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Question #43645

Difficulty: 8/10 | 🟢 Free

The Question

A small spherical particle of mass $m$ and charge $+q$ is immersed in a highly viscous liquid with a drag coefficient $b$ (Stokes' drag $\vec{F}_d = -b\vec{v}$ ). A fixed point charge $-Q$ is located at the origin. A uniform external electric field $\vec{E} = E_0 \hat{i}$ is applied. The particle has an equilibrium position at $x = x_0$ ( $x_0 > 0$ ).

The particle is slightly displaced to a new position $x = x_0 + \delta x$ (where $\delta x > 0$ and $\delta x \ll x_0$ ). In the overdamped regime, the inertial term ( $m\ddot{x}$ ) is negligible, and the particle's velocity $\vec{v}$ is determined entirely by the instantaneous balance of the conservative forces and the viscous drag.

Find the velocity vector $\vec{v}$ of the particle at this displaced position. (Given: $k = \frac{1}{4\pi\epsilon_0}$ )

Options

A.
$\frac{2}{b} \sqrt{\frac{q^2 E_0^3}{k Q}} \delta x \hat{i}$
B.
$-\frac{2}{b} \sqrt{\frac{q^2 E_0^3}{k Q}} \delta x \hat{i}$
C.
$\frac{1}{b} \sqrt{\frac{q^2 E_0^3}{k Q}} \delta x \hat{i}$
D.
$-\frac{1}{b} \sqrt{\frac{q^2 E_0^3}{k Q}} \delta x \hat{i}$

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

Coulombic Central Force|Equilibrium Stability|Viscosity and Stokes' Law

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to perform a linear stability analysis of an equilibrium point in a dissipative medium.

The primary trap is the assumption that all equilibria are stable.

Because the central force is attractive ( $-1/r^2$ ) and balanced by a constant outward force, the effective spring constant is negative ( $k_{eff} < 0$ ), making the equilibrium unstable.

Students who blindly assume a restoring force will incorrectly choose a velocity vector pointing back towards the origin ( $-\hat{i}$ ).

A secondary trap is forgetting the factor of 2 that arises from the Taylor expansion of the $1/x^2$ Coulombic force.

Finally, students must correctly apply the overdamped approximation ( $F_{net} = 0 \implies F_{conservative} + F_{drag} = 0$ ) rather than attempting to use energy conservation, which is invalid due to the non-conservative Stokes' drag.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Coulombic Central Force
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free