Before You Begin

Question #43020

Difficulty: 8/10 | 🟢 Free

The Question

A thin spherical shell of radius $R$ carries a non-uniform surface current density $\vec{K} = K_0 \sin\theta \hat{\phi}$ , where $\theta$ is the polar angle measured from the $z$ -axis, $\hat{\phi}$ is the azimuthal unit vector, and $K_0$ is a positive constant. Which of the following represents the net magnetic field vector $\vec{B}$ at the center of the spherical shell?

Options

A.
$\frac{2}{3} \mu_0 K_0 \hat{k}$
B.
$\mu_0 K_0 \hat{k}$
C.
$\frac{4}{3} \mu_0 K_0 \hat{k}$
D.
$\frac{1}{2} \mu_0 K_0 \hat{k}$

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

Magnetic Field on Axis vs. Center|Current Density Vector

Why This Question Is Tricky

Common Mistakes

Students incorrectly apply the standard magnetic field formula for a single loop to a continuous spherical distribution.

A common trap is writing the elemental field as $dB = \frac{\mu_0 dI}{2R}$ , ignoring that the elemental rings have varying radii $r = R\sin\theta$ and are located at varying axial distances $z = R\cos\theta$ from the center.

This flawed assumption leads to an incorrect integration of $\sin\theta$ instead of $\sin^3\theta$ , yielding the trap answer $\mu_0 K_0$ .

The problem tests the rigorous application of the Biot-Savart law and superposition principle over a 2D curved surface, requiring careful geometric parameterization.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Magnetic Field on Axis vs. Center
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free