Q.60 An infinitely long solenoid of radius r and number of turns per unit length n carries a steady current I. The ratio of the magnetic fields at a point on the axis of the solenoid to a point r/2 from the axis is _______ .

The magnetic field inside an infinitely long solenoid is uniform throughout its cross-section, making the ratio of the fields at the specified points equal to 1.

Solution Derivation

For an infinitely long solenoid with radius $r$, turns per unit length $n$, and current $I$, Ampère’s law gives the axial magnetic field magnitude as B=μ0nIB=μ0nI at all points inside the solenoid (where μ0μ0 is the permeability of free space).

This uniformity arises because the field lines are parallel and equidistant within the solenoid’s interior, independent of radial distance from the axis.

A point on the axis (radial distance 0) and a point at $r/2$ from the axis both lie inside the solenoid, so Baxis=Br/2=μ0nIBaxis=Br/2=μ0nI.

Thus, the ratio Baxis/Br/2=1Baxis/Br/2=1.

Why Uniformity Holds

Consider an Amperian rectangular loop parallel to the solenoid axis: the enclosed current is $nIl$ (for length $l$), yielding ∮B⋅dl=Bl=μ0nIl∮B⋅dl=Bl=μ0nIl, so B=μ0nIB=μ0nI.

Symmetry ensures no radial or azimuthal field components inside, and the field strength does not vary with position for infinite length.

Outside ($>r$), $B\approx 0$, but both queried points are inside.

Common Misconceptions

Finite solenoids show slight axial variation near ends, but infinite length eliminates this.

Unlike toroids (where $B\propto 1/\rho$), solenoids have constant $B$ inside regardless of $r$.

No options exist in this fill-in-the-blank, but wrong answers might assume radial dependence (ratio ≠1) or external field confusion.

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The infinitely long solenoid magnetic field ratio between a point on the axis and r/2 from the axis equals 1, as the field remains uniform B=μ0nIB=μ0nI everywhere inside. This key result from electromagnetism applies to ideal solenoids in physics exams like CSIR NET.

Uniform Field Inside Solenoid

An infinitely long solenoid produces a homogeneous axial magnetic field independent of radial position. Using Ampère’s law on a rectangular path inside confirms B=μ0nIB=μ0nI, constant for all points within radius $r$.

At the axis (distance 0) and r/2 (inside), both experience identical $B$, yielding ratio 1.

Derivation Steps

1. Apply Ampère’s circuital law: ∮B⋅dl=μ0Ienc∮B⋅dl=μ0Ienc.

2. For internal loop: Bl=μ0(nlI)Bl=μ0(nlI), so B=μ0nIB=μ0nI.

3. Infinite length ensures no end effects or radial variation.

Exam Relevance

CSIR NET questions test this uniformity—ratio ≠1 implies finite solenoid error. Field drops to zero outside, half at finite ends.