Before You Begin

Question #42276

Difficulty: 8/10 | 🟢 Free

The Question

A satellite of mass $m$ is in a quasi-circular orbit of initial radius $r_0$ and initial velocity $v_0$ around a planet. A uniform magnetic field perpendicular to the orbital plane exists in the region, varying with time as $B(t) = \frac{B_0}{\sqrt{1 + \alpha t}}$ , where $B_0$ and $\alpha$ are positive constants. The satellite deploys a straight conducting tether of length $L$ ( $L \ll r_0$ ) oriented radially outward. The tether forms a local closed electrical circuit of constant total resistance $R$ with the surrounding plasma. Assuming the circuit's enclosed area is negligible (so the induced EMF is purely motional) and neglecting all other forms of drag, which of the following expressions gives the orbital velocity $v(t)$ of the satellite as a function of time?

Options

A.
$v_0 (1 + \alpha t)^{\frac{B_0^2 L^2}{m R \alpha}}$
B.
$v_0 (1 + \alpha t)^{-\frac{B_0^2 L^2}{m R \alpha}}$
C.
$v_0 \exp\left( - \frac{B_0^2 L^2}{m R} t \right)$
D.
$v_0 (1 + \alpha t)^{\frac{2 B_0^2 L^2}{m R \alpha}}$

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

Kepler’s Third Law (Time Period)|Motional EMF in Rotating Rods|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

This question tests the 'satellite paradox' coupled with electrodynamics.

Students intuitively assume that a dissipative drag force (like eddy current braking) will directly decrease the velocity of the object, leading them to choose an option with a negative exponent or exponential decay.

However, in orbital mechanics, a non-conservative drag force reduces the total orbital energy, which causes the orbital radius to shrink.

By Kepler's Third Law (or the virial theorem), a smaller orbital radius corresponds to a higher orbital velocity.

The trap exploits the failure to recognize that the velocity vector is an output of the system's own dissipation and orbital energy constraints, not simple linear kinematics.

Quick Facts About This Question

Question Information

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