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: $-\hat{j}+\hat{k},-\hat{j}-\hat{k}$ (Option 4, corresponding to the labels in the image -ĵ + k̂, -ĵ - k̂).

Underlying Physics Principles:
For any plane electromagnetic wave propagating in a given direction, the direction of wave propagation is given by the direction of the Poynting vector, which is parallel to the cross product of the electric field vector ($\overrightarrow{E}$) and the magnetic field vector ($\overrightarrow{B}$):
$\text{Direction of Propagation}=\overrightarrow{E}\times \overrightarrow{B}$
Furthermore, in a transverse electromagnetic wave, both the electric field and the magnetic field must be perpendicular to the direction of propagation, and they must also be perpendicular to each other:
$\overrightarrow{E}·\overrightarrow{B}=0$

Step-by-Step Derivation and Calculation:
Given that the wave propagates along the positive x-direction, the propagation unit vector is:
$\hat{n}=\hat{i}$
Therefore, the cross product of the directions of $\overrightarrow{E}$ and $\overrightarrow{B}$ must point in the positive $\hat{i}$ direction:
$\widehat{e_E}\times \widehat{e_B}=C\hat{i}$ (where $C>0$)

Let's check the given correct option where the direction vectors of $\overrightarrow{E}$ and $\overrightarrow{B}$ are proportional to:
$\widehat{e_E}=-\hat{j}+\hat{k}$
$\widehat{e_B}=-\hat{j}-\hat{k}$

1. Verify Orthogonality:
Let us compute the dot product:
$\widehat{e_E}·\widehat{e_B}=(-\hat{j}+\hat{k})·(-\hat{j}-\hat{k})$
$\widehat{e_E}·\widehat{e_B}=(-1)(-1)+\left(1\right)(-1)=1-1=0$
Since the dot product is zero, the electric and magnetic field directions are perpendicular to each other.

2. Verify Direction of Propagation:
Let us compute the cross product:
$\widehat{e_E}\times \widehat{e_B}=(-\hat{j}+\hat{k})\times (-\hat{j}-\hat{k})$
Expanding the cross product terms:
$\widehat{e_E}\times \widehat{e_B}=(-\hat{j})\times (-\hat{j})+(-\hat{j})\times (-\hat{k})+\left(\hat{k}\right)\times (-\hat{j})+\left(\hat{k}\right)\times (-\hat{k})$
Using vector identities ($\hat{j}\times \hat{j}=0$, $\hat{k}\times \hat{k}=0$, $\hat{j}\times \hat{k}=\hat{i}$, and $\hat{k}\times \hat{j}=-\hat{i}$):
$\widehat{e_E}\times \widehat{e_B}=0+(\hat{j}\times \hat{k})-(\hat{k}\times \hat{j})-0$
$\widehat{e_E}\times \widehat{e_B}=\hat{i}-(-\hat{i})=2\hat{i}$

The resulting vector points along the positive x-direction ($\hat{i}$). This confirms that the given combination provides the correct possible directions for the electromagnetic wave propagation in the x-direction.