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Question #89124

Difficulty: 5/10 | 🟢 Free

The Question

A solid dielectric sphere of radius $R$ is centered at the origin and possesses a uniform permanent dipole polarization $\vec{P} = P_0 \hat{k}$ . A closed spherical Gaussian surface $S$ of radius $R$ is centered at $(0, 0, R)$ . What is the net electric flux through the surface $S$ ?

Options

A.
$\frac{3 \pi R^2 P_0}{4 \epsilon_0}$
B.
$\frac{\pi R^2 P_0}{\epsilon_0}$
C.
$\frac{\pi R^2 P_0}{2 \epsilon_0}$
D.
$\frac{3 \pi R^2 P_0}{8 \epsilon_0}$

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

Charged Conducting Sphere|Flux and Surface Geometry|Gauss Law|Integration

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the student's ability to apply Gauss's Law to a system with variable bound charge density and asymmetric boundaries.

A common trap is assuming the Gaussian surface $S$ (centered at $z=R$ ) encloses the entire upper hemisphere of the dielectric ( $z>0$ ), which would incorrectly yield $\pi R^2 P_0 / \epsilon_0$ .

Another trap is attempting to calculate the flux of a point dipole field, ignoring that the Gaussian surface intersects the physical charge distribution itself.

The problem demands rigorous spatial reasoning to find the correct intersection plane ( $z=R/2$ ) and accurate integration of the non-uniform surface charge density $\sigma_b = P_0 \cos \theta$ over the enclosed spherical cap.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Charged Conducting Sphere
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free