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Question #43308

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The Question

A toroidal inductor is constructed with a non-magnetic core of uniform height $h$ . The core is densely wound with $N$ turns of wire carrying a current $I$ . The inner boundary of the core is a cylinder of constant radius $a$ . The outer boundary of the core has a variable radius $b(\theta)$ that depends on the azimuthal angle $\theta$ as: $b(\theta) = \begin{cases} a e^{k \theta} & \text{for } 0 \leq \theta \leq \pi \\ a e^{k (2\pi - \theta)} & \text{for } \pi < \theta \leq 2\pi \end{cases}$ where $k$ is a positive dimensionless constant. Assuming the magnetic field inside the core at a distance $r$ from the central axis is given by $B = \frac{\mu_0 N I}{2\pi r}$ , what is the self-inductance $L$ of this toroid?

Options

A.
$\frac{\mu_0 N^2 h k}{4}$
B.
$\frac{\mu_0 N^2 h k}{2}$
C.
$\frac{\mu_0 N^2 h k}{\pi}$
D.
$\frac{\mu_0 N^2 h}{2\pi} \ln(1 + k\pi)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Energy Stored in Inductor|Toroidal Solenoid Boundary Conditions|Infinite Wire vs. Finite Wire

Why This Question Is Tricky

Common Mistakes

This question tests the student's fundamental understanding of inductance by breaking the symmetry required for standard formulas.

The primary trap relies on students over-relying on the standard formula for a rectangular toroid, $L = \frac{\mu_0 N^2 h}{2\pi} \ln(b/a)$ .

When faced with a variable outer radius $b(\theta)$ , students often incorrectly substitute the maximum radius ( $b_{max} = a e^{k\pi}$ ), yielding Option B, or attempt to find an average radius, yielding Option D.

The rigorous physical approach requires recognizing that inductance is fundamentally a measure of magnetic energy storage per unit current squared ( $L = 2U/I^2$ ).

By forcing the integration of the magnetic energy density over a non-uniform volume, the question filters out students who treat inductance as a static table-lookup value rather than a field-energy derivative.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Energy Stored in Inductor
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free