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Question #43396

Difficulty: 8/10 | 🟢 Free

The Question

A straight cylindrical wire of radius $R = 1.0 \text{ mm}$ and finite length $L = 6.0 \text{ cm}$ carries a steady current. The current density within the wire varies radially as $J(r) = J_0 \left(\frac{r}{R}\right)$ , where $r$ is the distance from the wire's axis and $J_0 = \frac{3}{\pi} \times 10^5 \text{ A/m}^2$ . Point $P$ is located on the perpendicular bisector of the wire at a distance $d = \sqrt{3} \text{ cm}$ from its axis. Assuming $R \ll d$ so that the wire's thickness can be neglected when calculating the magnetic field at $P$ , what is the magnitude of the magnetic field at point $P$ ? (Take $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$ )

Options

A.
$2.0 \text{ } \mu\text{T}$
B.
$3.0 \text{ } \mu\text{T}$
C.
$\frac{4}{\sqrt{3}} \text{ } \mu\text{T}$
D.
$2\sqrt{3} \text{ } \mu\text{T}$

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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

Infinite Wire vs. Finite Wire|Continuous Charge Distribution|Integration

Why This Question Is Tricky

Common Mistakes

This question tests two common misconceptions.

First, students often incorrectly assume a uniform current distribution ( $I = J_0 A$ ) instead of integrating the radially varying current density $J(r)$ , leading to an incorrect total current.

Second, students frequently misapply Ampere's Law ( $B = \mu_0 I / 2\pi d$ ) by treating the wire as infinite.

Because the observation distance $d$ is comparable to the wire's length $L$ , the infinite wire approximation fails.

Students must transition to the Biot-Savart Law for a finite wire, explicitly accounting for the angle-subtended geometry ( $\sin\theta_1 + \sin\theta_2$ ) to find the correct magnetic field.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Infinite Wire vs. Finite Wire
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free