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Question #42964

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The Question

A parallel plate capacitor consists of two square plates of area $A$ separated by a distance $d$ . The plates carry constant free surface charge densities $+\sigma_0$ at $x=0$ and $-\sigma_0$ at $x=d$ . The space between the plates is completely filled with a solid linear material that is both dielectric and diamagnetic. The relative permittivity of the material varies spatially as $\epsilon_r(x) = \frac{1}{1 - \alpha x}$ (where $\alpha d < 1$ ). The material has a uniform diamagnetic susceptibility $\chi_m = -\chi_0$ (where $0 < \chi_0 \ll 1$ ). An external non-uniform magnetic field $\vec{B}_{ext} = B_0 \sqrt{1 - \alpha x} \hat{k}$ is applied parallel to the plates. Calculate the total interaction potential energy $U_{int}$ of the induced electric and magnetic dipoles with the local fields, defined as $U_{int} = -\frac{1}{2} \int (\vec{P} \cdot \vec{E}) dV - \frac{1}{2} \int (\vec{M} \cdot \vec{B}) dV$ , where $\vec{E}$ and $\vec{B}$ are the total local fields inside the material. (Assume $\chi_0$ is small enough that $1 - \chi_0 \approx 1$ ).

Options

A.
$\frac{A d}{2} \left[ \frac{\chi_0 B_0^2}{\mu_0} \left( 1 - \frac{\alpha d}{2} \right) - \frac{\sigma_0^2 \alpha d}{2 \epsilon_0} \left( 1 - \frac{2\alpha d}{3} \right) \right]$
B.
$\frac{A d}{2} \left[ \frac{\chi_0 B_0^2}{\mu_0} \left( 1 - \frac{\alpha d}{2} \right) + \frac{\sigma_0^2 \alpha d}{2 \epsilon_0} \left( 1 - \frac{2\alpha d}{3} \right) \right]$
C.
$\frac{A d}{2} \left[ \frac{\chi_0 B_0^2}{\mu_0} \left( 1 - \frac{\alpha d}{2} \right) - \frac{\sigma_0^2}{\epsilon_0} \left( 1 - \frac{\alpha d}{2} \right) \right]$
D.
$A d \left[ \frac{\chi_0 B_0^2}{\mu_0} \left( 1 - \frac{\alpha d}{2} \right) - \frac{\sigma_0^2 \alpha d}{2 \epsilon_0} \left( 1 - \frac{2\alpha d}{3} \right) \right]$

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Correct Answer

Step-by-Step Solution

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

Diamagnetic Property of Materials|Non-Uniform Dielectric Filling|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly superimpose electric and magnetic interaction energies in a non-uniform medium.

The primary trap lies in failing to recognize that a spatially varying dielectric constant $\epsilon_r(x)$ creates a spatial gradient in the polarization density $\vec{P}$ , which must be explicitly integrated as $\vec{P} \cdot \vec{E}$ rather than incorrectly using the total field energy $\vec{D} \cdot \vec{E}$ .

Furthermore, the 'lethal' element is the coupling of the diamagnetic property: students often miss the sign reversal caused by the negative susceptibility ( $\chi_m = -\chi_0$ ), which makes the magnetic interaction energy positive, acting in opposition to the standard negative electric interaction energy.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Diamagnetic Property of Materials
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free