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Question #42798

Difficulty: 8/10 | 🟢 Free

The Question

A point charge $+Q$ is fixed at the origin. It is surrounded by a concentric, uncharged dielectric spherical shell of inner radius $R$ and outer radius $2R$ , having a uniform dielectric constant $K$ . A thin conducting spherical shell of radius $3R$ , carrying a net free charge $+2Q$ , is placed concentrically around the dielectric. Assuming the electric potential at infinity is zero, what is the total electric potential at a distance $r = R/2$ from the origin? (Here, $\epsilon_0$ is the permittivity of free space).

Options

A.
$\frac{Q(13K + 3)}{24\pi\epsilon_0 K R}$
B.
$\frac{Q(7K + 9)}{24\pi\epsilon_0 K R}$
C.
$\frac{Q(9K + 3)}{24\pi\epsilon_0 K R}$
D.
$\frac{Q(10K + 6)}{24\pi\epsilon_0 K R}$

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

Potential of Concentric Shells|Dielectric Constant (K) Shift|Gauss Law Applicability

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply Gauss's Law and the Superposition Principle in piecewise dielectric environments.

The 'lethal' trap is assuming the dielectric attenuates the potential globally or failing to account for the external conducting shell's charge in the outermost region.

Students frequently misapply the vacuum formula $V = kQ/r$ or incorrectly assume the dielectric constant $K$ applies to the inner vacuum cavity.

By forcing a piecewise integration of the electric field $V = -\int E \cdot dr$ from infinity, the problem exposes errors in identifying the correct enclosed free charge and local permittivity for each distinct spatial region.

Quick Facts About This Question

Question Information

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