Magnetic field of a finite solenoid with a linear permeable core

The integral form of Ampere’s law applied to an infinitely long solenoid gives a good first approximation to the magnetic field of finite air-cored solenoids with length L to width w ratios (aspect ratios) significantly greater than 1. It is less well known that this approximation requires much larger aspect ratios to be applicable to solenoids whose relative core permeability $μr$ is much greater than that of the surrounding medium. An exact expression is derived for the magnetic field and inductance of a linear permeable core spheroidal solenoid, and an inequality is obtained for the validity of the infinite solenoid approximation. It is demonstrated that the magnetic field and inductance of a spheroidal solenoid with a linear core approximate well with those of a cylindrical solenoid of the same length and width.