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Question #43636

Difficulty: 8/10 | 🟢 Free

The Question

At $t=0$ , a particle of mass $m$ and charge $+q$ enters a region filled with a viscous medium. At this instant, it is at a very large distance from a fixed heavy nucleus of charge $+Q$ , moving with an initial speed $v_0$ and an impact parameter $b$ relative to the nucleus. The medium exerts a drag force $\vec{F}_d = -\gamma \vec{v}$ on the particle, where $\gamma$ is a positive constant and $\vec{v}$ is its instantaneous velocity. The nucleus exerts a Coulombic repulsive force $\vec{F}_c = \frac{kQq}{r^2}\hat{r}$ . What is the total area swept by the position vector of the particle (measured from the nucleus) from $t=0$ to $t \to \infty$ ?

Options

A.
$\frac{m v_0 b}{2 \gamma}$
B.
$\frac{k Q q}{\gamma v_0}$
C.
$\frac{m v_0 b}{\gamma} + \frac{k Q q}{2 \gamma v_0}$
D.
$\frac{m v_0 b}{2 \gamma} e^{-\frac{k Q q}{m v_0^2 b}}$

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

Impact Parameter Geometry|Projectile Path Equation|Variable Force Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to decouple complex radial dynamics from angular dynamics by identifying the nature of the applied forces.

Students are heavily conditioned by standard Rutherford scattering problems to assume angular momentum is conserved and to focus on the hyperbolic trajectory dictated by the Coulomb force.

The trap lies in attempting to integrate the variable Coulomb force or find the exact path equation.

By recognizing that a central force exerts zero torque, students can isolate the effect of the non-conservative drag force.

The drag force creates a torque strictly proportional to the angular momentum itself, leading to a simple exponential decay of areal velocity that is entirely independent of the nucleus's charge or the exact trajectory.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Impact Parameter Geometry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free