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

The correct option is:
$-\hat{j}+\hat{k},-\hat{j}-\hat{k}$

Step-by-step Explanation:

1. Identify the direction of wave propagation:
The question states that the plane electromagnetic wave is propagating in the positive $x$-direction. Thus, the unit vector in the direction of propagation is:
$\hat{s}=\hat{i}$

2. Recall the properties of electromagnetic waves:
For a transverse electromagnetic wave propagating in free space:
• The electric field vector $\overrightarrow{E}$ and the magnetic field vector $\overrightarrow{B}$ are perpendicular to each other, meaning their dot product is zero:
$\overrightarrow{E}·\overrightarrow{B}=0$
• The cross product $\overrightarrow{E}\times \overrightarrow{B}$ points in the direction of wave propagation $\hat{s}$:
$\overrightarrow{E}\times \overrightarrow{B}=C\hat{i}$ (where $C>0$)

3. Analyze the given correct combination:
Let $\overrightarrow{E}=-\hat{j}+\hat{k}$ and $\overrightarrow{B}=-\hat{j}-\hat{k}$.

• Check the dot product:
$\overrightarrow{E}·\overrightarrow{B}=(-\hat{j}+\hat{k})·(-\hat{j}-\hat{k})$
$=(-1)(-1)+\left(1\right)(-1)=1-1=0$
Since the dot product is $0$, the electric and magnetic fields are perpendicular.

• Check the direction of the cross product:
$\overrightarrow{E}\times \overrightarrow{B}=(-\hat{j}+\hat{k})\times (-\hat{j}-\hat{k})$
Using the distributive property of the cross product:
$=(-\hat{j})\times (-\hat{j})+(-\hat{j})\times (-\hat{k})+\hat{k}\times (-\hat{j})+\hat{k}\times (-\hat{k})$
Since $\hat{j}\times \hat{j}=0$ and $\hat{k}\times \hat{k}=0$:
$=0+(\hat{j}\times \hat{k})-(\hat{k}\times \hat{j})+0$
Using the cyclic cross-product relationships ($\hat{j}\times \hat{k}=\hat{i}$ and $\hat{k}\times \hat{j}=-\hat{i}$):
$=\hat{i}-(-\hat{i})=2\hat{i}$
The resulting vector $2\hat{i}$ is along the positive $x$-direction, which perfectly matches the wave propagation direction.