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Question #42767

Difficulty: 8/10 | 🟢 Free

The Question

An isolated, uncharged thin conducting spherical shell of radius $R$ provides electrostatic shielding for its internal cavity. Embedded along the equator of this shell is a thin superconducting ring of radius $R$ and self-inductance $L$ . At the center of the cavity, an electron (charge $-e$ ) revolves in a circular orbit of radius $r$ ( $r \ll R$ ) in the equatorial plane with a constant speed $v$ . Initially, the superconducting ring carries no current. The electron is then completely removed from the cavity and taken to infinity. Let $\mu_{\text{ind}}$ be the magnitude of the final induced magnetic dipole moment of the superconducting ring, and $\Delta \Phi_E$ be the change in the net outward electric flux through a concentric spherical Gaussian surface of radius $2R$ . Which of the following is correct?

Options

A.
$\mu_{\text{ind}} = \frac{\mu_0 e v r \pi R}{4 L}, \Delta \Phi_E = \frac{e}{\epsilon_0}$
B.
$\mu_{\text{ind}} = \frac{\mu_0 e v r \pi R}{2 L}, \Delta \Phi_E = -\frac{e}{\epsilon_0}$
C.
$\mu_{\text{ind}} = \frac{\mu_0 e v r R}{4 \pi L}, \Delta \Phi_E = \frac{e}{\epsilon_0}$
D.
$\mu_{\text{ind}} = \frac{\mu_0 e v r^2 \pi}{2 L R}, \Delta \Phi_E = 0$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Cavity Field Shielding|Magnetic Moment of Orbiting Electron|Dipole Potential Angular Dependence

Why This Question Is Tricky

Common Mistakes

This question tests dual misconceptions regarding electromagnetic shielding and induction.

First, students often falsely assume an isolated, uncharged conducting shell perfectly shields the electric field of an internal charge from the outside world, leading them to incorrectly conclude the external electric flux is always zero ( $\Delta \Phi_E = 0$ ).

In reality, only a grounded shell eliminates the external monopole field.

Second, students treat the shell purely as a passive Faraday cage, ignoring dynamic magnetic induction.

By embedding a superconducting ring, the problem forces the application of Lenz's Law and flux conservation ( $\Delta \Phi_B = 0$ ).

The revolving electron acts as a magnetic dipole, and its removal triggers an induced EMF, converting the static Bohr magneton into a macroscopic induced magnetic moment via the ring's self-inductance.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Cavity Field Shielding
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free