Before You Begin

Question #42620

Difficulty: 8/10 | 🟢 Free

The Question

An electric dipole of dipole moment $p$ and moment of inertia $I$ undergoes small angular oscillations in a uniform electric field $E$ . The surrounding medium exerts a viscous drag torque $\tau = -b \dot{\theta}$ , where $\dot{\theta}$ is the angular velocity and $b$ is a positive constant. Let $A(t)$ denote the area of the phase space ellipse $(\theta, L_\theta)$ corresponding to the instantaneous total mechanical energy of the oscillator. Assuming weak damping ( $b \ll \sqrt{pEI}$ ), which of the following expressions gives $A(t)$ in terms of its initial value $A_0$ ?

Options

A.
$A_0 e^{-bt/I}$
B.
$A_0 e^{-bt/2I}$
C.
$A_0 e^{-2bt/I}$
D.
$A_0$

Check Your Answer

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

Phase Space Trajectory|Conservative Field Path Independence|Work Done in Rotating Dipole

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of non-conservative work and phase space dynamics in dissipative systems.

A common trap is assuming that the phase space area remains constant (Liouville's Theorem) or acts as an adiabatic invariant, which only holds for conservative Hamiltonian systems.

Another frequent mistake is confusing the decay rate of the angular amplitude ( $e^{-bt/2I}$ ) with the decay rate of the energy and phase space area ( $e^{-bt/I}$ ).

By requiring the calculation of the non-conservative work done by the drag torque, the problem forces students to link the macroscopic phase space area directly to the time-averaged energy dissipation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Phase Space Trajectory
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free