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Question #42675

Difficulty: 8/10 | 🟢 Free

The Question

An isolated, uncharged solid conducting sphere of radius $R$ is placed with its center at the origin $(0,0,0)$ . Two point charges, $q_1 = +2Q$ and $q_2 = +Q$ , are initially fixed at positions $(-d, 0, 0)$ and $(d, 0, 0)$ respectively, where $d \gg R$ . The charge $q_2$ is then displaced by a small distance $\delta$ ( $\delta \ll d$ ) along the positive x-axis to a new position $(d+\delta, 0, 0)$ , while $q_1$ remains fixed. What is the new electrostatic potential at a point $P$ located inside the conducting sphere at $(R/2, 0, 0)$ ? (Assume all given variables are in SI units and $\epsilon_0$ is the permittivity of free space).

Options

A.
$\frac{Q}{4\pi\epsilon_0 d} \left( 3 - \frac{\delta}{d} \right)$
B.
$\frac{Q}{4\pi\epsilon_0 d} \left( 3 - \frac{\delta}{d} - \frac{R}{2d} \right)$
C.
$\frac{Q}{4\pi\epsilon_0 d} \left( 3 + \frac{\delta}{d} \right)$
D.
$\frac{Q}{4\pi\epsilon_0 d} \left( 3 - \frac{\delta}{d} + \frac{R}{2d} \right)$

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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

Conductor Interior Potential|Neutral Equilibrium Points|Approximations (Binomial)

Why This Question Is Tricky

Common Mistakes

This question tests the 'Equipotential Fallacy' in the context of conductors and the superposition principle.

Students often incorrectly assume that the potential at a specific point inside a conductor can be found by simply calculating the potential due to external charges at that exact coordinate, ignoring the crucial role of induced surface charges.

By asking for the potential at an off-center point $P(R/2, 0, 0)$ , the trap lures students into calculating $V_{ext}(R/2)$ , which yields an incorrect extra term $-\frac{R}{2d}$ (Option B).

The correct physical framework requires recognizing that the conductor is an equipotential volume, meaning the potential everywhere inside equals the potential at the center.

At the center, the symmetric distance to all induced charges ensures their net potential contribution is zero, allowing the total potential to be calculated solely from the external charges.

Quick Facts About This Question

Question Information

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