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Question #43404

Difficulty: 8/10 | 🟢 Free

The Question

A tightly wound finite solenoid of length $L$ and radius $R$ ( $R \ll L$ ) carries a steady current $I$ . Let $B_{\text{exact}}$ be the magnetic field on the axis of the solenoid at a distance $x = L$ from its geometric center. Let $B_{\text{dipole}}$ be the magnetic field at the same point calculated by assuming the solenoid acts as an ideal point magnetic dipole located at its center. Evaluate the ratio $\frac{B_{\text{exact}}}{B_{\text{dipole}}}$ to the lowest non-vanishing order in $R/L$ .

Options

A.
$\frac{16}{9}$
B.
$\frac{32}{9}$
C.
$\frac{4}{3}$
D.
$1$

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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

Magnetic Field on Axis vs. Center|Bar Magnet as Equivalent Solenoid|Binomial Approximation

Why This Question Is Tricky

Common Mistakes

Students often blindly apply the point-dipole approximation $B \propto 1/r^3$ for any magnet or solenoid, forgetting that this is only the leading term of a multipole expansion valid strictly for $r \gg L$ .

This question forces them to confront the breakdown of the dipole approximation when the observation distance is comparable to the physical dimensions of the source ( $x = L$ ).

By requiring the exact integration of the Biot-Savart law for a finite solenoid (or the equivalent magnetic pole model) and comparing it to the ideal dipole formula, the problem tests their understanding of field decay scaling and the geometric limits of the $1/r^3$ approximation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Magnetic Field on Axis vs. Center
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free