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Question #42822

Difficulty: 8/10 | 🟢 Free

The Question

An infinite homogeneous dielectric medium of relative permittivity $K$ contains a spherical vacuum cavity of radius $R$ . A point charge $q$ is fixed at the center of the cavity. Another point charge $Q$ is embedded in the dielectric medium at a distance $d$ from the center of the cavity. Assuming $d \gg R$ , what is the magnitude of the net electrostatic force experienced by the charge $q$ ?

Options

A.
$\frac{3qQ}{4\pi\epsilon_0 (2K+1) d^2}$
B.
$\frac{qQ}{4\pi\epsilon_0 K d^2}$
C.
$\frac{qQ}{4\pi\epsilon_0 d^2}$
D.
$\frac{3qQ}{4\pi\epsilon_0 K(K+2) d^2}$

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

Dielectric Constant (K) Shift|Cavity Field Shielding|Coulomb's Law Vector Form

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of electric fields across dielectric boundaries and the superposition of macroscopic fields with local cavity fields.

A common trap is blindly applying Coulomb's law with the dielectric constant $K$ (yielding Option B) or assuming the vacuum cavity completely removes the dielectric's effect (yielding Option C).

Another lethal trap is misapplying the formula for a dielectric sphere in a vacuum ( $3/(K+2)$ ) to a vacuum cavity in a dielectric (yielding Option D).

Students often fail to realize that while the macroscopic field of $Q$ is reduced by $K$ , the spherical boundary of the cavity develops a non-uniform polarization charge density (dipolar distribution) due to this external field.

This induced surface charge creates an additional uniform field inside the cavity, modifying the net field to $E_{cav} = \frac{3K}{2K+1}E_{macro}$ .

The problem requires transitioning from naive $1/K$ scaling to rigorous boundary-value field analysis.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Dielectric Constant (K) Shift
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free