In `vecE` and `vecK` represent electric field and propagation vectors of the EM waves in vacuum,then magneticfield vector is given by :(ω-angular frequency):

Options

A. `1/ω(barK times bar E)`
B. `ω (barK times barE)`
C. `barKtimes barE`
D. `ω(barE times barK)`

Answer

To find the magnetic field vector \(\vec{B}\) in terms of the electric field vector \(\vec{E}\) and the propagation vector \(\vec{K}\) for electromagnetic waves in a vacuum, we can follow these steps:

### Step 1: Understand the relationship between \(\vec{E}\), \(\vec{B}\), and \(\vec{K}\)
In electromagnetic waves, the electric field \(\vec{E}\), magnetic field \(\vec{B}\), and the propagation vector \(\vec{K}\) are mutually perpendicular to each other. This means that \(\vec{K}\) is in the direction of wave propagation, while \(\vec{E}\) and \(\vec{B}\) oscillate in planes perpendicular to \(\vec{K}\).

### Step 2: Use the right-hand rule for cross products
The relationship between these vectors can be expressed using the cross product. The direction of \(\vec{B}\) can be determined using the right-hand rule: if you point your fingers in the direction of \(\vec{E}\) and curl them towards \(\vec{K}\), your thumb will point in the direction of \(\vec{B}\). Mathematically, we can express this as:
\[ \vec{K} = \frac{\omega}{c} \hat{k} \]
where \(\omega\) is the angular frequency and \(c\) is the speed of light in vacuum.

### Step 3: Express \(\vec{B}\) in terms of \(\vec{E}\) and \(\vec{K}\)
From Maxwell's equations, we know that:
\[ \vec{B} = \frac{1}{c} \hat{k} \times \vec{E} \]
Since \(c = \frac{\omega}{k}\) (where \(k\) is the wave number), we can rewrite this as:
\[ \vec{B} = \frac{1}{\omega} \hat{k} \times \vec{E} \]

### Step 4: Final expression for \(\vec{B}\)
Thus, the magnetic field vector \(\vec{B}\) can be expressed as:
\[ \vec{B} = \frac{1}{\omega} (\hat{k} \times \vec{E}) \]

### Conclusion
The magnetic field vector \(\vec{B}\) in terms of the electric field vector \(\vec{E}\) and the propagation vector \(\vec{K}\) is given by:
\[ \vec{B} = \frac{1}{\omega} \vec{K} \times \vec{E} \]