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 $ 2r $ 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.

• A. P \( \rightarrow \) 3, Q \( \rightarrow \) 1, R \( \rightarrow \) 2, S \( \rightarrow \) 4
• B. P \( \rightarrow \) 4, Q \( \rightarrow \) 5, R \( \rightarrow \) 3, S \( \rightarrow \) 1
• C. P \( \rightarrow \) 2, Q \( \rightarrow \) 1, R \( \rightarrow \) 4, S \( \rightarrow \) 5
• D. P \( \rightarrow \) 2, Q \( \rightarrow \) 1, R \( \rightarrow \) 3, S \( \rightarrow \) 5

Hint:

For systems with dipoles, always analyze symmetry and orientation. Use standard results for electric field due to dipoles along axial and equatorial positions. At midpoints, superpose field vectors considering direction and magnitude.

Answer 1

The Correct Option is C

Step 1: Analyze configuration Q
In configuration Q, the dipoles are aligned vertically and point in opposite directions. Being symmetrically placed about point \( X \), their electric fields cancel each other at \( X \).
\[ \Rightarrow \vec{E} = 0 \quad \Rightarrow \text{Q} \rightarrow 1 \] Step 2: Analyze configuration P
In configuration P, the dipoles are vertically aligned and point in the same direction. The field at midpoint \( X \) is the vector sum of the two dipole fields, both pointing in the same direction along \( -\hat{j} \). Using dipole field on the axial line: \[ \vec{E} = - \frac{p}{2\pi \varepsilon_0 r^3} \hat{j} \quad \Rightarrow \text{P} \rightarrow 2 \] Step 3: Analyze configuration R
Here the dipoles are on the \( x \)-axis and point along \( \hat{i} \). They are symmetrically placed, and the net field at \( X \) has both \( \hat{i} \) and \( \hat{j} \) components due to angular symmetry. The resultant field is: \[ \vec{E} = - \frac{p}{4\pi \varepsilon_0 r^3} (\hat{i} - \hat{j}) \quad \Rightarrow \text{R} \rightarrow 4 \] Step 4: Analyze configuration S
In configuration S, the dipoles are aligned along the \( x \)-axis but in opposite directions. The resultant field at midpoint \( X \) is along \( \hat{i} \), and both contribute in same direction due to orientation. Thus: \[ \vec{E} = \frac{p}{\pi \varepsilon_0 r^3} \hat{i} \quad \Rightarrow \text{S} \rightarrow 5 \]