An electromagnetic wave is propagating in a vacuum with an electric field given by $\vec{E}=E_0\sin (kx-\omega t)\hat{j}$. What is the direction of the associated magnetic field $\vec{B}$?

Solution

1. The electromagnetic wave propagates in the direction of the Poynting vector $\vec{S}$, which is given by the cross product $\vec{E}\times \vec{B}$.
2. The electric field is oriented along $\hat{j}$ and the wave propagates in the $+x$ direction, represented by the unit vector $\hat{i}$.
3. Substituting these into the relation gives $\hat{i}=\hat{j}\times \vec{B}$.
4. Using the right-hand rule for cross products, $\hat{j}\times \hat{k}=\hat{i}$, which confirms that $\vec{B}$ must be in the $\hat{k}$ direction.
5. Hence the answer is (A).