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Question #42683

Difficulty: 8/10 | 🟢 Free

The Question

A dielectric medium contains $N$ molecules per unit volume ( $\text{m}^{-3}$ ). Each molecule is modeled as a classical Lorentz oscillator, where an electron of mass $m$ ( $\text{kg}$ ) and charge $-e$ ( $\text{C}$ ) is bound to a heavy nucleus by a harmonic restoring force with natural angular frequency $\omega_0$ ( $\text{rad/s}$ ). The electron also experiences a damping force $\vec{F}_d = -b \vec{v}$ , where $\vec{v}$ is its velocity ( $\text{m/s}$ ) and $b$ ( $\text{kg/s}$ ) is a positive constant. The medium is subjected to a uniform, time-varying external electric field $\vec{E}(t) = E_0 \cos(\omega t) \hat{i}$ ( $\text{V/m}$ ). Assuming the system is in a steady state, what is the average power dissipated per unit volume ( $\text{W/m}^3$ ) of the dielectric due to the external field?

Options

A.
$\frac{N e^2 E_0^2 b \omega^2}{2 [m^2(\omega_0^2 - \omega^2)^2 + b^2 \omega^2]}$
B.
0
C.
$\frac{N e^2 E_0^2 \omega}{2 \sqrt{m^2(\omega_0^2 - \omega^2)^2 + b^2 \omega^2}}$
D.
$\frac{N e^2 E_0^2 b \omega_0^2}{2 [m^2(\omega_0^2 - \omega^2)^2 + b^2 \omega^2]}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Dielectric Polarization Logic|Driven Oscillations and Resonance

Why This Question Is Tricky

Common Mistakes

This question tests the misconception that an induced dipole moment perfectly tracks an external time-varying electric field (i.e., zero phase lag).

If students assume the static relation $P = \epsilon_0 \chi E$ holds dynamically without accounting for complex susceptibility, they will conclude the displacement $x(t)$ is exactly in phase with $E(t)$ .

Consequently, the velocity $v(t)$ would be exactly $90^\circ$ out of phase with the driving force, leading to an erroneous average power dissipation of zero over a full cycle (Option B).

By introducing the damping force, the model forces a phase shift, which is the physical origin of dielectric loss (heating).

Option C traps students who calculate $F_{rms} v_{rms}$ but forget the power factor $\cos(\theta_v)$ , while Option D incorrectly uses the natural frequency instead of the driving frequency in the numerator.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Dielectric Polarization Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free