Editing Inductance (section)

=== Derivation of mutual inductance ===
The inductance equations above are a consequence of [[Maxwell's equations]]. For the important case of electrical circuits consisting of thin wires, the derivation is straightforward.

In a system of <math>K</math> wire loops, each with one or several wire turns, the [[flux linkage]] of loop {{nowrap|<math>m</math>,}} {{nowrap|<math>\lambda_m</math>,}} is given by
<math display=block>\displaystyle \lambda_m = N_m \Phi_m = \sum\limits_{n=1}^K L_{m,n}\ i_n\,.</math>

Here <math>N_m</math> denotes the number of turns in loop {{nowrap|<math>m</math>;}} <math>\Phi_m</math> is the [[magnetic flux]] through loop {{nowrap|<math>m</math>;}} and <math>L_{m,n}</math> are some constants described below. This equation follows from [[Ampere's law]]: ''magnetic fields and fluxes are linear functions of the currents''. By [[Faraday's law of induction]], we have

<math display=block>\displaystyle v_m = \frac{\text{d}\lambda_m}{\text{d}t} = N_m \frac{\text{d}\Phi_m}{\text{d}t} = \sum\limits_{n=1}^K L_{m,n}\frac{\text{d}i_n}{\text{d}t},</math>

where <math>v_m</math> denotes the voltage induced in circuit {{nowrap|<math>m</math>.}} This agrees with the definition of inductance above if the coefficients <math>L_{m,n}</math> are identified with the coefficients of inductance. Because the total currents <math>N_n\ i_n</math> contribute to <math>\Phi_m</math> it also follows that <math>L_{m,n}</math> is proportional to the product of turns {{nowrap|<math>N_m\ N_n</math>.}}