List-I shows four configurations, each consisting of a pair of ideal electric dipoles. Each dipole has a dipole moment of magnitude p, oriented as marked by arrows in the figures. In all the configurations the dipoles are fixed such that they are at a distance 2r apart along the x direction. The midpoint of the line joining the two dipoles is X. The possible resultant electric fields E at X are given in List-II.

Choose the option that describes the correct match between the entries in List-I to those in List-II.

To find the matching between List-I and List-II, we calculate the resultant electric field at the midpoint $X$ (origin, $(0,0)$) due to the two electric dipoles. The left dipole is located at $(-r,0)$ and the right dipole is located at $(r,0)$. The distance of both dipoles from the midpoint $X$ is $r$.

For a dipole with dipole moment vector $\overrightarrow{p}$:
1. At an axial point at a distance $d$, the electric field is in the direction of the dipole moment:
${\overrightarrow{E}}_{\text{axial}}=\frac{2\overrightarrow{p}}{4\pi {\epsilon }_0{d}^3}$
2. At an equatorial point at a distance $d$, the electric field is opposite to the direction of the dipole moment:
${\overrightarrow{E}}_{\text{equatorial}}=-\frac{\overrightarrow{p}}{4\pi {\epsilon }_0{d}^3}$

Let us evaluate each configuration:

Configuration (P):
Both dipoles are directed along the $+\hat{j}$ direction (${\overrightarrow{p}}_1=p\hat{j}$ and ${\overrightarrow{p}}_2=p\hat{j}$). The midpoint $X$ lies on the equatorial axis of both dipoles.
${\overrightarrow{E}}_1=-\frac{p\hat{j}}{4\pi {\epsilon }_0{r}^3}$
${\overrightarrow{E}}_2=-\frac{p\hat{j}}{4\pi {\epsilon }_0{r}^3}$
The net electric field is:
${\overrightarrow{E}}_P={\overrightarrow{E}}_1+{\overrightarrow{E}}_2=-\frac{p}{2\pi {\epsilon }_0{r}^3}\hat{j}$
This matches with entry (2) in List-II.

Configuration (Q):
The left dipole is directed along $+\hat{j}$ (${\overrightarrow{p}}_1=p\hat{j}$) and the right dipole is directed along $-\hat{j}$ (${\overrightarrow{p}}_2=-p\hat{j}$). The midpoint $X$ lies on the equatorial axis of both dipoles.
${\overrightarrow{E}}_1=-\frac{p\hat{j}}{4\pi {\epsilon }_0{r}^3}$
${\overrightarrow{E}}_2=-\frac{-p\hat{j}}{4\pi {\epsilon }_0{r}^3}=\frac{p\hat{j}}{4\pi {\epsilon }_0{r}^3}$
The net electric field is:
${\overrightarrow{E}}_Q={\overrightarrow{E}}_1+{\overrightarrow{E}}_2=0$
This matches with entry (1) in List-II.

Configuration (R):
The left dipole is directed along $+\hat{j}$ (${\overrightarrow{p}}_1=p\hat{j}$), so $X$ is on its equatorial axis. The right dipole is directed along $+\hat{i}$ (${\overrightarrow{p}}_2=p\hat{i}$), so $X$ is on its axial line.
${\overrightarrow{E}}_1=-\frac{p\hat{j}}{4\pi {\epsilon }_0{r}^3}$
${\overrightarrow{E}}_2=\frac{2p\hat{i}}{4\pi {\epsilon }_0{r}^3}$
The net electric field is:
${\overrightarrow{E}}_R={\overrightarrow{E}}_1+{\overrightarrow{E}}_2=\frac{p}{4\pi {\epsilon }_0{r}^3}(2\hat{i}-\hat{j})$
This matches with entry (4) in List-II.

Configuration (S):
Both dipoles are directed along the $+\hat{i}$ direction (${\overrightarrow{p}}_1=p\hat{i}$ and ${\overrightarrow{p}}_2=p\hat{i}$). The midpoint $X$ lies on the axial line of both dipoles.
${\overrightarrow{E}}_1=\frac{2p\hat{i}}{4\pi {\epsilon }_0{r}^3}$
${\overrightarrow{E}}_2=\frac{2p\hat{i}}{4\pi {\epsilon }_0{r}^3}$
The net electric field is:
${\overrightarrow{E}}_S={\overrightarrow{E}}_1+{\overrightarrow{E}}_2=\frac{4p\hat{i}}{4\pi {\epsilon }_0{r}^3}=\frac{p}{\pi {\epsilon }_0{r}^3}\hat{i}$
This matches with entry (5) in List-II.

Therefore, the correct matching is: P→2, Q→1, R→4, S→5.