A coil C of N turns is placed around a long solenoid S of radius R and n turns per unit length a

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

A coil $\mathrm{C}$ of $N$ turns is placed around a long solenoid $\mathrm{S}$ of radius $R$ and $n$ turns per unit length as in Fig. $32-29 .$ Show that the mutual inductance for the coil-solenoid combination is given by $M=\mu_{0} \pi R^{2} n N .$ Explain why $M$ does not depend on the shape, size, or possible lack of close-packing of the coil.

Key Concepts

Uniform Magnetic Field in a Solenoid

A long solenoid produces a nearly uniform magnetic field within its interior. This uniformity means that the magnetic flux through any loop encircling the solenoid depends only on the cross-sectional area of the solenoid and not on the specific details of the loop's geometry, making calculations simpler and more general.

Ampere's Law

Ampere's Law relates the circulating magnetic field in space to the electric current that produces it, and is particularly useful in deriving the magnetic field inside a solenoid. For a long solenoid, it provides the uniform field expression that is foundational for calculating the enclosed flux.

Geometric Independence of Mutual Inductance

The mutual inductance in this configuration does not depend on the shape, size, or packing of the external coil because the induced voltage is determined solely by the flux through the solenoid's fixed cross-sectional area. Since the solenoid’s magnetic field is uniform inside, only the total number of turns of the external coil matters with respect to capturing the same flux, regardless of its precise spatial configuration.

Mutual Inductance

Mutual inductance is a measure of the voltage induced in one circuit due to the time-varying current in another circuit. It is defined as the ratio of the magnetic flux through one circuit produced by the current in the other circuit, and it quantifies the degree of electromagnetic coupling between the two circuits.

Magnetic Flux Linkage

Magnetic flux linkage refers to the total magnetic flux that is linked or 'captured' by a coil or circuit, taking into account the number of turns. When calculating inductance, the total flux through all the turns of the coil is considered, which is why parameters like the number of turns are important.

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