As shown above, the magnetic flux density within the coil is practically constant and is given by
where μ0 is the magnetic constant, the number of turns, the current and the length of the coil. Ignoring end effects, the total magnetic flux through the coil is obtained by multiplying the flux density by the cross-section area :
When this is combined with the definition of inductance,
it follows that the inductance of a solenoid is given by:
A table of inductance for short solenoids of various diameter to length ratios has been calculated by Dellinger, Whittmore, and Ould.
This, and the inductance of more complicated shapes, can be derived from Maxwell's equations. For rigid air-core coils, inductance is a function of coil geometry and number of turns, and is independent of current.
Similar analysis applies to a solenoid with a magnetic core, but only if the length of the coil is much greater than the product of the relative permeability of the magnetic core and the diameter. That limits the simple analysis to low-permeability cores, or extremely long thin solenoids. The presence of a core can be taken into account in the above equations by replacing the magnetic constant μ0 with μ or μ0μr, where μ represents permeability and μr relative permeability. Note that since the permeability of ferromagnetic materials changes with applied magnetic flux, the inductance of a coil with a ferromagnetic core will generally vary with current.