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Question #43293

Difficulty: 8/10 | 🟢 Free

The Question

A non-ideal tangent galvanometer is constructed using a uniformly wound cylindrical solenoid of length $L = 2\sqrt{3}R$ and radius $R$ , having a total of $N$ turns. It carries a steady current $I$ . The coil is aligned such that its axis is perpendicular to the magnetic meridian. The horizontal component of Earth's magnetic field is $B_H$ . To analyze the field at the center, the coil is modeled using the Gilbert model, where it is equivalent to a cylinder with uniform magnetization $M = \frac{NI}{L}$ along its axis, and its two flat circular ends act as fictitious magnetic surface charge densities $\pm \sigma_m$ . A small magnetic compass needle placed at the exact center of the coil shows a steady deflection of $60^\circ$ . Assuming the needle aligns with the net magnetic induction $\mathbf{B}$ at the center, find the correct expression for the current $I$ . ( $\mu_0$ is the vacuum permeability)

Options

A.
$\frac{4\sqrt{3} R B_H}{\mu_0 N}$
B.
$\frac{2\sqrt{3} R B_H}{\mu_0 N}$
C.
$\frac{12 R B_H}{\mu_0 N (2-\sqrt{3})}$
D.
$\frac{12 R B_H}{\mu_0 N (2+\sqrt{3})}$

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

Tangent Galvanometer Principle|Gauss’s Law for Magnetism|Continuous Charge Distribution

Why This Question Is Tricky

Common Mistakes

Students often blindly apply the ideal tangent galvanometer formula $B = \frac{\mu_0 N I}{2R}$ , which assumes a single loop or an infinitely short coil, ignoring flux leakage in a finite solenoid.

Furthermore, when introduced to the Gilbert (magnetic charge) model, a critical conceptual trap awaits: students mistakenly assume the magnetic induction $\mathbf{B}$ is solely generated by the fictitious pole charges (i.e., $B = \mu_0 H_{poles}$ ).

They fail to realize that inside the equivalent magnetic material, the fundamental relation is $\mathbf{B} = \mu_0 (\mathbf{H} + \mathbf{M})$ .

The poles generate a demagnetizing field $\mathbf{H}$ that opposes the magnetization $\mathbf{M}$ , and the net induction $\mathbf{B}$ requires accounting for both.

This question penalizes rote memorization and tests the rigorous application of field theory and boundary conditions.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Tangent Galvanometer Principle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free