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Question #42712

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The Question

A long straight wire carrying a steady current $I_0$ is placed along the $x$ -axis. A long rectangular conducting slab of width $w$ and thickness $t$ (where $t \ll a$ ) is placed in the $xy$ -plane such that its length is parallel to the $x$ -axis, and its two longitudinal edges are located at $y = a$ and $y = a + w$ . The slab carries a steady current $I$ in the $+x$ -direction. The number density of free electrons in the slab is $n$ , and the elementary charge is $e$ . Assuming steady-state conditions, what is the magnitude of the Hall potential difference $\Delta V$ developed across the width of the slab?

Options

A.
$\frac{\mu_0 I_0 I}{2 \pi n e t w} \ln\left(1 + \frac{w}{a}\right)$
B.
$\frac{\mu_0 I_0 I}{2 \pi n e t \left(a + \frac{w}{2}\right)}$
C.
$\frac{\mu_0 I_0 I}{2 \pi n e t a}$
D.
$\frac{\mu_0 I_0 I}{2 \pi n e t w} \ln\left(\frac{a+w}{w}\right)$

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|Moving Charge in Non-Uniform B|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from algebraic formulas to calculus-based physics in the context of the Hall effect.

A common trap is to blindly apply the standard uniform-field Hall voltage formula ( $\Delta V = v_d B w$ ) by substituting a single representative value for the magnetic field, such as the field at the center of the slab ( $B_{avg}$ ) or at the near edge.

This fundamentally ignores the spatial dependence of the Lorentz force.

Because the magnetic field $B(y)$ from the wire decays as $1/y$ , the local transverse electric field required to maintain equilibrium also varies as $1/y$ .

Consequently, calculating the true macroscopic potential difference requires integrating this spatially varying electric field across the entire width of the conductor.

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