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Question #42957

Difficulty: 8/10 | 🟢 Free

The Question

A toroidal ring of mean circumference $L = 0.50 \text{ m}$ and uniform cross-sectional area is made of a ferromagnetic material with relative permeability $\mu_r = 2000$ . A narrow air gap of width $d = 1.0 \text{ mm}$ is cut into the ring. A coil of $N = 1000$ turns is wound uniformly around the ring and carries a steady current $I = 1.0 \text{ A}$ . Assuming the magnetic flux is entirely confined within the core and the gap (ignoring fringing effects), what is the magnitude of the magnetic field $B$ in the air gap?

(Given: Permeability of free space $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$ )

Options

A.
$0.32\pi \text{ T}$
B.
$1.60\pi \text{ T}$
C.
$0.40\pi \text{ T}$
D.
$0.16\pi \text{ T}$

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

Magnetic Permeability and Susceptibility|Toroidal Solenoid Boundary Conditions|Approximation (Binomial)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from an ideal Amperean loop to a magnetic circuit model with series reluctances.

The primary trap is ignoring the air gap's reluctance because its physical width $d$ is very small compared to the ring's circumference $L$ .

However, because the core's relative permeability $\mu_r$ is very large, the effective magnetic length of the gap ( $d$ ) is actually greater than the effective magnetic length of the entire core ( $L/\mu_r$ ).

Students who blindly apply the ideal toroid formula $B = \mu_0 \mu_r N I / L$ fail to recognize that the tiny air gap dominates the total reluctance of the circuit, leading to a massive overestimation of the magnetic field (Option B).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Magnetic Permeability and Susceptibility
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free