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Question #81910

Difficulty: 5/10 | 🟢 Free

The Question

A satellite of mass $m$ is in a circular orbit of radius $r$ around a planet of mass $M$ . The planet has a magnetic dipole moment $\mu$ aligned with its rotation axis. The satellite orbits in the planet's equatorial plane. A conducting tether of arbitrary shape connects the main satellite to a small probe. The probe is located at a distance $l$ ( $l \ll r$ ) directly below the satellite along the local vertical. The tether completes an electrical circuit through the planet's ionosphere, maintaining a constant total electrical resistance $R$ . The magnetic field in the equatorial plane is $B = \frac{\mu_0 \mu}{4\pi r^3}$ . Assuming the tether's mass is negligible, what is the instantaneous rate of change of the orbital radius, $\frac{dr}{dt}$ ?

Options

A.
$- \frac{\mu_0^2 \mu^2 l^2}{8 \pi^2 m R r^5}$
B.
$- \frac{\mu_0^2 \mu^2 l^2}{4 \pi^2 m R r^5}$
C.
$- \frac{\mu_0^2 \mu^2 l^2}{8 \pi^2 m R r^4}$
D.
$+ \frac{\mu_0^2 \mu^2 l^2}{8 \pi^2 m R r^5}$

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

Weightlessness in Orbit|Motional EMF for Arbitrary Shapes|Vector Cross-Product

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the coupled application of Faraday's Law for arbitrary shapes and Keplerian orbital mechanics.

A common trap is failing to recognize that the total mechanical energy of an orbit is E = -GMm/2r, not just the potential energy -GMm/r.

Students who equate the magnetic drag force directly to m(dv/dt) without accounting for the virial theorem (where half the lost potential energy goes into increasing kinetic energy) will miss the factor of 2, leading to Option B.

Additionally, the arbitrary shape of the tether tests the understanding that motional EMF in a uniform local field depends only on the vector displacement between the endpoints, while Option C serves as a dimensional analysis trap for students who do not verify units.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Weightlessness in Orbit
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free