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Question #43113

Difficulty: 8/10 | 🟢 Free

The Question

A point charge $+Q$ is placed at the exact center of a spherical cavity of radius $R$ located inside an uncharged, isolated solid conducting sphere of outer radius $3R$ . The cavity is completely filled with two different linear, isotropic dielectric materials: the upper hemisphere ( $z > 0$ ) has a relative permittivity $\epsilon_1 = 2$ , and the lower hemisphere ( $z < 0$ ) has a relative permittivity $\epsilon_2 = 4$ . Let $u_1$ be the electrostatic energy density in the upper hemisphere at a radial distance $r = R/2$ , and $u_2$ be the electrostatic energy density in the lower hemisphere at the same radial distance $r = R/2$ . Let $u_3$ be the electrostatic energy density at a radial distance $r = 2R$ (inside the solid conductor). Which of the following represents the correct ratio $u_1 : u_2 : u_3$ ?

Options

A.
$1 : 2 : 0$
B.
$2 : 1 : 0$
C.
$1 : 1 : 0$
D.
$1 : 2 : 1$

Check Your Answer

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

Medium Dependency|Energy Density Locality|Cavity Field Shielding

Why This Question Is Tricky

Common Mistakes

This question tests the misconception that the displacement field D is always spherically symmetric for a point charge, regardless of the surrounding medium's angular dependence.

Students often incorrectly assume D is uniform and calculate energy density as u = D^2 / (2 * epsilon), leading to the inverted ratio 2:1.

The correct physical principle requires applying the boundary condition that the tangential electric field must be continuous across the dielectric interface, which, combined with the equipotential nature of the cavity wall, forces the electric field E to be spherically symmetric instead.

Consequently, the energy density is directly proportional to the local permittivity.

Furthermore, it reinforces the absolute nature of electrostatic shielding within the bulk of a conductor, regardless of the complex polarization occurring inside the cavity.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Medium Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free