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Question #43317

Difficulty: 8/10 | 🟢 Free

The Question

An n-type semiconducting circular ring of radius $R$ (in $\text{m}$ ) lies in the $xy$ -plane, centered at the origin. It has a uniform initial electron concentration $n_0$ (in $\text{m}^{-3}$ ) and electron mobility $\mu$ (in $\text{m}^2\cdot\text{V}^{-1}\cdot\text{s}^{-1}$ ). At $t=0$ , a time-varying magnetic field is applied, inducing an electric field whose tangential component along the ring is $E_\theta(\theta) = E_0 \sin^2(\theta)$ (in $\text{V}\cdot\text{m}^{-1}$ ), where $\theta$ is the azimuthal angle and $E_0$ is a positive constant. Assuming the initial hole concentration is negligible, what is the initial rate of change of the electron concentration $\frac{\partial n}{\partial t}$ (in $\text{m}^{-3}\cdot\text{s}^{-1}$ ) at $\theta = \pi/6$ and $t=0^+$ ?

Options

A.
$\frac{\sqrt{3} n_0 \mu E_0}{2 R}$
B.
$-\frac{\sqrt{3} n_0 \mu E_0}{2 R}$
C.
0
D.
$\frac{n_0 \mu E_0}{2 R}$

Check Your Answer

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

Induced Electric Fields|Drift vs Diffusion Current|Transient Current in RL Circuit

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between transient transport physics and steady-state circuit theory.

A common trap is assuming that because the ring is a closed series circuit, the current must be uniform everywhere instantaneously ( $\nabla \cdot J = 0$ ), leading to the incorrect conclusion that $\frac{\partial n}{\partial t} = 0$ .

However, in a semiconductor subjected to a non-uniform induced electric field, the initial drift current is non-uniform.

Because the carrier concentration is initially uniform, diffusion cannot immediately balance this.

Students must correctly apply the continuity equation to the drift flux, accounting for the negative charge of electrons which reverses the flux direction relative to the electric field, to find the local rate of charge accumulation before the steady-state drift-diffusion balance is established.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Induced Electric Fields
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free