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.

Question

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.

Knowee AI · Accepted Answer