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Question #42753

Difficulty: 8/10 | 🟢 Free

The Question

Two identical positive point charges, each of magnitude $+Q$ , are fixed at coordinates $(2a, 0)$ and $(-2a, 0)$ in a gravity-free space. An electron of mass $m$ and charge $-e$ is projected from the point $(0, a)$ along the positive $x$ -direction with speed $v_0$ . A student attempts to find the value of $v_0$ such that the electron moves in a perfect circular path of radius $a$ centered at the origin. Which of the following statements is correct regarding the required projection speed $v_0$ ?

Options

A.
$v_0 = \sqrt{\frac{Qe}{10\sqrt{5}\pi\epsilon_0 m a}}$
B.
$v_0 = \sqrt{\frac{Qe}{5\sqrt{5}\pi\epsilon_0 m a}}$
C.
$v_0 = \sqrt{\frac{Qe}{4\pi\epsilon_0 m a}}$
D.
No such value of v_0 exists because a circular orbit is physically impossible in this potential.

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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

Superposition of Potentials|Coulombic Central Force|Orbital Velocity Energy Balance

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between an instantaneous radius of curvature and a stable orbital path.

The lethal trap occurs when students blindly apply the centripetal force formula $F_c = mv^2/r$ at the point of projection, assuming that matching the instantaneous radial force guarantees a circular orbit.

They fail to recognize that the superposition of two point charges creates a non-central force field.

Because the electric field is not spherically symmetric, the force at a general point on the circle is neither purely radial nor constant in magnitude.

Consequently, torque about the origin is non-zero, angular momentum is not conserved, and a closed circular orbit cannot be maintained without external constraints.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Superposition of Potentials
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free