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Question #43445

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical magnetic material of radius $R$ and length $L$ is modeled as having a non-uniform azimuthal volume current density $\vec{J}(r) = J_0 \left(\frac{r}{R}\right)^2 \hat{\phi}$ , where $r$ is the radial distance from the central axis of the cylinder, $J_0$ is a positive constant, and $\hat{\phi}$ is the azimuthal unit vector. The cylinder is centered at the origin with its axis along the $z$ -axis. Which of the following represents the magnitude of the magnetic field $B$ at a point on the $z$ -axis at a distance $z$ from the center, where $z \gg L$ and $z \gg R$ ?

Options

A.
$\frac{\mu_0 J_0 L R^3}{10 z^3}$
B.
$\frac{\mu_0 J_0 L R^3}{6 z^3}$
C.
$\frac{\mu_0 J_0 L R^3}{20 z^3}$
D.
$\frac{\mu_0 J_0 L R^3}{5 z^3}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Bar Magnet as Equivalent Solenoid|Biot-Savart Law Vector Symmetry|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from the Biot-Savart law for a simple loop to a volume-distributed current density $J(r)$ , which models non-homogeneous magnetization.

The primary trap lies in incorrectly defining the differential magnetic moment.

Students often calculate the total current $I_{tot} = \int J(r) L dr$ and mistakenly multiply it by the total cross-sectional area $\pi R^2$ (yielding the incorrect option B), failing to recognize that the area enclosed by each differential current loop varies with $r$ .

Another common error is using the equatorial dipole formula instead of the axial one, omitting the factor of 2 (yielding option C).

By requiring the integration of $d\mu = J(r) \cdot \pi r^2 \cdot dr\,dz$ , the problem enforces a rigorous understanding of the superposition principle for magnetic dipoles.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Bar Magnet as Equivalent Solenoid
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free