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Question #42775

Difficulty: 8/10 | 🟢 Free

The Question

A solid, uncharged dielectric sphere of radius $R$ and dielectric constant $K$ is placed in a uniform external electric field of magnitude $E_0$ . The external field induces a uniform polarization $\vec{P}$ within the sphere, creating a bound surface charge distribution. Calculate the electrostatic self-energy $U_{\text{self}}$ of this induced bound charge distribution. (Note: $U_{\text{self}}$ is the energy required to assemble the bound charges against their own electric field, which is equivalent to the energy stored in the electric field produced solely by the bound charges).

Options

A.
$2\pi \epsilon_0 R^3 E_0^2 \left( \frac{K-1}{K+2} \right)^2$
B.
$\frac{2\pi}{9} \epsilon_0 R^3 E_0^2 \left( \frac{K-1}{K} \right)^2$
C.
$\frac{2\pi}{3} \epsilon_0 R^3 E_0^2 \left( \frac{K-1}{K+2} \right)^2$
D.
$2\pi \epsilon_0 R^3 E_0^2 \left( \frac{K-1}{K+2} \right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Dielectric Polarization Logic|Potential vs Field Relationship|Gravitational Self-Energy

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly model the internal electric field of a dielectric sphere and calculate the self-energy of a dipole charge distribution.

The primary trap (Option B) catches students who naively apply the 1D infinite capacitor formula $E_{\text{in}} = E_0/K$ , ignoring the boundary conditions that lead to the depolarization field $E_d = -P/(3\epsilon_0)$ in spherical geometries.

Another trap (Option C) catches those who only integrate the energy density of the internal depolarization field ( $\frac{1}{2}\epsilon_0 E_d^2$ ), forgetting that the bound surface charges also produce a dipole field outside the sphere that stores exactly twice as much energy as the internal field.

The gravitational analogy highlights that unlike a uniform mass sphere, the energy is stored in both internal and external fields of a surface distribution.

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