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Question #42814

Difficulty: 8/10 | 🟢 Free

The Question

A dielectric medium contains $n$ polarizable molecules per unit volume, each modeled as a pair of bound charges $+q$ and $-q$ . An alternating electric field $E(t) = E_0 \cos(\omega t)$ is applied to the medium. Due to internal viscous drag opposing the rotation and stretching of the molecular dipoles, the macroscopic polarization $P(t)$ lags the electric field by a phase angle $\delta$ , such that $P(t) = P_0 \cos(\omega t - \delta)$ . Which of the following expressions correctly gives the average thermal power dissipated per unit volume $\langle P_{th} \rangle$ due to the work done against the internal torque, and the maximum relative drift velocity $v_{d0}$ between the bound charges of a single molecule?

Options

A.
$\langle P_{th} \rangle = \frac{1}{2} E_0 P_0 \omega \sin\delta$ , $v_{d0} = \frac{P_0 \omega}{n q}$
B.
$\langle P_{th} \rangle = \frac{1}{2} E_0 P_0 \omega \cos\delta$ , $v_{d0} = \frac{P_0 \omega}{n q}$
C.
$\langle P_{th} \rangle = E_0 P_0 \omega \sin\delta$ , $v_{d0} = \frac{P_0 \omega}{2 n q}$
D.
$\langle P_{th} \rangle = 0$ , $v_{d0} = \frac{P_0 \omega}{n q}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Drift Velocity Mechanism|Work Done in Rotating Dipole|Dielectric Polarization Logic

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to bridge macroscopic dielectric properties with microscopic dipole dynamics, specifically focusing on dielectric loss.

A common trap is treating a dielectric purely as an ideal capacitive element where polarization is perfectly in phase with the electric field ( $\delta = 0$ ), leading to the incorrect conclusion that no net work is done per cycle ( $\langle P_{th} \rangle = 0$ ).

By forcing the integration of $E \cdot dP$ (or $E \cdot J_p$ ), the problem highlights that the phase lag $\delta$ , caused by internal viscous torque during dipole rotation, results in a non-conservative resistive heat loss.

Furthermore, it tests the definition of polarization current to correctly extract the microscopic drift velocity of bound charges.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Drift Velocity Mechanism
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free