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Question #43392

Difficulty: 8/10 | 🟢 Free

The Question

A long straight solid cylindrical conductor of radius $R$ carries a steady current. The current density $\vec{J}$ is parallel to the axis of the cylinder and its magnitude varies with the radial distance $r$ from the axis as $J(r) = J_0 \left(\frac{r}{R}\right)$ , where $J_0$ is a positive constant. Let $B_1$ be the magnitude of the magnetic field at a radial distance $r = R/2$ from the axis, and $B_2$ be the magnitude of the magnetic field at a radial distance $r = 2R$ from the axis. Which of the following correctly represents the pair $(B_1, B_2)$ ?

Options

A.
$\left( \frac{\mu_0 J_0 R}{12}, \frac{\mu_0 J_0 R}{6} \right)$
B.
$\left( \frac{\mu_0 J_0 R}{8}, \frac{\mu_0 J_0 R}{4} \right)$
C.
$\left( \frac{\mu_0 J_0 R}{16}, \frac{\mu_0 J_0 R}{8} \right)$
D.
$\left( \frac{\mu_0 J_0 R}{4}, \frac{\mu_0 J_0 R}{4} \right)$

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

Ampere’s Circuital Law Symmetry|Gauss Law Applicability|Integration (Radial)

Why This Question Is Tricky

Common Mistakes

This question tests the integral form of Ampere's Circuital Law, which is mathematically analogous to Gauss's Law for electrostatics.

The primary trap lies in how students calculate the enclosed current for a non-uniform distribution.

A common misconception is to simply multiply the local current density by the cross-sectional area ( $I_{enc} = J(r) \times \pi r^2$ ), which incorrectly yields $B_1 = \mu_0 J_0 R / 8$ and $B_2 = \mu_0 J_0 R / 4$ .

To solve this correctly, students must recognize that the current density varies radially and set up the area integral $I_{enc} = \int J \cdot dA$ using the differential area element of a ring, $dA = 2\pi r dr$ .

Failing to account for this variable cross-sectional area weighting leads to incorrect polynomial dependencies and wrong coefficients.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Ampere’s Circuital Law Symmetry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free