List-I shows four configurations, each consisting of a pair of ideal electric dipoles. Each dipole has a dipole moment of magnitude , oriented as marked by arrows in the figures. [2025]

Question

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 $2 r$ apart along the $x$ direction. The midpoint of the line joining the two dipoles is $X$ . The possible resultant electric fields $\vec{E}$ 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. [2025]

	List-I		List-II
(P)		(1)	$\vec{E} = 0$
(Q)		(2)	$\vec{E} = \frac{p}{2 π ε_{0} r^{3}} \hat{j}$
(R)		(3)	$\vec{E} = - \frac{p}{4 π ε_{0} r^{3}} (\hat{i} - \hat{j})$
(S)		(4)	$\vec{E} = \frac{p}{4 π ε_{0} r^{3}} (2 \hat{i} - \hat{j})$
		(5)	$\vec{E} = \frac{p}{π ε_{0} r^{3}} \hat{i}$

	List-I		List-II
(P)		(1)	$\vec{E} = 0$
(Q)		(2)	$\vec{E} = \frac{p}{2 π ε_{0} r^{3}} \hat{j}$
(R)		(3)	$\vec{E} = - \frac{p}{4 π ε_{0} r^{3}} (\hat{i} - \hat{j})$
(S)		(4)	$\vec{E} = \frac{p}{4 π ε_{0} r^{3}} (2 \hat{i} - \hat{j})$
		(5)	$\vec{E} = \frac{p}{π ε_{0} r^{3}} \hat{i}$

Smart Achievers · Accepted Answer

(3)

(P) ${\vec{E}}_{net} = \frac{- 2 k P}{r^{3}} \hat{j} = \frac{- P \hat{j}}{2 π ε_{0} r^{3}} \to (2)$

(Q) ${\vec{E}}_{net} = 0 \to (1)$

(R) ${\vec{E}}_{net} = \frac{2 P \hat{i}}{4 π ε_{0} r^{3}} - \frac{P \hat{j}}{4 π ε_{0} r^{3}}$

$= \frac{P}{4 π ε_{0} r^{3}} (2 \hat{i} - \hat{j}) \to (4)$

(S)

${\vec{E}}_{net} = \frac{4 k P \hat{i}}{r^{3}} = \frac{4 P \hat{i}}{4 π ε_{0} r^{3}} = \frac{P}{π ε_{0} r^{3}} \hat{i} \to (5)$

Solution

1 (P) → (3), (Q) → (1), (R) → (2), (S) → (4)

2 (P) → (4), (Q) → (5), (R) → (3), (S) → (1)

3 (P) → (2), (Q) → (1), (R) → (4), (S) → (5)

4 (P) → (2), (Q) → (1), (R) → (3), (S) → (5)

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