A plane electromagnetic wave is propagating along the direction $\dfrac{\hat{i} + \hat{j}}{\sqrt{2}}$, with its polarization along the direction $\hat{k}$. The correct form of the magnetic field of the wave would be (here $B_0$ is an appropriate constant):
Answer: (A) $B_0\dfrac{\hat{i} - \hat{j}}{\sqrt{2}}\cos\left(\omega t - k\dfrac{x + y}{\sqrt{2}}\right)$
A wave travelling along $\hat{n} = \dfrac{\hat{i} + \hat{j}}{\sqrt{2}}$ has phase $\omega t - k\,\hat{n}\cdot\vec{r} = \omega t - k\dfrac{x + y}{\sqrt{2}}$. This rules out (B).
$\vec{B}$ must be perpendicular to both $\vec{E}$ (along $\hat{k}$) and $\hat{n}$, which rules out (C) and (D). So $\vec{B}$ is along $\pm\dfrac{\hat{i} - \hat{j}}{\sqrt{2}}$.
Check the sign with $\vec{E}\times\vec{B}$ along $\hat{n}$: $\hat{k}\times\dfrac{\hat{i} - \hat{j}}{\sqrt{2}} = \dfrac{\hat{j} + \hat{i}}{\sqrt{2}}$. Correct, so the answer is (A).
Solution by Sreeraj P, M.Sc Physics