The magnetic field in a plane electromagnetic wave is given by $\vec{B}=B_0\sin (kx+\omega t)\hat{k}$. What is the expression for the electric field vector $\vec{E}$?

Solution

1. The wave propagates in the $-\hat{i}$ direction, as indicated by the phase term $(kx+\omega t)$.
2. The relationship between the electric field $\vec{E}$, magnetic field $\vec{B}$, and propagation direction is given by the direction of the Poynting vector $\vec{S}\propto \vec{E}\times \vec{B}$.
3. Substituting the known directions, we have $-\hat{i}=\hat{E}\times \hat{k}$, which requires $\hat{E}=-\hat{j}$.
4. Using the magnitude relation $E_0=c{B}_0$, we obtain $\vec{E}=-c{B}_0\sin (kx+\omega t)\hat{j}$.
5. Hence the answer is (B).