Editing Inductance (section)

===Mutual inductance of two wire loops===
This is the generalized case of the paradigmatic two-loop cylindrical coil carrying a uniform low frequency current; the loops are independent closed circuits that can have different lengths, any orientation in space, and carry different currents. Nonetheless, the error terms, which are not included in the integral are only small if the geometries of the loops are mostly smooth and convex: They must not have too many kinks, sharp corners, coils, crossovers, parallel segments, concave cavities, or other topologically "close" deformations. A necessary predicate for the reduction of the 3-dimensional manifold integration formula to a double curve integral is that the current paths be filamentary circuits, i.e. thin wires where the radius of the wire is negligible compared to its length.

The mutual inductance by a filamentary circuit <math>m</math> on a filamentary circuit <math>n</math> is given by the double integral ''[[Franz Ernst Neumann|Neumann]] formula''<ref>
{{cite journal
 | last=Neumann | first=F.E. |author-link=Franz Ernst Neumann
 | year=1846
 | title=Allgemeine Gesetze der inducirten elektrischen Ströme  | language=de
 | trans-title=General rules for induced electric currents
 | journal=Annalen der Physik und Chemie
 | volume=143  | issue=1  | pages=31–44
 | issn=0003-3804  | doi=10.1002/andp.18461430103
 | bibcode = 1846AnP...143...31N
 | url = https://zenodo.org/record/1423608
 | publisher=Wiley
}}
</ref>
: <math display=block> L_{m,n} = \frac{\mu_0}{4\pi} \oint_{C_m} \oint_{C_n} \frac{\mathrm{d}\mathbf{x}_m\cdot \mathrm{d}\mathbf{x}_n}{\ \left| \mathbf{x}_m - \mathbf{x}_n \right|\ }\ ,</math>

where
: <math>C_m</math> and <math>C_n</math> are the curves followed by the wires.
: <math>\mu_0</math> is the [[permeability of free space]] ({{nowrap|4{{pi}}×{{10^|−7}} H/m}})
: <math>\mathrm{d}\mathbf{x}_m</math> is a small increment of the wire in circuit {{mvar|C}}{{sub|m}}
: <math>\mathbf{x}_m</math> is the position of <math>\mathrm{d}\mathbf{x}_m</math> in space
: <math>\mathrm{d}\mathbf{x}_n</math> is a small increment of the wire in circuit {{mvar|C}}{{sub|n}}
: <math>\mathbf{x}_n</math> is the position of <math>\mathrm{d}\mathbf{x}_n</math> in space.