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 = μ_oN₁I / l
. : magnetic flux through each turn of the primary is
    BA = μ_oN₁IA / 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} = μ_oN₁IA / l
Total Magnetic flux through N2 turn of the secondary} = Φ = μ_oN₁IAN₂ / l

By definition of mutual inductance,
    M = Φ/1 = μ_oN₁IAN₂ / l
If the core is an material of permeability μr, then
    M = μ_oμ_rN₁IAN₂ / l henry
If there are a number of cores of areas of cross-section A₁, A₂, A₃, etc., and relative permeabilities μr₁, μ_r2, μ_r3, etc.
    M = μ_oN₁N₂ / l [μ_r1A₁ + μ_r2A₂ + μ_r3A₃ + …] henry