Question

For an electromagnetic wave propagating through vacuum, \( \vec{k},\vec{E}\) and \( \omega \) represent propagation vector, electric field and angular frequency respectively. The magnetic field associated with this wave is represented by :

• A. \( \dfrac{\vec{E}\times\vec{k}}{\omega} \)
• B. \( \dfrac{\vec{k}\times\vec{E}}{\omega} \)
• C. \( \omega(\vec{E}\times\vec{k}) \)
• D. \( \omega(\vec{k}\times\vec{E}) \)

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the relationship between the magnetic field vector (\(\vec{B}\)), the electric field vector (\(\vec{E}\)), the propagation vector (\(\vec{k}\)), and the angular frequency (\(\omega\)) for an electromagnetic wave in vacuum.
Step 2: Key Formula or Approach:
In a plane electromagnetic wave, the relationship between \(\vec{E}\), \(\vec{B}\), and the propagation vector \(\vec{k}\) is derived from Maxwell's equations.
The standard relation is:
\[ \vec{B} = \frac{\vec{k} \times \vec{E}}{\omega} \]
Step 3: Detailed Explanation:
The propagation direction is defined by the cross product of the electric and magnetic fields (\(\vec{E} \times \vec{B}\)).
Also, from Faraday's law in the frequency domain for a plane wave:
\[ \vec{k} \times \vec{E} = \omega \vec{B} \]
By rearranging this equation to isolate \(\vec{B}\):
\[ \vec{B} = \frac{\vec{k} \times \vec{E}}{\omega} \]
Step 4: Final Answer:
Comparing this result with the given options, option (B) is the correct representation.