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Question #42926

Difficulty: 8/10 | 🟢 Free

The Question

A long cylindrical region of radius $a$ contains a spatially uniform, time-varying magnetic field $\vec{B}(t) = B_0 \sin(\omega t) \hat{k}$ , where $\hat{k}$ is along the axis of the cylinder. A solid conducting sphere of radius $R$ ( $R \ll a$ ) carrying a net static charge $Q$ is placed such that its center is at a distance $d$ ( $R \ll d < a$ ) from the cylinder's axis. Assume the sphere is small enough that the induced electric field from the changing magnetic field is approximately uniform across its volume. Which of the following expressions gives the maximum magnitude of the displacement current density, $\|\vec{J}_d\|_{\text{max}}$ , evaluated at any point just outside the surface of the sphere?

Options

A.
$\frac{3}{2} \epsilon_0 d B_0 \omega^2$
B.
$\frac{1}{2} \epsilon_0 d B_0 \omega^2$
C.
$\frac{3}{2} \epsilon_0 d B_0 \omega^2 + \frac{Q \omega}{4 \pi R^2}$
D.
$\frac{1}{2} \epsilon_0 R B_0 \omega^2$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Charged Conducting Sphere|Induced Electric Fields|Maxwell's Correction to Ampere's Law|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple Faraday's Law of induction with electrostatic boundary conditions and the Maxwell-Ampere law.

The primary trap lies in treating the displacement current purely based on the ambient induced electric field ( $J_d = \epsilon_0 \partial E_{\text{ind}} / \partial t$ ), ignoring the fact that the conducting sphere must polarize to remain an equipotential volume.

This polarization modifies the local electric field by a factor of 3 at the poles.

A secondary trap is incorrectly assuming that the static charge $Q$ contributes to the displacement current, failing to recognize that only the time-varying component of the electric field generates $J_d$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Charged Conducting Sphere
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free