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Question #42272

Difficulty: 8/10 | 🟢 Free

The Question

A satellite of mass $m$ and constant magnetic dipole moment $\vec{M} = M_0 \hat{i}$ orbits a planet. The planet's magnetic field in the orbital plane ( $xy$ -plane) is given by $\vec{B} = B_0 \left( \frac{x^2 - y^2}{R^2} \hat{i} - \frac{2xy}{R^2} \hat{j} \right)$ , where $R$ is a positive constant. At the instant the satellite passes through the point $P \left( \frac{R}{\sqrt{2}}, \frac{R}{\sqrt{2}} \right)$ , what is the magnitude of the rate of change of its areal velocity vector about the origin?

Options

A.
$\frac{M_0 B_0}{m}$
B.
$\frac{M_0 B_0}{2m}$
C.
$\frac{3 M_0 B_0}{2m}$
D.
0

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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

Kepler’s Second Law (Areal Velocity)|Magnetic Dipole Moment Orientation|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the distinction between orbital and spin angular momentum when a magnetic dipole interacts with a non-uniform magnetic field.

A common trap is assuming Kepler's Second Law holds globally (yielding 0) or incorrectly using the spin torque $\vec{\tau} = \vec{M} \times \vec{B}$ to calculate the change in areal velocity.

However, areal velocity is strictly governed by the orbital angular momentum, which is altered only by the moment of the translational magnetic force $\vec{r} \times \nabla(\vec{M} \cdot \vec{B})$ , not the internal spin torque.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Kepler’s Second Law (Areal Velocity)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free