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Question #43046

Difficulty: 8/10 | 🟢 Free

The Question

A long solid non-magnetic cylinder of radius $R$ is oriented with its axis along the $z$ -axis. It carries a steady current with a non-uniform current density vector given by $\mathbf{J} = J_0 (1 + \alpha \cos \theta) \hat{k}$ , where $J_0$ and $\alpha$ are positive constants, $\theta$ is the azimuthal angle measured from the positive $x$ -axis, and $\hat{k}$ is the unit vector along the $z$ -axis. What is the magnetic field vector $\mathbf{B}$ at the central axis of the cylinder ( $x=0, y=0$ )?

Options

A.
$-\frac{\mu_0 J_0 \alpha R}{2} \hat{j}$
B.
$\mathbf{0}$
C.
$\frac{\mu_0 J_0 R}{2} \hat{j}$
D.
$\frac{\mu_0 J_0 \alpha R}{4} \hat{i}$

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

Toroidal Solenoid Boundary Conditions|Current Density Vector|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the limits of Ampere's Circuital Law and the assumption of symmetry.

Students are heavily conditioned to apply $\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc}$ to cylindrical geometries, pulling $B$ out of the integral by assuming it is purely tangential and constant.

However, the non-uniform current density $\mathbf{J}(\theta)$ breaks azimuthal symmetry.

If a student blindly applies Ampere's law to a vanishingly small loop at the center, they will incorrectly conclude the field is zero (Option B).

To solve this, the student must recognize the broken symmetry and revert to the fundamental Biot-Savart law, integrating the contributions of infinitesimal line currents over the cylinder's cross-section.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Toroidal Solenoid Boundary Conditions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free