Before You Begin

Question #43784

Difficulty: 8/10 | 🟢 Free

The Question

An electron (mass $m$ , charge $-e$ ) is constrained to move along a frictionless, non-conducting circular ring of radius $r$ . Initially, there is no magnetic field, and the electron is in a quantized stationary state with a mechanical angular momentum $\vec{L}_i = n\hbar \hat{k}$ (where $n$ is a positive integer and $\hbar = h/2\pi$ ). A uniform time-varying magnetic field $\vec{B}(t) = B(t) \hat{k}$ is then applied across the region, slowly increasing from $B=0$ to a constant final value $B=B_0$ . Which of the following expressions represents the final mechanical angular momentum $\vec{L}_f$ of the electron?

Options

A.
$n\hbar \hat{k}$
B.
$\left( n\hbar + \frac{e B_0 r^2}{2} \right) \hat{k}$
C.
$\left( n\hbar - \frac{e B_0 r^2}{2} \right) \hat{k}$
D.
$\left( n\hbar + e B_0 r^2 \right) \hat{k}$

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

Quantization of Angular Momentum|Motional EMF for Arbitrary Shapes|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the misconception that Bohr's quantization condition ( $L = n\hbar$ ) strictly and permanently applies to the mechanical angular momentum ( $mvr$ ) even in the presence of a time-varying magnetic field.

Students falling for this trap will incorrectly choose the static option.

The core physical principle is that a changing magnetic flux induces a non-conservative electric field (Faraday's Law) which exerts a tangential torque on the electron, doing work and altering its mechanical angular momentum.

In advanced quantum mechanics, it is the canonical angular momentum ( $L_{can} = L_{mech} + q\vec{r}\times\vec{A}$ ) that remains quantized to $n\hbar$ , forcing the mechanical angular momentum to shift by exactly the angular impulse delivered by the induced EMF.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Quantization of Angular Momentum
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free