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Question #42935

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ and positive charge $q$ is placed in a two-dimensional electric field. The electric potential is given by $V(x, y) = \frac{k}{2}(x^2 - y^2)$ , where $k$ is a positive constant with units of $\text{V m}^{-2}$ . The particle experiences a velocity-dependent drag force $\vec{F}_d = -b\vec{v}$ , where $b$ is a positive damping constant with units of $\text{kg s}^{-1}$ , and $\vec{v}$ is the velocity of the particle. At time $t = 0$ , the particle is at the origin $(0,0)$ and is given a small initial velocity $\vec{v}_0 = v_0 \hat{j}$ (where $v_0 > 0$ ). Which of the following expressions represents the asymptotic relationship between the particle's velocity $v_y$ and its position $y$ as $t \to \infty$ ?

Options

A.
$v_y = \left( \frac{-b + \sqrt{b^2 + 4mqk}}{2m} \right) y$
B.
$v_y = 0$
C.
$v_y = \left( \frac{-b - \sqrt{b^2 + 4mqk}}{2m} \right) y$
D.
$v_y = \sqrt{\frac{qk}{m}} y$

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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

Neutral Equilibrium Points|Damped Oscillation Equation|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between stable and unstable equilibria in the presence of non-conservative damping forces.

A common trap is the blind application of the Damped Harmonic Oscillator (DHO) model, where students assume that any velocity-dependent drag ( $F = -bv$ ) will eventually dissipate the system's energy and bring the particle to rest at the equilibrium point ( $v_y = 0$ ).

However, Earnshaw's theorem dictates that a 2D quadrupole field has a saddle point, meaning the y-axis presents an anti-restoring force ( $F = +qky$ ).

The damping modifies the characteristic roots but cannot overcome the exponential divergence caused by the negative curvature of the potential energy surface.

Students must correctly set up Newton's Second Law with the anti-restoring force and analyze the asymptotic dominance of the positive exponential root.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Neutral Equilibrium Points
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free