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Question #43275

Difficulty: 8/10 | 🟢 Free

The Question

A magnetic scalar potential $V_m(r, \theta)$ is established in the vacuum region $R \le r \le 3R$ between two concentric spherical shells. The potential is given by $V_m(r, \theta) = B_0 \left( \frac{R^3}{r^2} + r \right) \cos\theta$ where $B_0$ is a positive constant with units of Tesla, $r$ is the radial distance from the origin, and $\theta$ is the polar angle measured from the z-axis. The magnetic field is defined as $\vec{B} = -\nabla V_m$ . Calculate the magnitude of the magnetic field $\vec{B}$ at a point $P$ located at $r = R$ and $\theta = \pi/3$ .

Options

A.
$\frac{\sqrt{13}}{2} B_0$
B.
$\frac{\sqrt{7}}{2} B_0$
C.
$B_0$
D.
$\frac{\sqrt{7} + 2}{2} B_0$

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

Magnetic Field on Axial vs Equatorial Line|Dipole Potential Angular Dependence|Potential of Concentric Shells

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply the gradient operator in spherical polar coordinates to a scalar potential that is a superposition of a dipole field (the $1/r^2$ term) and a uniform field (the $r$ term).

The primary trap exploits a common misconception where students calculate the magnitudes of the magnetic fields produced by the two potential terms separately and add them algebraically ( $|\vec{B}_1| + |\vec{B}_2|$ ), leading to Option D.

This fails to recognize that the fields must be added vectorially via their orthogonal $\hat{r}$ and $\hat{\theta}$ components.

Secondary traps include ignoring the transverse angular derivative entirely or relying on memorized formulas for purely axial/equatorial positions, which do not apply at an arbitrary angle like $\theta = \pi/3$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Magnetic Field on Axial vs Equatorial Line
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free