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Question #42730

Difficulty: 8/10 | 🟢 Free

The Question

A constant potential difference $V$ is applied across the ends of a long straight cylindrical solid wire of radius $R$ and length $L$ . The resistivity of the material of the wire varies with the radial distance $r$ from its central axis as $\rho(r) = \rho_0 \frac{R}{r}$ , where $\rho_0$ is a positive constant. What is the magnitude of the magnetic field $B$ at a radial distance $r = \frac{R}{2}$ from the axis of the wire?

Options

A.
$\frac{\mu_0 V R}{6 \rho_0 L}$
B.
$\frac{\mu_0 V R}{12 \rho_0 L}$
C.
$\frac{\mu_0 V R}{8 \rho_0 L}$
D.
$\frac{\mu_0 V R}{4 \rho_0 L}$

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

Resistivity vs Resistance|Biot-Savart Law Vector Symmetry|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to link the microscopic form of Ohm's Law with Ampere's Law in the presence of a non-uniform current distribution.

The primary trap lies in calculating the magnetic field inside the conductor.

Students frequently calculate the total equivalent resistance of the wire, find the total current $I_{tot}$ , and then incorrectly apply the standard formula for a uniform current density $B = \frac{\mu_0 I_{tot} r}{2\pi R^2}$ .

This bypasses the fundamental requirement of Ampere's Law to integrate the local current density $J(r) = E/\rho(r)$ to find the exact enclosed current $I_{enc}$ within the specific Amperian loop.

By making resistivity radially dependent, the problem punishes rote application of the uniform-wire magnetic field formula and rewards a first-principles approach.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Resistivity vs Resistance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free