A coil C of N turns is placed around a long solenoid S of...

A coil $C$ of $N$ turns is placed around a long solenoid S of radius $R$ and $n$ turns per unit length, as in Fig. $30-69$ . (a) Show that the mutual inductance for the coil-solenoid combination is given by $M=$ $\mu_{0} \pi R^{2} n N .$ (b) Explain why $M$ does not depend on the shape, size, or possible lack of close packing of the coil.

A coil $C$ of $N$ turns is placed
around a long solenoid S of radius
$R$ and $n$ turns per unit length, as in
Fig. $30-69$ . (a) Show that the mutual
inductance for the coil-solenoid
combination is given by $M=$
$\mu_{0} \pi R^{2} n N .$ (b) Explain why $M$ does
not depend on the shape, size, or possible lack of close packing of the coil.

Key Concepts

Independence from Coil Geometry

The mutual inductance in this setup is independent of the geometric details (such as shape and winding density) of the secondary coil because it is solely determined by the total flux from the primary source (the solenoid) and the total number of turns in the secondary coil. As long as the coil encompasses the uniform magnetic field region, variations in its distribution do not affect the overall flux linkage, emphasizing the inherent reciprocity in electromagnetic induction.

Magnetic Field in a Solenoid

The magnetic field inside a long solenoid can be derived using Ampère’s law. This field is uniform and given by the product of the permeability of free space, the solenoid’s turn density, and the current flowing through it. This uniform field is central to calculating the flux through the area enclosed by the surrounding coil and thereby determining mutual inductance.

Mutual Inductance

Mutual inductance is a measure of the coupling between two circuits, defined as the ratio of the magnetic flux linked with one circuit to the current flowing in the other. It quantifies how effectively a change in current in one circuit induces an electromotive force in the other circuit, and is foundational to the operation of transformers and inductive sensors.

Magnetic Flux

Magnetic flux represents the total magnetic field passing through a given area. In the context of inductance, the flux through a coil is determined by integrating the magnetic field over the area enclosed by the coil. The calculation assumes that only the field within the region of the source (e.g., inside the solenoid) contributes significantly to the flux, which leads to the derived expression for the mutual inductance.

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