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 magnetic field vector $\vec{B}$ associated with this wave?

Solution

1. The electromagnetic wave propagates in the $+x$ direction, denoted by the unit vector $\hat{i}$, and the electric field oscillates in the $\hat{j}$ direction.
2. The direction of propagation is given by the Poynting vector direction, which follows the relation ${\hat{k}}_{prop}=\hat{E}\times \hat{B}$.
3. Substituting the known directions, we have $\hat{i}=\hat{j}\times \hat{B}$.
4. Using the right-hand rule for cross products, $\hat{j}\times \hat{k}=\hat{i}$, which confirms that the magnetic field vector $\vec{B}$ must point in the $\hat{k}$ direction.
5. Hence the answer is (A).