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Question #43781

Difficulty: 8/10 | 🟢 Free

The Question

A solid, isolated conducting sphere of radius $R$ is fixed in space and carries a net positive charge $Q = \frac{7}{18}q$ . A point particle of mass $m$ and positive charge $q$ is projected from infinity directly towards the center of the sphere. What is the minimum initial kinetic energy $K_0$ required for the particle to eventually collide with the surface of the sphere? \n\n(Assume the system is in vacuum with permittivity $\epsilon_0$ and ignore gravity. The potential energy of the particle must account for the polarization of the conducting sphere).

Options

A.
$\frac{11 q^2}{288 \pi \epsilon_0 R}$
B.
$\frac{7 q^2}{72 \pi \epsilon_0 R}$
C.
$\frac{17 q^2}{288 \pi \epsilon_0 R}$
D.
$\frac{7 q^2}{144 \pi \epsilon_0 R}$

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

Distance of Closest Approach|Charged Conducting Sphere|Dimensional Consistency and Transcendental Functions

Why This Question Is Tricky

Common Mistakes

Students often incorrectly treat a charged conducting sphere purely as a point charge located at its center, ignoring the polarization of the sphere caused by the approaching charge.

This leads them to calculate the required kinetic energy simply as the point-charge potential at the surface ( $K = kQq/R$ ), missing the fact that the induced negative charge on the near side creates a strong attractive force at close distances.

The method of images reveals a potential barrier at $x = 2R$ .

Once the particle passes this barrier, the attractive polarization force dominates, pulling it into the sphere.

The required energy is the height of this barrier, not the potential at the surface.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Distance of Closest Approach
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free