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Question #42694

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor with plate area $A$ and separation $d$ is charged to a total free charge $Q$ and then electrically isolated. A dielectric slab of the same area $A$ and thickness $d$ is slowly inserted between the plates. The dielectric constant of the slab varies with the distance $x$ from one of its faces as $K(x) = \frac{K_0}{1 + \alpha x}$ , where $K_0$ and $\alpha$ are positive constants. Assume $K_0 > 1 + \frac{\alpha d}{2}$ and that the electric field is strictly normal to the plates. What is the magnitude of the work done by the external agent to completely insert the slab? (Here, $\epsilon_0$ is the vacuum permittivity).

Options

A.
$\frac{Q^2 d}{2 \epsilon_0 A} \left( 1 - \frac{2 + \alpha d}{2 K_0} \right)$
B.
$\frac{Q^2 d}{2 \epsilon_0 A} \left( 1 - \frac{\alpha d}{K_0 \ln(1+\alpha d)} \right)$
C.
$\frac{Q^2 d}{2 \epsilon_0 A} \left( 1 - \frac{\ln(1+\alpha d)}{K_0 \alpha d} \right)$
D.
$\frac{Q^2 d}{\epsilon_0 A} \left( 1 - \frac{2 + \alpha d}{2 K_0} \right)$

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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 Slab Displacement|Conductor Interior Potential|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of Gauss's Law in non-uniform dielectrics and the work-energy theorem.

A common trap is treating the non-uniform dielectric by averaging its dielectric constant $K(x)$ over the distance, or incorrectly applying the series capacitance formula by integrating $K(x)$ directly instead of its reciprocal (elastance).

By defining $K(x)$ such that its reciprocal is linear, students are forced to either correctly integrate the infinitesimal elastances $d(1/C) = dx / (\epsilon_0 A K(x))$ or use Gauss's Law for the displacement field $\vec{D}$ to find the true potential difference.

The problem also verifies the correct sign convention for the work done by an external agent against the electrostatic pull.

Quick Facts About This Question

Question Information

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