Question

In $\vec{E}$ and $\vec{K}$ represent electric field and propagation vectors of the EM waves in vacuum, then magnetic field vector is given by:$(\omega-$ angular frequency):

• A. \(\frac{1}{\omega}(\vec{K}\times\vec{E})\)
• B. \(\omega(\vec{E}\times\vec{K})\)
• C. \(\vec{K}\times\vec{E}\)
• D. \(\omega(\vec{ K } \times \vec{ E })\)

Answer 1

The Correct Option is A

Electromagnetic (EM) waves consist of oscillating electric and magnetic fields that are perpendicular to each other and to the direction of wave propagation. The following relationships hold for the electric field (\(\vec{E}\)), magnetic field (\(\vec{B}\)), and wave vector (\(\vec{K}\)):
1. The wave vector \(\vec{K}\) indicates the direction of wave propagation.
2. The electric field \(\vec{E}\) oscillates perpendicular to \(\vec{K}\).
3. The magnetic field \(\vec{B}\) is perpendicular to both \(\vec{E}\) and \(\vec{K}\).
The magnitude and direction of \(\vec{B}\) are determined using the cross product: \[ \vec{B} = \frac{\vec{K} \times \vec{E}}{\omega} \] where \(\omega\) is the angular frequency of the wave.
Explanation of Options: Option (1): Correct. This represents the magnetic field as proportional to the cross product of \(\vec{K}\) and \(\vec{E}\), divided by the angular frequency.
Option (2): Incorrect. The direction of \(\vec{B}\) would be wrong as the cross product order matters (\(\vec{K} \times \vec{E} \neq \vec{E} \times \vec{K}\)).
Option (3): Incorrect. Multiplying by \(\omega\) gives a dimensionally incorrect magnetic field.
Option (4): Incorrect. The dot product results in a scalar, not a vector.
Thus, the correct option is (1).