Figure 32-30 shows, in cross section, two coaxial solenoids....

Figure $32-30$ shows, in cross section, two coaxial solenoids. Show that the mutual inductance $M$ for a length $l$ of this solenoid-solenoid combination is given by $M=\pi R_{1}^{2} l \mu_{0} n_{1} n_{2}$ in which $n_{1}$ and $n_{2}$ are the respective numbers of turns per unit length and $R_{1}$ is the radius of the inner solenoid. Why does $M$ depend on $R_{1}$ and not $\quad$ FIGURE $32-30=$ Problem 17 . on $R_{2} ?$

Key Concepts

Mutual Inductance

Mutual inductance is a measure of how much magnetic flux generated by one circuit links with another circuit when a current flows in the first circuit. It is determined by the geometry and the number of turns of the coils involved. In the context of the solenoid-solenoid system, the mutual inductance quantifies the coupling between the magnetic fields of the inner and outer solenoids.

Magnetic Flux

Magnetic flux refers to the total magnetic field passing through a given area. In calculating mutual inductance, the flux produced by one coil that goes through the area encompassed by another coil is critical. The inner solenoid’s cross-sectional area directly affects the amount of flux captured, which is why the radius of the inner solenoid is important in the formula.

Magnetic Field of a Solenoid

The magnetic field inside an ideal solenoid is uniform and primarily depends on the number of turns per unit length and the current flowing through the coil. This predictable field is used to calculate the flux through the solenoid's cross-sectional area. Understanding this allows the derivation of mutual inductance by integrating the static field contributions over the area of the inner solenoid.

Coaxial Solenoids

Coaxial solenoids are two cylinders with aligned axes, where one solenoid is placed inside another. The mutual inductance between coaxial solenoids is defined by how effectively the inner solenoid captures the uniform magnetic field produced by the outer solenoid. The dependence on the inner solenoid’s radius stems from the effective 'capture area' for the magnetic flux produced by the outer coil.

Dependence on Inner Geometry

The mutual inductance depends on the inner solenoid's radius (R1) because the inner solenoid’s cross-sectional area directly governs the amount of magnetic flux linking with it. Even if the outer solenoid has a larger radius, only the flux crossing the inner radius contributes to the mutual inductance, which is why R1 appears in the mutual inductance formula rather than the outer solenoid's radius (R2).

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