Self Inductance of Co-axial Cylinder

A cable contains two cylindrical conductors with a common axis. Let a and b be the radii of cylindrical conductors (b<a). The current in these cylinders are equal and opposite. Let l be the current. There is a magnetic flux between the two conductors.

The magnetic inductions B at a distance r from the axis is
        B = μ₀I/2πr

Consider an area of unit length between distances r and r + dr.
The flux through the area is
        dΦ = Bdr
         = μ₀I/2πr

Total flux through the area between cylinders of unit length is Φ.
        Φ = ʃ μ₀I/2πr dr/r = μ₀I/2π log_e b/a
But        Φ = LI
Hence the self-inductance per unit length L = μ₀/2πr log_e b/a



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