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Question #42951

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor consists of two solid circular conducting plates of radius $R$ separated by a small gap $d$ ( $d \ll R$ ). The space between the plates is filled with a solid dielectric material of dielectric constant $K$ , except for a cylindrical cavity of radius $a$ ( $a < R$ ). The axis of the cavity is parallel to the central axis of the capacitor but displaced by a distance $c$ (such that $c + a < R$ ). A time-varying potential difference $V(t)$ is applied across the plates such that $\frac{dV}{dt} = \alpha$ , where $\alpha$ is a positive constant. Neglecting edge effects, what is the magnitude of the magnetic field at the center of the cylindrical cavity?

Options

A.
$\frac{\mu_0 K \epsilon_0 \alpha c}{2 d}$
B.
$\frac{\mu_0 \epsilon_0 \alpha c}{2 d}$
C.
$\frac{\mu_0 \epsilon_0 \alpha}{2 d} \left[ K c - (K-1) a \right]$
D.
$\frac{\mu_0 \epsilon_0 \alpha c}{2 d} \left[ \frac{K R^2 - (K-1) a^2}{R^2} \right]$

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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

Superposition of Magnetic Fields|Displacement Current Source|Equipotential Surface Geometry|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the superposition of magnetic fields generated by displacement currents in a non-uniform medium, combined with equipotential geometry.

The primary trap (Option D) catches students who calculate the total capacitance, find the total displacement current, and incorrectly assume it is uniformly distributed across the entire cross-section (using an average $J_d$ ).

Another trap (Option B) catches those who assume the magnetic field at a point only depends on the local displacement current density at that exact point.

The correct approach requires recognizing that equipotential plates enforce a uniform electric field, leading to piecewise uniform displacement currents, and then applying the vector superposition of these cylindrical current distributions (analogous to the classic cavity-in-a-wire magnetostatics problem).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Superposition of Magnetic Fields
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free