Editing Solenoid (section)

=== Inductance ===
{{See also|Inductance with physical symmetry}}

As shown above, the magnetic flux density <math>B</math> within the coil is practically constant and given by
:<math>B = \mu_0 \frac{NI}{l},</math>
where ''μ''<sub>0</sub> is the [[Permeability (electromagnetism)|magnetic constant]], <math>N</math> the number of turns, <math>I</math> the current and <math>l</math> the length of the coil. Ignoring end effects, the total [[magnetic flux]] through the coil is obtained by multiplying the flux density <math>B</math> by the cross-section area <math>A</math>:
:<math>\Phi = \mu_0 \frac{NIA}{l}.</math>

Combining this with the definition of [[inductance]]
:<math>L = \frac{N \Phi}{I},</math>
the inductance of a solenoid follows as
:<math>L = \mu_0 \frac{N^2A}{l}.</math>

A table of inductance for short solenoids of various diameter to length ratios has been calculated by Dellinger, Whittmore, and Ould.<ref>{{cite book |url= https://books.google.com/books?id=Xn8KbsgeFrwC&pg=PA248 |title=Radio Instruments and Measurements |author1=D. Howard Dellinger |author2=L. E. Whittmore |author3=R. S. Ould |name-list-style=amp |year= 1924 |journal=NBS Circular |volume= C74 |access-date=7 September 2009 |isbn=9780849302527 }}</ref>

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 (electromagnetism)|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 ''μ<sub>0</sub>'' with ''μ'' or ''μ<sub>0</sub>μ<sub>r</sub>'', where ''μ'' represents permeability and ''μ<sub>r</sub>'' [[Permeability (electromagnetism)|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.