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:
Answer: (B) $\dfrac{\vec{k} \times \vec{E}}{\omega}$
Concept. For a plane wave in vacuum, $\vec B=\dfrac1c\,\hat k\times\vec E$: $\vec B$ is perpendicular to both, with magnitude $E/c$.
Step 1. Write with $\vec k$. $\hat k=\dfrac{\vec k}{k}$ and $k=\dfrac\omega c$, so $\dfrac1c\hat k\times\vec E=\dfrac{\vec k\times\vec E}{ck}=\dfrac{\vec k\times\vec E}{\omega}$.
Step 2. Check the order. $\vec E\times\vec B$ must point along $\vec k$; with $\vec B\propto\vec k\times\vec E$ this holds (option A would reverse it).
Answer. $\dfrac{\vec k\times\vec E}{\omega}$ (Option B).
Solution by Sreeraj P, M.Sc Physics
