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Physics Question on Electromagnetic waves

For a plane electromagnetic wave propagating in xx-direction, which one of the following combination gives the correct possible directions for electric field (E) and magnetic field (B) respectively?

A

j^+k^,j^+k^\hat{j}+\hat{k}, \hat{j}+\hat{k}

B

−j^+k^,−j^−k^-\hat{j}+\hat{k},-\hat{j}-\hat{k}

C

j^+k^,−j^−k^\hat{j}+\hat{k},-\hat{j}-\hat{k}

D

−j^+k^,−j^+k^-\hat{j}+\hat{k},-\hat{j}+\hat{k}

Answer

−j^+k^,−j^−k^-\hat{j}+\hat{k},-\hat{j}-\hat{k}

Explanation

Solution

(E⃗×B⃗)(\vec{ E } \times \vec{ B }) gives the direction of propagation
E⃗=−j^+k^,B⃗=−j^−k^\vec{ E }=-\hat{ j }+\hat{ k }, \vec{ B }=-\hat{ j }-\hat{ k }
E⃗×B⃗=∣i^j^k^ 0−11 0−1−1∣=2i^\vec{E} \times \vec{B}=\begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\\ 0 & -1 & 1 \\\ 0 & -1 & -1\end{vmatrix}=2 \hat{i}
Also E⃗⋅B⃗=(−j^+k^)(−j^−k^)=0\vec{E} \cdot \vec{B}=(-\hat{j}+\hat{k})(-\hat{j}-\hat{k})=0