Before You Begin

Question #42236

Difficulty: 8/10 | 🟢 Free

The Question

A solid non-magnetic cylinder of mass $M$ and cross-sectional area $A$ floats vertically in a non-magnetic liquid of density $\rho$ . A tightly wound circular coil of $N$ turns and radius $R$ ( $R \ll \sqrt{A}$ ) is fixed at the base of the cylinder. The system is placed in an external, axially symmetric magnetic field whose vertical component on the axis is given by $B_z = B_0(1 - \alpha z)$ , where $z$ is measured upwards from the undisturbed liquid surface. A steady current $I$ is passed through the coil such that its magnetic moment is directed upwards. Assuming the liquid surface remains flat and ignoring the magnetic field produced by the coil itself, what is the additional depth $\Delta h$ to which the cylinder sinks?

Options

A.
$\frac{N I \pi R^2 B_0 \alpha}{\rho A g}$
B.
$\frac{N I \pi R^2 B_0 \alpha}{\rho A g} + \frac{B_0^2}{2\mu_0 \rho g}$
C.
$\frac{N I \pi R^2 B_0 \alpha}{\rho A g} - \frac{B_0^2}{2\mu_0 \rho g}$
D.
$\frac{N I \pi R^2 B_0}{\rho A g \alpha}$

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

Archimedes' Buoyancy Dynamics|Gauss’s Law for Magnetism|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize Gauss's Law for Magnetism with Archimedes' principle while avoiding the 'magnetic pressure' trap.

A common misconception is that an external magnetic field alters the hydrostatic pressure of a non-magnetic fluid via $P_m = B^2/2\mu_0$ , leading students to incorrectly add a magnetic pressure head to the buoyancy equation (Option B or C).

Furthermore, students must correctly apply $\nabla \cdot \mathbf{B} = 0$ to find the radial magnetic field component $B_r$ responsible for the vertical Lorentz force on the current loop, distinguishing between the localized force on the coil and the macroscopic fluid boundary conditions.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Archimedes' Buoyancy Dynamics
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free