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Question #42758

Difficulty: 8/10 | 🟢 Free

The Question

A rigid extended dipole consists of two point charges $+q$ and $-q$ separated by a fixed distance $2a$ . The center of the dipole is held stationary at a distance $r$ (where $r > a$ ) from a fixed point charge $+Q$ . Initially, the dipole is in stable equilibrium. An external agent slowly rotates the dipole by $180^\circ$ about its center so that it reaches unstable equilibrium. Calculate the work done by the external agent. (Assume the system is in vacuum and $\epsilon_0$ is the permittivity of free space).

Options

A.
$\frac{Q q a}{\pi \epsilon_0 (r^2 - a^2)}$
B.
$\frac{Q q a}{\pi \epsilon_0 r^2}$
C.
$\frac{Q q a}{\pi \epsilon_0 (r - a)^2}$
D.
$\frac{Q q a}{2 \pi \epsilon_0 (r^2 - a^2)}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Work Done in Rotating Dipole|Coulombic Central Force|Potential vs Field Relationship

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between an ideal point dipole in a uniform field and an extended physical dipole in a non-uniform Coulombic field.

The primary trap lies in blindly applying the standard uniform-field formula $W = pE(\cos\theta_1 - \cos\theta_2)$ .

Because the electric field from the point charge $Q$ varies significantly over the length $2a$ of the dipole, the torque is not simply $pE\sin\theta$ .

Students who use the point-dipole approximation will evaluate the field only at the center ( $E = Q/4\pi\epsilon_0 r^2$ ) and incorrectly obtain $W = 2pE = Qqa/(\pi\epsilon_0 r^2)$ .

The rigorous approach requires abandoning the point-dipole shortcut and calculating the exact electrostatic potential energy of the individual charges in the initial and final states.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Work Done in Rotating Dipole
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free