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Question #43785

Difficulty: 8/10 | 🟢 Free

The Question

A spacecraft is moving in deep space with a constant acceleration $\vec{a} = a \hat{i}$ ( $a > 0$ ). Inside the spacecraft, a nucleus of charge $+Ze$ is rigidly fixed at the origin $(0,0,0)$ of the coordinate system attached to the spacecraft. Consider an electron (mass $m$ , charge $-e$ ) moving in this non-inertial frame. The effective force on the electron is the vector sum of the Coulombic central force and the inertial pseudo force. There exists a unique critical point (a saddle point of the effective potential) where the electron can be in a state of equilibrium. Assuming the Coulomb potential energy is zero at infinity and the potential energy associated with the pseudo force is zero at the origin, which of the following correctly gives the position vector $\vec{r}_0$ of this equilibrium point and the effective potential energy $U_{eff}$ of the electron at this point?

Options

A.
$\vec{r}_0 = -\sqrt{\frac{Z e^2}{4\pi\epsilon_0 m a}} \hat{i}, \quad U_{eff} = -2\sqrt{\frac{Z e^2 m a}{4\pi\epsilon_0}}$
B.
$\vec{r}_0 = +\sqrt{\frac{Z e^2}{4\pi\epsilon_0 m a}} \hat{i}, \quad U_{eff} = -2\sqrt{\frac{Z e^2 m a}{4\pi\epsilon_0}}$
C.
$\vec{r}_0 = -\sqrt{\frac{Z e^2}{4\pi\epsilon_0 m a}} \hat{i}, \quad U_{eff} = 0$
D.
$\vec{r}_0 = +\sqrt{\frac{Z e^2}{4\pi\epsilon_0 m a}} \hat{i}, \quad U_{eff} = +2\sqrt{\frac{Z e^2 m a}{4\pi\epsilon_0}}$

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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

Coulombic Central Force|Neutral Equilibrium Points|Pseudo Force Directionality|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply pseudo forces and construct effective potentials in non-inertial frames.

A common trap is misidentifying the direction of the pseudo force (taking it as $+ma \hat{i}$ instead of $-ma \hat{i}$ ), which incorrectly places the equilibrium point on the positive x-axis.

Furthermore, students frequently misapply the concept of potential energy at an equilibrium point; they assume that because the net force is zero, the effective potential energy must also be zero, or they incorrectly integrate the pseudo force (yielding $U_p = -max$ ), failing to recognize that the pseudo potential adds constructively to the depth of the Coulomb well at the saddle point.

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