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Question #42777

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The Question

A parallel plate capacitor consists of two circular plates of radius $R$ separated by a distance $d$ ( $d \ll R$ ). The space between the plates is filled with a non-magnetic dielectric material whose relative permittivity varies radially as $K(r) = 1 + \alpha r$ , where $r$ is the distance from the central axis of the plates and $\alpha$ is a positive constant. A time-varying potential difference $V(t) = \beta t$ (where $\beta > 0$ is a constant) is applied across the plates. Neglecting edge effects, what is the magnitude of the Poynting vector $\vec{S}$ at the edge of the capacitor ( $r = R$ ) at time $t$ ?

Options

A.
$\varepsilon_0 \frac{\beta^2 t}{d^2} \left( \frac{R}{2} + \frac{\alpha R^2}{2} \right)$
B.
$\varepsilon_0 \frac{\beta^2 t}{d^2} \left( \frac{R}{2} + \frac{\alpha R^2}{4} \right)$
C.
$\varepsilon_0 \frac{\beta^2 t}{d^2} \left( \frac{R}{2} + \frac{\alpha R^2}{3} \right)$
D.
$\varepsilon_0 \frac{\beta^2 t}{d^2} (1 + \alpha R) R$

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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

Energy Density Locality|Poynting Vector and Intensity|Medium Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of the Poynting vector and displacement current in a spatially inhomogeneous medium.

A common lethal trap is treating the capacitor as a lumped parameter and assuming the magnetic field at the edge can be calculated using the local displacement current density $J_d(R)$ as if it were uniform ( $B = \mu_0 J_d(R) R / 2$ ), which leads to Option A.

Another trap is incorrectly averaging the dielectric constant linearly without the proper $2\pi r dr$ area weighting, leading to Option B.

Students must recognize that the non-uniform dielectric forces an integration of the local displacement current density over the entire circular area to satisfy the boundary conditions for the global power flow.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Energy Density Locality
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free