Before You Begin

Question #42755

Difficulty: 8/10 | 🟢 Free

The Question

An isolated parallel-plate capacitor has a uniform electric field of magnitude $E_0$ in the vacuum between its plates. A large block of linear, isotropic dielectric material with dielectric constant $K$ is then inserted, completely filling the space between the plates. Inside the dielectric block, there is a small, empty spherical cavity. A rigid permanent electric dipole of dipole moment $p$ is placed at the center of this cavity. Initially, the dipole is in its stable equilibrium position. What is the minimum work done by an external agent to slowly rotate the dipole by $180^\circ$ ?

Options

A.
$\frac{6 p E_0}{2K+1}$
B.
$\frac{2 p E_0}{K}$
C.
$\frac{2(K+2) p E_0}{3K}$
D.
$2 p E_0$

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|Dielectric Constant (K) Shift|Potential Energy Reference

Why This Question Is Tricky

Common Mistakes

This question tests the profound distinction between the macroscopic electric field, the Lorentz local field, and the field inside a real macroscopic cavity.

A common trap is to assume the dipole simply interacts with the macroscopic field $E_{mac} = E_0/K$ , leading to the incorrect answer $2pE_0/K$ .

Advanced students might fall into the trap of using the Lorentz local field $E_{loc} = \frac{K+2}{3} E_{mac}$ , failing to realize it applies to a virtual molecular cavity where the surrounding dielectric does not relax, not a real empty cavity.

Furthermore, students must recognize that the reaction field of the dipole itself rotates with it, doing no net work, so the external work is purely against the cavity field.

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