbf{B}$ for electromagnetic waves in vacuum is given by: $\mathbf{B} = \frac{1}{c} (\mathbf{k} \times \mathbf{E}) = \frac{k}{\omega} \mathbf{k} \times \mathbf{E}$ This relationship highlights how the electric field and the propagation vector influence the magnetic field in an electromagnetic wave.