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Question #90406

Difficulty: 5/10 | 🟢 Free

The Question

A semi-infinite tightly wound solenoid of radius $R$ has $n$ turns per unit length and lies along the negative z-axis from $z = 0$ to $z = -\infty$ . A small flat circular coil of radius $r$ ( $r \ll R$ ) and $N$ turns is placed such that its center is at coordinates $(0, y, d)$ , where $d \gg R$ and $y \ll d$ . The plane of the small coil is parallel to the xy-plane. Assuming the medium is vacuum with permeability $\mu_0$ , what is the mutual inductance $M$ between the solenoid and the small coil? (Keep terms up to the lowest non-zero order in $(y/d)$ ).

Options

A.
$\frac{\mu_0 \pi n N R^2 r^2}{4 d^2} \left( 1 - \frac{3}{2} \frac{y^2}{d^2} \right)$
B.
$\frac{\mu_0 \pi n N R^2 r^2}{4 d^2} \left( 1 - \frac{y^2}{d^2} \right)$
C.
$\frac{\mu_0 \pi n N R^2 r^2}{2 d^2} \left( 1 - \frac{3}{2} \frac{y^2}{d^2} \right)$
D.
$\frac{\mu_0 \pi n N R^2 r^2}{2 d^2} \left( 1 - \frac{y^2}{d^2} \right)$

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

Mutual Inductance Reciprocity|Solenoid Fringe Fields|Approximations

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the student's ability to model the fringe field of a solenoid, a regime where the ideal $B = \mu_0 n I$ approximation fails completely.

The primary trap is calculating the flux using the magnitude of the magnetic field $|\vec{B}|$ instead of the perpendicular component $B_z$ , which leads to an incorrect binomial factor of $(1 - y^2/d^2)$ instead of $(1 - 3y^2/2d^2)$ .

Additionally, it penalizes students who fail to recognize the $1/4$ geometric factor arising from the semi-infinite integration (or the equivalent magnetic monopole surface charge at the open end), often confusing it with the $1/2$ factor of an infinite wire or a full dipole.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Solenoid Fringe Fields
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free