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Mutual Inductance between two Coaxial Solenoids

Consider a long air-cored solenoid with primary PP and secondary SS.

Let,     N1 = number of turns in the primary,
N2 = number of turns in the secondary,
    A = Area of cross section,
    L = length of the primary,
    L = current in the primary.

Magnetic field at any point inside the primary} = B = μoN1I / l
. : magnetic flux through each turn of the primary is
    BA = μoN1IA / l

Since the secondary is wound closely over the central portion of the primary, the same flux is also linked with each turn of the secondary.

Magnetic flux through each turn of the secondary} = μoN1IA / l
Total Magnetic flux through N2 turn of the secondary} = Φ = μoN1IAN2 / l

By definition of mutual inductance,
    M = Φ/1 = μoN1IAN2 / l
If the core is an material of permeability μr, then
    M = μoμrN1IAN2 / l henry
If there are a number of cores of areas of cross-section A1, A2, A3, etc., and relative permeabilities μr1, μr2, μr3, etc.
    M = μoN1N2 / l [μr1A1 + μr2A2 + μr3A3 + …] henry

Mutual Inductance

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