Before You Begin

Question #42815

Difficulty: 8/10 | 🟢 Free

The Question

A circuit consists of a battery of constant EMF $\mathcal{E}$ , an ideal inductor of inductance $L$ , and a resistor of resistance $R$ connected in series. The resistor is made of a uniform wire of cross-sectional area $A$ and free electron density $n$ . The switch is closed at $t=0$ . The electric field signal propagates through the circuit almost instantaneously, but the macroscopic current is governed by the circuit's inductance. Assuming the drift velocity of electrons is directly proportional to the instantaneous current, what is the net displacement of an average free electron in the resistor during the time interval from $t=0$ to $t = L/R$ ? (Here $q$ is the elementary charge and $e$ is the base of the natural logarithm).

Options

A.
$\frac{\mathcal{E} L}{n q A R^2 e}$
B.
$\frac{\mathcal{E} L}{n q A R^2} \left(1 - \frac{1}{e}\right)$
C.
$\frac{\mathcal{E} L}{n q A R^2}$
D.
$\frac{\mathcal{E} L}{n q A R^2} \left(1 + \frac{1}{e}\right)$

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

Drift Velocity Mechanism|Transient Current in RL Circuit|The Reaction Time Gap

Why This Question Is Tricky

Common Mistakes

This question tests the 'Reaction Time Gap' misconception in transient circuits.

Students often memorize that the electric field sets up in a circuit at the speed of light, leading to the false assumption that the drift velocity of electrons instantly reaches its steady-state value $v_d = \mathcal{E}/(nqAR)$ .

If they apply this, they will simply multiply the steady-state drift velocity by the time $L/R$ , arriving at the incorrect Option C.

The trap forces students to recognize that while the electric field signal propagates almost instantaneously, the actual magnitude of the macroscopic electric field inside the resistive wire (and consequently the drift velocity) is constrained by the inductive inertia of the circuit, growing as $1 - e^{-Rt/L}$ .

Option B traps students who incorrectly integrate the transient term, and Option D traps those who make a sign error when evaluating the lower limit of the integral.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Drift Velocity Mechanism
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free