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Question #43423

Difficulty: 8/10 | 🟢 Free

The Question

An ideal toroid of mean radius $R$ and circular cross-section of radius $r$ ( $r \ll R$ ) has a uniform winding that would consist of $N$ total turns if it were completely closed. However, a small angular section $\Delta \theta$ (where $\Delta \theta \ll 1 \text{ rad}$ ) is cut out and removed, leaving a gap. A steady current $I$ flows through the remaining turns. Assuming the windings are perfectly poloidal (meaning there is no net azimuthal current along the toroid's circumference), what is the magnitude of the magnetic field at the geometric center $O$ of the toroid?

Options

A.
$\frac{\mu_0 N I r^2 \Delta \theta}{8 \pi R^3}$
B.
$\frac{\mu_0 N I r^2 \Delta \theta}{4 \pi R^3}$
C.
$\frac{\mu_0 N I \Delta \theta}{4 \pi R}$
D.
$0$

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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

Toroidal Solenoid Boundary Conditions|Moving Charge in Non-Uniform B|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the limits of Ampere's Law and the application of the Principle of Superposition in magnetostatics.

Students often reflexively assume that the magnetic field outside any toroidal geometry is zero, failing to recognize that the gap breaks the azimuthal symmetry (leading to Option D).

This forces a transition from the Amperian limit to a superposition approach, treating the gap as a 'negative' magnetic dipole.

Even if students realize the gap acts as a source, they frequently misidentify the missing section as a simple current arc (Option C) or incorrectly apply the axial dipole formula instead of the equatorial one (Option B), ignoring the perpendicular vector geometry of the magnetic moment relative to the observation point at the center.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Toroidal Solenoid Boundary Conditions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free