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Question #86986

Difficulty: 5/10 | 🟢 Free

The Question

An electron (mass $m$ , charge $-e$ ) moves in the $xy$ -plane in a time-varying, space-dependent magnetic field $\vec{B} = \beta t \left(\frac{r}{a}\right)^n \hat{k}$ , where $\beta, a, n$ are positive constants, $t$ is time, and $r$ is the distance from the origin. The electron's motion is a slow spiral, approximated as a circular orbit of slowly changing radius $r(t)$ centered at the origin. Assuming the radial acceleration is negligible compared to the centripetal acceleration, which of the following gives the magnitude of the induced electric field $E$ at radius $r$ , and the rate of change of the electron's orbital magnetic dipole moment $\frac{d\mu}{dt}$ ?

Options

A.
$E = \frac{\beta r^{n+1}}{a^n (n+2)}$ , $\frac{d\mu}{dt} = 0$
B.
$E = \frac{\beta r^{n+1}}{2 a^n}$ , $\frac{d\mu}{dt} = \frac{e^2 \beta r^{n+2}}{2 m a^n}$
C.
$E = \frac{\beta r^{n+1}}{a^n (n+2)}$ , $\frac{d\mu}{dt} = \frac{e^2 \beta r^{n+2}}{2 m a^n}$
D.
$E = \frac{\beta r^{n+1}}{a^n (n+1)}$ , $\frac{d\mu}{dt} = 0$

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

Magnetic Dipole Moment of Revolving Electron|Motional EMF for Arbitrary Shapes|Integration

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the rigorous application of Faraday's Law in a space-dependent magnetic field and the concept of adiabatic invariance.

A common trap is assuming the induced EMF can be calculated using $\Phi = B \cdot \pi r^2$ as if the field were uniform, leading to an incorrect electric field.

Furthermore, students often mistakenly assume the orbital radius remains constant as the magnetic field increases, directly differentiating $\mu = \frac{e^2 B r^2}{2m}$ to find a non-zero $\frac{d\mu}{dt}$ .

In reality, the coupling of the induced tangential electric field and the radial drift velocity creates a torque that shrinks the orbit at exactly the rate required to conserve the magnetic flux through the orbit, rendering the magnetic dipole moment an exact invariant ( $\frac{d\mu}{dt} = 0$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Magnetic Dipole Moment of Revolving Electron
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free