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 representing the possible directions for the electric field ($\overrightarrow{E}$) and the magnetic field ($\overrightarrow{B}$) is:

$\overrightarrow{E}=-\hat{j}+\hat{k}$, $\overrightarrow{B}=-\hat{j}-\hat{k}$

Step-by-Step Explanation:

1. Transverse Nature and Propagation Direction of Electromagnetic Waves:
In a plane electromagnetic wave, the electric field vector $\overrightarrow{E}$ and the magnetic field vector $\overrightarrow{B}$ are perpendicular to each other, and both are perpendicular to the direction of wave propagation. The direction of propagation is given by the direction of the Poynting vector, which points in the direction of the cross product of the electric and magnetic fields:

$\text{Direction of propagation}\propto \overrightarrow{E}\times \overrightarrow{B}$

Since the wave is propagating in the positive x-direction, the cross product $\overrightarrow{E}\times \overrightarrow{B}$ must point along the positive x-axis (represented by the unit vector $\hat{i}$).

2. Verifying Mutual Perpendicularity ($\overrightarrow{E}·\overrightarrow{B}=0$):
Let us check the dot product of the fields in the correct option to confirm they are perpendicular to each other:

$\overrightarrow{E}·\overrightarrow{B}=(-\hat{j}+\hat{k})·(-\hat{j}-\hat{k})$

$\overrightarrow{E}·\overrightarrow{B}=(-1)(-1)+\left(1\right)(-1)=1-1=0$

Since the dot product is zero, the electric and magnetic fields are mutually perpendicular.

3. Verifying the Direction of Propagation ($\overrightarrow{E}\times \overrightarrow{B}$):
Now, let us calculate the cross product of the fields:

$\overrightarrow{E}\times \overrightarrow{B}=(-\hat{j}+\hat{k})\times (-\hat{j}-\hat{k})$

Expanding the cross product terms:

$\overrightarrow{E}\times \overrightarrow{B}=(-\hat{j})\times (-\hat{j})+(-\hat{j})\times (-\hat{k})+\hat{k}\times (-\hat{j})+\hat{k}\times (-\hat{k})$

Using the properties of unit vector cross products where $\hat{j}\times \hat{j}=0$ and $\hat{k}\times \hat{k}=0$:

$\overrightarrow{E}\times \overrightarrow{B}=0+(\hat{j}\times \hat{k})-(\hat{k}\times \hat{j})+0$

Using the cyclic cross product relations $\hat{j}\times \hat{k}=\hat{i}$ and $\hat{k}\times \hat{j}=-\hat{i}$:

$\overrightarrow{E}\times \overrightarrow{B}=\hat{i}-(-\hat{i})=2\hat{i}$

The vector $2\hat{i}$ points in the positive x-direction, which matches the wave propagation direction specified in the question. Thus, this combination represents a physically correct pair of directions for the electric and magnetic fields.