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Question #42684

Difficulty: 8/10 | 🟢 Free

The Question

An uncharged solid dielectric sphere of radius $R$ and relative permittivity $\epsilon_r = 2$ is placed in a non-uniform external electric field. The external electric potential is given by $V_{\text{ext}}(x,y,z) = -k(2z^2 - x^2 - y^2)$ , where $k$ is a positive constant with units of $\text{V m}^{-2}$ . The origin of the coordinate system coincides with the center of the sphere. What is the electric potential produced solely by the induced bound charges at a point on the positive $z$ -axis at a distance $r = 2R$ from the center?

Options

A.
$\frac{1}{14} k R^2$
B.
$\frac{1}{8} k R^2$
C.
$\frac{1}{4} k R^2$
D.
0

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Dielectric Polarization Logic|Superposition of Potentials|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle dielectric polarization in a non-uniform external field, specifically a quadrupole field.

The primary trap is the 'Uniform Field Assumption.' Students often memorize the induced dipole moment formula for a dielectric sphere in a uniform field and attempt to apply it by evaluating the external field at a specific point (like the surface or the center).

Since the external field at the center is zero, they might incorrectly conclude the induced potential is zero.

Alternatively, they might incorrectly use the $l=1$ boundary condition denominator $(\epsilon_r + 2)$ instead of the correct $l=2$ denominator $(2\epsilon_r + 3)$ .

Solving this requires recognizing the $l=2$ Legendre polynomial symmetry and correctly applying Laplace's equation boundary conditions for higher-order multipoles.

Quick Facts About This Question

Question Information

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