Match List I with List II: List I (Wave Equation) List II...

The direction of propagation of an electromagnetic wave is given by the direction of the cross product $\vec{E} \times \vec{B}$.

(A) $\vec{E}$ is along $\hat{i}$ and the wave propagates along $+z$ direction ($\hat{k}$).
$\hat{E} \times \hat{B} = \hat{v} \Rightarrow \hat{i} \times \hat{B} = \hat{k}$. Since $\hat{i} \times \hat{j} = \hat{k}$, $\vec{B}$ is along $\hat{j}$ (Matches III).

(B) $\vec{B}$ is along $\hat{i}$ and the wave propagates along $-y$ direction ($-\hat{j}$).
$\hat{E} \times \hat{B} = \hat{v} \Rightarrow \hat{E} \times \hat{i} = -\hat{j}$. Since $(-\hat{k}) \times \hat{i} = -\hat{j}$, $\vec{E}$ is along $-\hat{k}$ (Matches IV).

(C) $\vec{E}$ is along $\hat{j}$ and the wave propagates along $+x$ direction ($\hat{i}$).
$\hat{E} \times \hat{B} = \hat{v} \Rightarrow \hat{j} \times \hat{B} = \hat{i}$. Since $\hat{j} \times \hat{k} = \hat{i}$, $\vec{B}$ is along $\hat{k}$ (Matches I).

(D) $\vec{B}$ is along $\hat{j}$ and the wave propagates along $-z$ direction ($-\hat{k}$).
$\hat{E} \times \hat{B} = \hat{v} \Rightarrow \hat{E} \times \hat{j} = -\hat{k}$. Since $(-\hat{i}) \times \hat{j} = -\hat{k}$, $\vec{E}$ is along $-\hat{i}$ (Matches II).

Therefore, the correct matching is (A)-(III), (B)-(IV), (C)-(I), (D)-(II).

Answer: (A)-(III), (B)-(IV), (C)-(I), (D)-(II)