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Question #42061

Difficulty: 8/10 | 🟢 Free

The Question

A steady current density $\vec{J} = J_0 \frac{x}{R} \hat{j}$ is maintained within a spherical region of radius $R$ centered at the origin, where $J_0$ is a positive constant. What is the magnetic field $\vec{B}$ at the origin?

Options

A.
$\frac{\mu_0 J_0 R}{6} \hat{k}$
B.
$\frac{\mu_0 J_0 R}{3} \hat{k}$
C.
$-\frac{\mu_0 J_0 R}{4} \hat{i}$
D.
\vec{0}

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Biot-Savart Law Vector Symmetry|Vector Decomposition Ambiguity|Integration

Why This Question Is Tricky

Common Mistakes

Students often rely on Ampere's Law and assume that highly symmetric geometries (like a sphere) imply zero magnetic field at the center or allow for a simple scalar loop equation.

Here, the non-uniform, Cartesian-dependent current density $\vec{J} = J_0 (x/R) \hat{j}$ breaks standard azimuthal symmetry.

The trap forces students to abandon Ampere's Law and perform a full 3D Biot-Savart vector integration.

A common mistake is incorrectly evaluating the cross product $\vec{J} \times \hat{r}$ or assuming the field cancels out entirely ( $\vec{0}$ ) due to the anti-symmetric current flow in the left and right hemispheres.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Vector Decomposition Ambiguity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free